Business Statistics, Canadian Edition, 4th edition Norean R Sharpe Solution Manuals

Instructor Resource Guide Alexei Gokhman and Ioulia Kim Humber College

Business Statistics Fourth Canadian Edition Norean R. Sharpe St. John’s University

Richard D. De Veaux Williams College

Paul F. Velleman Cornell University

David Wright University of Ottawa

Contents About This Guide 0-1 Part I: Exploring and Collecting Data Chapter 1

An Introduction to Statistics 1-1

Chapter 2

Data 2-1

Chapter 3

Surveys and Sampling 3-1

Chapter 4

Displaying and Describing Categorical Data 4-1

Chapter 5

Displaying and Describing Quantitative Data 5-1

Chapter 6

Scatterplots, Association, and Correlation 6-1

Chapter 7

Introduction to Linear Regression 7-1

Part II: Understanding Probability Distributions and Statistical Inference Chapter 8

Randomness and Probability 8-1

Chapter 9

Random Variables and Probability Distributions 9-1

Chapter 10

Sampling Distributions 10-1

Chapter 11

Confidence Intervals for Proportions 11-1

Chapter 12

Testing Hypotheses About Proportions 12-1

Chapter 13

Confidence Intervals and Hypothesis Tests for Means 13-1

Chapter 14

Comparing Two Means 14-1

Chapter 15

Design of Experiments and Analysis of Variance, ANOVA 15-1

Chapter 16

Interference for Counts: Chi-Square Tests 16-1

Chapter 17

Nonparametric Methods 17-1

Part III: Exploring Relationships Among Variables Chapter 18

Inference for Regression 18-1

Chapter 19

Understanding Regression Residuals 19-1

Chapter 20

Multiple Regression 20-1

Chapter 21

Building Multiple Regression Models

Part IV: Using Statistics for Decision Making Chapter 22

Time Series Analysis 22-1

Chapter 23

Decision Making and Risk 23-1

Chapter 24

Quality Control 24-1 . ii

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Chapter 1 An Introduction to Statistics What’s It About?

This chapter is about Statistics – what it is and why we care. It also describes the important features of the text.

Comments Do expect your students to read the book. Give them specific reading assignments, starting with this first chapter. We’ve tried to make the book engaging. You can assign a few pages of reading along with some problems each night. Chapter 1 makes a good opening assignment.

This is the students’ first look at the style of the book, and we do lay it on more heavily here than we will when discussing, say, confidence intervals. We want to shake things up. We want them to notice that this is not the same old math or science textbook they’ve seen before. And we’d like to get them on our side. That’s the reason for the humour and self-deprecating remarks.

Every Statistics text starts with a definition of Statistics. We do too, but ours is different. And the difference matters. We say that Statistics is a way of reasoning and that the goal is to help us understand the world. We’ve found it helpful to reinforce this idea throughout the book, especially when we get into the methods sections of the course. This book is first and primarily about statistical thinking. Methods, definitions, and skills are all here, but each is presented with the purpose of understanding the world so we can make better business decisions. That’s why every guided example follows the Plan, Do, Report pattern, starting with careful reasoning and concluding with a sentence or two telling what we’ve learned about the world.

It is easy to be drawn into a focus on definitions, on algorithms, and on getting the “right answer.”

Those are easier to teach and certainly easier to grade. Please resist the temptation. If you can help your students to stay focused on statistical thinking and understanding, this course can change the way they view the world.

We have included a section on “20th and 21st Century Statistics” that describes how statistics has evolved, particularly today with its uses in analytics.

Since this text is designed for business students, we have also included a section on how the various business disciplines use statistics. This may help to address the commonly heard question of “why do I need to know this?”

A Note on Teaching We don’t spend an entire class period on this chapter; in fact, sometimes we just start right in on Chapter 2. Chapter 1 is meant to be read by students not to them. Discuss the important definitions, collect some data, and move on as fast as you can. We’ve got a lot to cover and can’t afford a full class just to get moving.

Looking Ahead Most of our own students actually do the reading. (Yes, we were surprised at first.) If you can hook students on reading the text, there are big benefits later on. As they read they learn that the .

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Part I Exploring and Collecting Data

book is a valuable resource to help them understand sticky concepts, to help lead them through writing complete and clear solutions, and to help them avoid common mistakes.

Class Do’s One of our favourite definitions says “Statistics is the art of distilling meaning from data.” Data have a story to tell. Our objective is to uncover that story. Be sure to distinguish the differences between data and statistics. Collect some data in class, and ask students to look for interesting facts hiding there. Or have students gather current economic or business data from the internet.

Encourage students to think about the concepts and definitions in this chapter. For example, why do we talk about “a statistic” when we don’t discuss “a mathematic” or “a physic”? Statistics is a whole that is made up of many parts, and each of those parts has its own meaning and its own story to tell. Ask students about how different companies may utilise large amounts of data. For example, what data might a large company such as The Bay or Fairmont Hotels & Resorts look at and for what purpose?

The Importance of What You Don’t Say One of the reasons Statistics can be difficult to teach is that we often deal with vague concepts.

Students and instructors both like clear definitions; they’re easier to teach, learn, and grade. But reasonable people can disagree about whether a histogram is symmetric or skewed, whether a straggling point is really an outlier or just the largest value, or whether the spread really is the same in two groups we want to compare. It is important to allow students their own opinions and insights into data.

Avoid the temptation to lead students in any particular direction or give them hints about how to examine the data you collected. They know some things to do—they’ll find percents and averages, maybe draw some simple graphs, and write a few sentences about what they see. Just let it happen.

You’ll have plenty of chances to suggest, lead, and modify later on.

The motivation of hoping to understand the world is quite sufficient to justify thinking beyond the numbers at hand. This is not the time to introduce technical concepts of inference. We’d rather not see these ideas yet (and you won’t find them this early in the text). We all know that we’ll be heading that way eventually, but if we make reasoning about patterns in the data too formal too soon, we’ll stifle students’ interest and enthusiasm and burden them with a lot of terms they don’t have a context for just yet.

Class Examples Take a quick class survey. Make sure that you have a good mix of quantitative and categorical variables. Some possible variables are gender, Program of Study (Acct’g, Fn’c, Mrkt’g, etc.) number of siblings, number of provinces visited, and shoe size. Start passing a survey sheet around at the very beginning of class; it can circulate quickly while you are introducing yourself and talking about the course, expectations, and so on. Passing out slips of paper with a list of states for each student to check off will speed things up, so we provide that, too. After you’ve entered the data into the computer, give the students the data file and ask them what story it tells about their class. You’ll find that a lot of important issues will surface during discussions. As new methods are introduced throughout the course, you can continue using the data for examples.

Hints: Data are rarely as simple as they seem. Suggest the variables above, and then pause for some discussion. Does touching down at an airport qualify as “visiting” a province? Does an only .

Chapter 1 An Introduction to Statistics

1-3

child count herself when counting siblings? Should shoe sizes be adjusted because men’s and women’s size 7 are different sizes? Give them a chance and they’ll find other issues—and they’ll be developing a healthy skepticism for data. That’s just what we want, so they’ll value the tools that help them look at data more carefully.

If you don’t specify units for height, you will almost certainly get some values in centimeters.

Alternatively, if you specify inches, you may get a “55” from someone who meant 5’5”. Those outliers make for good class discussion.

Resources Decisions Through Data*

• Video Unit 1: What is Statistics ActivStats† Have your students work through the initial lesson in ActivStats now (which is available at www.pearsonmylabandmastering.com/northamerica/mystatlab/) to be sure there are no glitches.

Assignment: Read Chapter 1. Explore the data collected, and make some observations about the class. What could you report to someone who asked you to describe your class? What did you find interesting (or perhaps surprising) about the class?

*

Decisions Through Data; COMAP (1-800-77COMAP), 1992.

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Part I Exploring and Collecting Data

An Example of a Survey Sheet Student #

Gender (F/M)

Program of Study (Acct’g, Fn’c, Mrkt’g, etc.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

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# of Siblings

# of Provinces Visited

Shoe Size

Height

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Chapter 2 Data What’s It About?

In this chapter we introduce students to data. We talk about the importance of context (the W’s), about variables, and make the distinction between categorical and quantitative data. We begin to introduce the vocabulary of Statistics.

Comments It is valuable to get students involved with data from the start. We don’t take a “big picture” approach at this time. There will be plenty of time to build models and draw inferences later. For now, let’s just get our hands dirty playing with the data. When students have a good sense of what kinds of things data can say to us, they learn to expect to listen to the data. Throughout the course, we insist that no analysis of data is complete without telling what it means. This is where that understanding starts.

Rather than head directly for the “real purpose” of the course in the inference chapters, we prefer to emphasize the connection between our work with data and what they tell us about the world. No analysis is complete without a connection back to the real-world circumstances. Setting that stage is the underlying motivation for this chapter. We’ll spend the next 5 chapters or so looking at and exploring data without making formal inferences.

Looking Ahead You might have the students thumb through the book and read the opening of some chapters. Each one starts with a story about a company or business sector and data, and most have additional stories and more data inside. Statistics is about the real world. Among other topics, we’ll be discussing Future Shop, MBNA, PotashCorp, Whole Foods Market, and even Canada’s Wonderland. We need to get students thinking about the context of data and able to make the distinction between categorical and quantitative data. These are fundamental skills for everything that follows, and they’ll be used throughout the course.

Class Do’s Get the class thinking about what the term “data” means. Students need to understand that data are not just numbers and that they must have a context (the W’s). When data are quantitative, they should also have units. There are two ways we treat data: categorical and quantitative. Don’t get distracted by worrying about ratio, interval, and other distinctions. These are problematic and don’t matter for the concepts and methods discussed in this book. Emphasize that the distinction between treating data as categorical or quantitative may be more about how we display and analyze data than it is about the variable itself. The variable “sex” is data, but just because we might label the males as 1 and the females as 0 doesn’t mean that it’s quantitative. On the other hand, taking the average of those 0’s and 1’s does give us the percentage of males. How about age? It is often quantitative, but could be categorical if broken down only into child, adult, and senior. Students should recognize that every discipline has its own vocabulary, and Statistics is no exception. They’ll need to understand and use that vocabulary properly. Unfortunately, many Statistics words have a common everyday usage that’s not quite the same. We’ll be pointing those out as we go along.

Emphasize vocabulary words as they come up. One of the first should be variable. Make the point that it does not mean exactly the same thing as it did in Algebra. There, we call “x” a variable, but .

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Part I Exploring and Collecting Data

often that means that we just don’t currently know its value. In Statistics a variable is an attribute or characteristic of an individual or object whose value varies from case to case.

Point out that summaries of data can be verbal, visual, and numerical. All are important. In fact, any complete analysis of data almost always includes all three of these.

Hope that someone objects to finding an overall average shoe size or to comparing men’s and women’s sizes—shoe sizes are inconsistent in terms of units. This adds emphasis to the importance of units and the W’s.

The Importance of What You Don’t Say We are laying a foundation here. Stretching up to the attic at this point just makes everyone feel unsafe. Many fundamental Statistics terms are left unmentioned in this chapter. You’ll find it best to leave it that way. We’ll get to them when the students have a safe place to file them along with their other knowledge. So we have an unusually long list of terms we recommend leaving for later in the course. In particular, avoid saying the following: Hypothesis, Inference. These are certainly important in this course, but we have no background for discussing them honestly now, so they would just be confusing and intimidating.

Nominal, Ordinal, Interval, Ratio. “Nominal” is used by some software packages as a synonym for “categorical” as “continuous” is used for “quantitative.” These distinctions arise from studies of measurement scales. But it isn’t correct to claim that each variable falls into one of these categories. It is the use to which the data are put that determines what properties the variable must have. Ordinal categorical data may come up, but there are no special techniques for dealing with ordered categories in this course. And any differences between interval- and ratio-scaled data are commonly ignored in statistical analyses. If any of these terms were mentioned now, they’d never come up again anyway.

Random, Probability, Correlation. Everyone has some intuitive sense of these terms, and we’ll deal with them formally—but not for a while. Students may want to use these terms, but at this early stage in the course, we don’t need them. Without background and careful definition, they are likely to be misused and can simply be frightening.

Class Examples 1. Ask students to tell some things they learned about the class from inspecting the data collected in the opening day’s survey. You can use that discussion to develop several of the important points of the chapter.

2. Consider 17, 21, 44, and 76. Are those data? Context is critical—they could be test scores, ages in a golf foursome, or uniform numbers of the starting backfield on the football team. In each case, our reaction changes.

3. Run through some other examples of data, asking about the W’s, the variables (what are they, what type is each used as, and what are the units), and so on.

•A Consumer Reports article on energy bars gave the brand name, flavor, price, number of calories, and grams of protein and fat.

•A report on a charity run in Toronto listed each runner’s gender, country, age, and time.

Solution: .

Chapter 2 Data

2-3

Consumer Reports Who: energy bars What: brand name, flavor, price, calories, protein, fat When: not specified Where: not specified How: not specified. Are data collected from the label? Are independent tests performed?

Why: information for potential consumers Categorical variables: brand name, flavor Quantitative variables: price ($), number of calories (calories), protein (grams), fat(grams) Charity Run Who: charity runners What: gender, country, age, time When: not specified Where: Toronto How: not specified. Presumably, the data were collected from registration information.

Why: race result reporting Categorical variables: gender, country Quantitative variables: age (years), time (hours, minutes, seconds) Resources ActivStats* • Start with Lesson 1 to let students familiarize themselves with the features of the software.

Lesson 2 examines types of data and context.

Web Links • The Data and Story Library (DASL; http://lib.stat.cmu.edu/DASL/) is a source of data for student projects and classroom examples.

• Statistics Canada has a wealth of information that is free to educational institutions if accessed through the institutional library, or you may want to visit the Statistics Canada website for educators and students at: http://www.statcan.gc.ca/edu/power-pouvoir/ch1/5214857-eng.htm This website features not only a vast amount of data by different subjects, but also exercises that you can use with your students as well.

Other • Read polls, studies, or other reports in newspaper and magazine articles. It’s always interesting to see how well (or poorly) they provide information about the W’s.

If you have a computer and projection capabilities in class, you can find daily surveys at Gallup and other polling organizations. Current data are often particularly interesting to students. But don’t use results of voluntary-response online surveys. We’ll be making the point that these are fatally flawed—but we can’t say that clearly without concepts and terms that we haven’t developed yet.

* ActivStats is available at www.pearsonmylabandmastering.com/northamerica/mystatlab/

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Basic Exercises 1.

The following data show responses to the question “What is your primary source for news?” from a sample of students at the University of Manitoba in June 2015.

Internet Newspaper Newspaper Internet TV a. b. c.

2.

Newspaper TV TV Internet Internet

Internet Internet TV Internet Internet

Internet TV TV Internet TV

Identify the W’s for this data Is the data categorical or quantitative?

Is the data cross sectional or time series?.

In September 2015, a cable company surveyed its customers in Burnaby, BC, and asked how likely they were to watch more than 2 hours of TV per day. The following data show the responses.

Very Likely Likely Unlikely Very Likely Unlikely a. b. c.

Unlikely Unlikely Unlikely Unlikely Unlikely

Unlikely Likely Likely Unlikely Unlikely

Identify the W’s for this data Is the data categorical or quantitative?

Is the data cross sectional or time series?.

ANSWERS 1. a.

TV Internet Newspaper Internet TV

Who: students What: primary source of news When: June 2015 Where: University of Manitoba How: not specified.

Why: not specified

b.

Categorical

c.

Cross-sectional

2. a.

Who: Cable company customers .

Very Likely Likely Likely Very Likely Likely

Chapter 2 Data What: likelihood of watching more than 2 hours of TV per day When: September 2015 Where: Burnaby, BC How: survey Why: not specified b.

Categorical

c.

Cross-sectional

.

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Chapter 3 Surveys and Sampling What’s It About?

With the ultimate goal of uncovering truths about a population, we discuss how to collect useful data from representative samples. We consider polls, surveys, and other means of gathering data, and introduce terminology and notation about populations and parameters. We examine the importance of random selection and sample size. We discuss several sampling designs, and many kinds of bias that can render our results meaningless.

Comments We now begin our discussion of study design. Statistical investigations usually begin with a question about the real world, and we set out in pursuit of truth. To have any chance of finding the truth, we need good data. How does that happen? This is the key and critical step. How can we know whether a poll’s result bears any resemblance to the actual preference of the voters? That depends on how the people were chosen, what questions they were asked, and many other factors.

Designing a good study is extremely difficult. Polls, surveys, observational studies, and experiments all require careful planning if the results are to lead to any meaningful conclusions.

Looking Ahead In this chapter, we discuss sampling error, the fact that every sample gives a slightly different snapshot of the population. We will see in Chapter 11 that this variability is quantifiably predictable. That predictability will allow us to place margins of error on our estimates about population model parameters.

Our discussion of statistical inference in Chapters 12 through 17 examines the conclusions we can and cannot draw about populations based on what we observe in samples.

One important issue in designing any study is determining the size of the sample or number of subjects to use. In this chapter we point out that the precision of our results depends only on the size of the sample, not the size of the population. We’ll see how to calculate the sample size needed for a desired margin of error in Chapter 11 (for proportions) and again in Chapter 13 (for means).

Class Do’s This chapter introduces the concepts of population and sample. Be careful to use the terms precisely and insist that students do so, too. While any subset of a population can be considered a “sample,” only a randomized representative sample is likely to be useful for Statistics, so we like to restrict the term to such samples.

This chapter has an unusually large amount of new vocabulary, dealing with the various sampling designs and the many bias issues. Insist that students use the words properly. Clarity of communication is important, and there is plenty of room for confusion. For example, response bias, voluntary response bias, and nonresponse bias are not the same.

It’s important that students understand the difference between bias and error. Unfortunately “error” sounds worse, as though it was a mistake and thus potentially preventable. It isn’t. Sampling error might also be called sampling variation. Sampling error simply describes the natural variability in results that will be observed from one sample to the next, none of them exactly capturing the truth in the population. Every sample produces a different estimate about the population. Luckily, as .

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we’ll see later on, this sample-to-sample variation is predictable, as in a poll of which the report says “sampling error of  4%.” We don’t have to be able to claim the result is perfect; it’s enough to know how close we probably are.

Bias is quite another matter. While error is found in every sample, individual samples can’t be biased. Bias is found in the sampling method. Bias means that something about the design systematically distorts the results so that they are unlikely to reflect reality. This problem is fatal.

When bias is present, we cannot draw any conclusions about the population. Imagine shooting arrows at a target. Even if we aim at the bull’s-eye, we usually miss, sometimes high or low, left or right. But at least we know that the arrow strikes somewhere near the intended spot. That’s error.

When our sights are out of adjustment, though, the arrows don’t even cluster around the bull’s-eye.

That’s bias, clearly a more serious problem. “Error” is okay; bias is bad!

Beware: Because we talk so often about simple random samples (SRS), students often believe that any random sample is an SRS. Get them to stop paying attention to the word “random” and instead look at the word “simple.” If the sampling strategy allows any subset of the population and equal chance to be chosen, it’s simple. When other restrictions are present (such as choosing an equal number of males and females), the sample may still be random, but it is no longer simple. With stratified sampling, we point out that weighting is important and that a simple arithmetic mean may not be appropriate. You may want to ask your students why this is the case.

Notation now becomes increasingly important. Students need to understand that there is nothing casual about the notation we use. It is part of the way we communicate. In Algebra they were free to use whatever letter they wanted to represent the base of the rectangle—x, b, w, l, or anything else. In Statistics, many of the letters we use stand for something very precise, to allow us to distinguish between similar and potentially confusing values. The mean of a sample is written y .

Do not confuse this with the mean of a population,  . A sample mean is a statistic calculated from actual observed data. A population mean is a model parameter, a theoretical concept whose actual value we do not and cannot know.

It’s important for students to realize that characteristics of populations aren’t merely unknown; they are usually unknowable. This is why we are forced to rely on statistics. We understand in concept what we mean when we talk about the mean height of all Canadian adults. We can estimate what that mean is. We just cannot know. Never mind the national average, we can’t even know the mean for all students in one city. How would we measure them all? When would we do it? How accurate would the measurements be? We emphasize that population parameter refers to a model. It’s not the population of adults in Canada that has a mean height, but our model of that group that does. The physical population isn’t well defined. It’s constantly changing as someone dies and someone else becomes an adult on turning 21. It’s our model that has a parameter, and our sample is taken to estimate that. The challenge, of course, is to be sure that the sample accurately represents the population.

The Importance of What You Don’t Say Don’t say “population statistics.” Population models have parameters; samples have statistics. We calculate the sample statistics from our data. We can only make guesses about the parameters of the models we think describe the population. Don’t slip and say that we are trying to estimate the sample mean. We can compute the sample mean exactly once we have the sample. It’s the population mean we are estimating. .

Chapter 3 Surveys and Sampling

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Don’t speak of sampling a percent of the population. Try to speak of samples in terms of their size (n) rather than in terms of what fraction of the population they represent. Don’t slip and suggest that it is the fraction of the population we sample that determines the precision of our estimates.

All that matters is the sample size.

Class Examples 1. You need to convince students of the necessity of random selection. The fact that bias will exist even if we consciously try to prevent it can be demonstrated convincingly using the Random Rectangles activity from Activity-Based Statistics.* Keep silent about the objective of the exercise.

• Hand out the page face down. Explain to students that they will see a population of rectangles (numbered 1 to 100) presented as grids – little checkerboards scattered here and there. Some are small, some very large, some square, others long and narrow. Their goal is to estimate the average size (area) of the rectangle population. First we want just their impression, akin to our impressions about the world around us. (About what percent of all cars are red? We all have a hunch based on casual observation.) Let them see the rectangles for 5 seconds. Have them write down their guesses for the mean area (nearest integer). Use a show of hands to make a quick histogram. Typically these guesses will be quite spread out and centered around 10 square units.

• Now let them to do it again, more carefully. Ask students to look over the population and select a sample of five rectangles that they judge to be representative of the others. Have each student calculate the average area of his or her chosen sample. Make another histogram. Compare the center and spread of these “judgment samples” to the pure guesses.

• Finally, ask them to use random numbers (calculator or table) to select the sample of five rectangles and find the average size of this random sample. They will be surprised to see this histogram of estimates centered much lower, around 7 or 8. The punch line is that the actual average is 7.42 square units. Random sampling worked where the other methods failed. (Our minds tend to be impressed with the large rectangles and overlook the fact that many are very small.) Not only were our guesses wrong, we could do little better when we consciously tried to be fair. Both histograms were centered at the wrong place – that’s bias.

Only the random samples produced a histogram centered in the right place, with the sampling variability we expect to always be present. Moral: only random selection can combat bias. Second moral: sampling error is ever-present, and less problematic than bias.

2. Ask students what kind of design problems can bias a survey. They will think of many examples, allowing you to name under-coverage, judgment sample, convenience sample, voluntary response, nonresponse, and response bias. Be sure they understand that response bias can be produced by the wording of the question or by the way the data are collected (people may just be unwilling to share personal information). A good example of a “poll” gone awry is provided in Activity-Based Statistics, the Popular Science Nuclear Power Poll. In addition, most local news programs run viewer call-in “polls,” and these make good discussion points.

* Activity-Based Statistics, 2nd Ed., Schaeffer, Gnanadesekan, Watkins, and Witmer; Key College Press, 2003.

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3. Discuss random sampling strategies. The activity “Estimating a Total” (in Activity-Based Statistics) is excellent for seeing the differences among SRSs, stratified samples, and cluster samples.

• Dots on the page represent diseased trees in a forest. They are randomly scattered throughout the forest – perhaps the disease is spread via flying insects. We want a count of the diseased trees without having to inspect every tree. Students can divide the page into a 5x10 grid of squares, randomly select any 5 squares, count the dots in the sample, and multiply by 10. (Make tick marks along the sides of the page to speed up the process.) An SRS works fine.

• The second “forest” represents a disease that must be spread by contact – it clusters in some regions and is almost unseen in others. A simple random sample might significantly over- or under-estimate the problem. Some students will propose stratifying into regions of equal prevalence, then randomly choosing one square from each. Others will simply choose at random one strip that spans all of the types of regions, a cluster sample.

4. Systematic sampling is convenient when two requirements are met. The population must be ordered randomly with respect to the variable being investigated, and we must choose a random starting point. Customs officers may inspect the luggage of every 10th person crossing a border, but they should not always start with (or avoid) the first person. And it’s much easier to pick every 50th name off a long alphabetized list (as long as what you measure is not related to the name itself) than to number the entire list and randomly select a sample by drawing numbers.

5. How could you get an SRS of the students at your school? How about stratified or cluster samples? Or a systematic sample?

To obtain an SRS of the students at your school, you could number the names in the student directory, and choose your sample by generating random numbers and matching the numbers with the students in your sample. This may raise some issues, however. Are all students included in the directory? If not, do the students excluded differ in any systematic way from the students in the directory?

A stratified sample may consist of a certain number of students in first, second, third and fourth years. Other strata may be males and females, athletes and non-athletes, or any other groupings that you feel would decrease variability in each stratum. If the sample sizes of each stratum are different, then this will be a good exercise to show students why the weighted mean should be used.

You might choose a cluster sample by choosing a class at random and sampling all of the students in that class.

A systematic sample could be obtained from a list of all students at the school. Suppose you want to sample 50 students from a school of 500. Number the students 001 to 500. Generate a random number from 001 to 010, and start with that student. Every 10th student in the list becomes part of your sample.

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Chapter 3 Surveys and Sampling

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6. Most people find it counterintuitive that the precision of survey or poll results is determined by the size of the sample regardless of the population size. By sampling 1000 voters we can estimate the outcome of an election with the same margin of error, whether it is a mayoral election in a city, a provincial election or a federal election. (This is true as long as the population is much larger than the sample—if there are only 1200 voters in the city, then our sample will provide much more accurate results, of course.) How can this be? Consider:

• You walk into the local ice cream shop and notice that they have a new flavour—Peanut Fudge. Sounds good, but before you buy a whole cone, you’d like a taste. They hand you a spoon and an open pint container. You scoop out one spoonful as your sample, savour the marvellous flavour, and buy an extra large cone. You base your decision on the assumption that your random spoonful is representative of all the peanut fudge ice cream the shop makes. (You chose a cluster sample randomized by the stirring process when the ice cream was made.) The servers in the ice cream shop will scoop your cone from a large 3-gallon tub. Just because that “population” is bigger won’t give you justification for a bigger sample. (Nice try, though!) And out back where they make the ice cream, they have a huge vat full of the stuff. If they let you sample from that massive population, you’d still need only a spoonful to learn what it tastes like. The issue is the size and randomness of the sample, not the size of the population.

Resources ActivStats†

• Lesson 3 – “Sample Surveys.” The simulations in this chapter give students a chance to see how samples reflect a known population, and how they nevertheless vary one from another.

That’s an experience that is hard to get from real-world sampling, but one that’s easily seen with the computer.

Decisions Through Data‡

• Video Unit 17: Census and Sampling • Video Unit 18: Sample Surveys Web Links

• Statistics Canada’s link provides valuable information on topics such as survey design, data collection , and organisation: http://www.statcan.gc.ca/edu/power-pouvoir/toctdm/5214718-eng.htm The Gallup and Nanos websites offers detailed information and FAQs about polling: http://www.gallup.com/178685/methodology-center.aspx http://www.nanosresearch.com/

• The Nielsen Media Research website offers information and FAQs about television ratings: http://www.nielsen.com/us/en.html

† ActivStats is available at www.pearsonmylabandmastering.com/northamerica/mystatlab/ ‡ Decisions Through Data; COMAP (1-800-77COMAP), 1992.

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Part I Exploring and Collecting Data

Basic Exercises 1.

Indicate whether each statement below is true or false. If false, explain why. a. We can eliminate sampling error by selecting an unbiased sample. b. The only truly representative sample is a census. c. Randomization helps to ensure that our sample is representative. d. Sampling error refers to sample differences and is also known as sampling variability.

2.

Suppose that the administration of a large urban hospital is interested in obtaining feedback from its employees regarding how their processes can be improved. For each of the following, identify the sampling method. a. There are six categories of employees (administration, physicians, nurses, medical technologists, clerical and maintenance). Randomly select twenty individuals from each category. b. Each employee has an ID number. Randomly select 150 numbers. c. Randomly select a department within the hospital (e.g., Radiology) and survey all of the employees who work in that department. d. Select every tenth person from the complete list of the hospital employees.

3.

A consumer advocacy group is interested in estimating perceptions about food safety among professionals in the food industry. Specifically, they wish to determine the percentage of professional food preparers in Canada who believe food safety has improved.

They use an alphabetized list of members of the organization Chef’s Collaborative and select every 10th member to include in the sample. a. What is the population? b. What is the sampling frame? c. What is the population parameter of interest? d. What sampling method is used?

4.

Indicate whether each statement below is true or false. If false, explain why. a. A local television news program that asks viewers to call in and give their opinion on an issue typically results in biased voluntary response sample. b. Convenience samples are generally not representative of the population. c. Measurement error is the same as sampling error. d. A pilot test is useful for identifying poorly worded questions on a survey.

ANSWERS 1.

a. false; unlike bias, sampling error cannot be avoided b. false; a census is a covering of the entire population; a sample is a subset of the population and if selected carefully can be representative of the population c. true d. true .

Chapter 3 Surveys and Sampling

3-7

2.

a. b. c. d.

stratified simple random cluster systematic

3.

a. professional food preparers in Canada b.

Chef’s Collaborative membership listing c. percentage of professional food preparers in Canada who believe food safety has improved d. systematic

4.

a. b. c. d.

true true false; measurement error refers to inaccurate responses true

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Chapter 4 Displaying and Describing Categorical Data What’s It About?

We introduce students to distributions of categorical variables. The mathematics is easy (summaries are just percentages) and the graphs are straightforward (pie charts and bar graphs).

We challenge students to uncover the story the data tell and to write about it in complete sentences in context.

Then we up the ante, asking them to compare distributions in two-way tables. Constructing comparative graphs, discussing conditional distributions, and considering (informally) the idea of independence give students a look at issues that require deeper thought, careful analysis, and clear writing.

To expand coverage of these topics you can discuss with the students Simpson’s paradox which often occurs in statistics related to medicine and social science.

Comments Most texts do not deal with conditional distributions, independence, and confounding (Simpson's paradox) so soon. Our experience is that students can get lulled into a false sense of security in the early part of this course, if all they see is things like means and histograms that they have dealt with since middle school. They think the course is going to be pretty easy, and they may not recognize the level of sophistication that is required until it’s too late. The ideas in Chapter 4 are not hard and are introduced only informally, but they do require some thought. Students will find it difficult to make clear explanations. We want these ideas to be interesting, to engage imaginations, and to challenge students. We hope the level of thought required will get their attention and arouse their interest.

It is probably beginning to dawn on your students that this isn’t a math class. At the very least, they are going to be expected to write often and clearly. In business, the numeric solution alone is never sufficient. There is always a real-world understanding or conclusion.

Looking Ahead There are many important skills and ideas here that prepare students for later topics. They need to think about the type of data, checking a condition before plunging ahead. They need to think about what comparisons will answer the questions posed, and write clear explanations in context. They begin to think about independence, one of the most important issues in Statistics. And, in Simpson’s paradox, they see the need to think more deeply to avoid being misled by lurking or confounding variables.

Class Do’s Continue to emphasize precision of vocabulary (and notation). These are an important part of clear communication and critical to success.

Emphasize Plan-Do-Report right from the start. The key to doing well in Statistics is to plan carefully by understanding the business questions on the table and ask what statistical techniques can address those issues before starting to write an answer. And then, after doing some calculations or other work, to write clear and concise report of what it all means. Your students may rebel at first at having to write sentences, much less paragraphs, in a course they may have thought was a math class. They are used to just doing the Do Report is at least 50% of each .

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Part I Exploring and Collecting Data

solution. If you make that point consistently right from the start of the course it becomes second nature soon, and puts each student in the right mindset for writing solid answers. Continually remind them: Answers are sentences, not numbers.

Weave the key step of checking the assumptions and conditions into the fabric of doing Statistics.

It’s easy: have students check that the data are being treated as categorical before they proceed with pie charts, conditional distributions, and the like. As the course goes on, thinking about assumptions and conditions will help students Plan appropriate statistical procedures. Start now.

Discuss categorical data and appropriate summaries: numerical (counts/percentages) and graphical (pie charts, bar graphs). Discuss distribution, frequency, relative frequency.

It gets more interesting when we make comparisons (using bivariate data): for example, political leanings by gender. Discuss two-way tables, marginal and conditional distributions. Political views may be interesting, but looking at the differences in political view by gender adds much more to the discussion. You can emphasize the vocabulary by asking things like “What is the marginal frequency distribution of gender?” vs. “What is the conditional relative frequency distribution of gender among Conservatives?”

Make sure students can correctly sort out answers to similar sounding questions: 1. What percent of the class are women with liberal political views?

2. What percent of the liberals are women?

3. What percent of the women are liberals?

Raise the issue of independence. It’s not formal independence yet, just the general idea that if gender and political view were independent, the percentages for either gender would mirror the class as a whole, or the percentages of Liberal, NDP, and Conservative would be the same for both genders. If they are not, we encounter what the politicians refer to as the “gender gap.”

Statisticians would say this indicates that voting preference is not independent of gender.

Pay attention in each chapter to the What Can Go Wrong? (WCGW) section. Helping students avoid common pitfalls is one of the keys to success in this course.

Simpson’s Paradox is fun, but don’t overemphasize it. It’s not a critical issue, but it’s a good discussion point about making valid comparisons, and not overlooking lurking or confounding variables.

The Importance of What You Don’t Say Probability. You can see that we are patrolling the perimeter of probability. Concepts like relative frequency, conditional relative frequency, and independence cry out for a formal discussion in probabilistic terms. Don’t heed the cry. You and we know that we are setting up the habits of thought that students will need for learning about probability. But this isn’t the time to discuss the formalities. Or even to say the word “probability” out loud. (Notice that the book doesn’t use the term in this chapter at all.) Talk about “relative frequency” instead. In this class probability is a relative frequency, so we are encouraging students to think about the concepts correctly.

We’ll get to the basics of probability in Chapter 8.

Class Examples 1. If you have collected class data about gender and political view, you can use it here. Help students develop their Plan-Do-Report skills with questions like: • What percent of the class are women with liberal political views? .

Chapter 4 Displaying and Describing Categorical Data

• • • • •

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What percent of the liberals are women?

What percent of the women are liberals?

What is the marginal frequency distribution of political views?

What is the conditional relative frequency distribution of gender among conservatives?

Are gender and political view independent?

2. Is the color distribution of M&Ms independent of the type of candy? Break open bags of plain and peanut M&Ms and count the colors. (Then eat the data…)

3. Refer to Simpson’s Paradox. Here’s a nice thought problem to pose to the class; give them a few minutes to work it out. Two companies have labour and management classifications of employees. Company A’s labourers have a higher average salary than company B’s, as do Company A’s managers. But overall company B pays a higher average salary. How can that be? And which is the better way to compare earning potential at the two companies?

Solution: Make sure you first point out that this example deals with quantitative variables, not categorical. The paradox can be explained when you realize that Company A must employ a greater percentage of labourers than Company B. Also, Company A must employ a smaller percentage of managers than Company B. If labourers earn salaries that are considerably lower than managers, the salaries of Company A’s labourers will pull the company average down, and the salaries of Company B’s managers will pull the company average up.

The proper way to compare 8 8 Mean Mean the companies is to use the 6 6 Salary Salary salaries that are broken $28000 $43000 4 4 down by job type. Using 2 the overall average salary 2 leads to a misleading 10000 30000 50000 10000 30000 50000 conclusion.

Company A Salaries ($)

Company B Salaries ($)

Resources ActivStats* • Lesson 4-1: Displaying Categorical Variables - examines displays of categorical data.

Basic Exercises 1.

The following data show responses to the question “What is your primary source of news?” from a sample of college students.

Internet Newspaper Newspaper Internet TV

Newspaper TV TV Internet Internet

Internet Internet TV Internet Internet

TV Internet Newspaper Internet TV

Internet TV TV Internet TV

* ActivStats is available at www.pearsonmylabandmastering.com/northamerica/mystatlab/

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Part I Exploring and Collecting Data a. b. c. d.

2.

Prepare a frequency table for these data.

Prepare a relative frequency table for these data.

Based on the frequencies, construct a bar chart.

Based on relative frequencies, construct a pie chart.

A cable company surveyed its customers and asked how likely they were to watch more than 2 hours of TV per day. The following data show the responses.

Very Likely Likely Unlikely Very Likely Unlikely a. b. c. d.

3.

Unlikely Likely Likely Unlikely Unlikely

Very Likely Likely Likely Very Likely Likely

Prepare a frequency table for these data.

Prepare a relative frequency table for these data.

Based on frequencies, construct a bar chart.

Based on relative frequencies, construct a pie chart.

A membership survey at a local gym asked whether weight loss or fitness was the primary goal for joining. Of 200 men surveyed, 150 responded fitness and the rest responded weight loss. Of 250 women surveyed, 175 responded weight loss and the rest responded fitness. a. b. c. d.

4.

Unlikely Unlikely Unlikely Unlikely Unlikely

Construct a contingency table.

How many members have fitness as their primary goal for joining the gym?

How many members have weight loss as their primary goal for joining the gym?

Based on the results, should the owner of the gym emphasize one goal over the other?

Explain.

The following contingency table shows students by major and home province for a university in New Brunswick.

Major Program of Study Home Biology Accounting History Education Province NB 80 65 55 100 AB 50 40 65 95 ON 75 50 45 80 BC 65 55 40 40 a. b. c. d.

Give the marginal frequency distribution for home province.

Give the marginal frequency distribution for major program of study.

What percentage of students major in accounting and come from NB? What percentage of students major in education and come from ON?

.

Chapter 4 Displaying and Describing Categorical Data 5.

The following contingency table shows students by major and home province for a university in New Brunswick.

Major Program of Study Home Biology Accounting History Education Province NB 80 65 55 100 AB 50 40 65 95 ON 75 50 45 80 BC 65 55 40 40 a. Find the conditional distribution (in percentages) of major distribution for the home province of AB. b. Find the conditional distribution (in percentages) of major distribution for the home province of BC. c. What can you say about these two conditional distributions?

ANSWERS 1.

a.

News Source Internet Newspaper TV

Number of Students 12 4 9

b.

News Source Internet Newspaper TV Total

% of Students 48 % 16 % 36 % 100%

c.

Chart of Count 12 10

Count

8 6 4 2 0

Internet

Newspaper Source

d.

.

TV

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Part I Exploring and Collecting Data Pie Chart of Source Category Internet Newspaper TV

TV 36.0%

Internet 48.0%

Newspaper 16.0%

a.

Response Unlikely Likely Very Likely

Number of Customers 10 6 4

b.

Response Unlikely Likely Very Likely Total

% of Customers 50 % 30 % 20 % 100%

c.

Chart of Count 10

8

Count

2.

6

4

2

0

Unlikely

Likely Response

.

Very Likely

Chapter 4 Displaying and Describing Categorical Data d.

Pie Chart of Response Category Unlikely Likely Very Likely

Very Likely 20.0%

Unlikely 50.0%

Likely 30.0%

3.

a.

Goal for Gym Membership Fitness Weight Loss 150 50 75 175 225 225

Gender Men Women Total b. c. d.

Total 200 250 450

225 225 No. 50% of the membership is pursuing each goal.

4.

Home Province NB AB ON BC Total a.

Biology 80 50 75 65 270

Major Program of Study Accounting History 65 40 50 55 210

Home Province NB AB ON BC .

55 65 45 40 205 Number of Students 300 250 250 200

Education

Total

100 95 80 40 315

300 250 250 200 1000

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Part I Exploring and Collecting Data b.

Major Biology Accounting History Education

Number of Students 270 210 205 315

c.

65/1000 = 0.065 = 6.5 %

d.

80/1000 = 0.08 = 8 %

5.

Home Province NB AB ON BC Total a.

b.

c.

Biology

Major Program of Study Accounting History

Education

Total

80 50 75 65 270

65 40 50 55 210

100 95 80 40 315

300 250 250 200 1000

Major Biology Accounting History Education Major Biology Accounting History Education

55 65 45 40 205

Conditional for AB 20 % 16 % 26 % 38 % Conditional for BC 32.5 % 27.5 % 20 % 20 %

Percentage of biology and accounting majors in BC is higher than in AB. Percentage of history and education majors in AB is higher than in BC.

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Chapter 5 Displaying and Describing Quantitative Data What’s It About?

Now we turn our attention to quantitative variables. First, we examine displays: histograms and stem-and-leaf displays. We want students to learn to describe the important features of a distribution: shape, centre, spread, and any unusual features such as outliers, gaps, or clusters.

Next, we learn how to summarize the centre and spread of a distribution. We discuss the problems that arise when these statistics are used for skewed data or in the presence of outliers and note that the median and IQR are more appropriate in those situations.

Then we move on to comparing distributions. The five-number summary provides a brief but powerful numerical description of data. The Outlier Rule gives students a guideline for nominating cases as possible outliers. Collectively these descriptions allow students to create and interpret boxplots, great displays for comparing several distributions at once. Another new plot, the time series plot, allows us to display the change in a quantitative variable over time. In an optional section, we discuss how to re-express skewed data to make its distribution more nearly symmetric.

In What Can Go Wrong? we include some important cautions along with an example of an inappropriate histogram.

Comments We continue to focus on understanding and interpretation. Many of the concepts in this chapter are vague—something students might not expect if they were taking a math class. For example, there’s no fixed criterion for declaring a distribution to be “skewed” or even “bimodal”. How far away must a point get before it is an outlier? Is a gap real if it disappears when we rescale the histogram?

The Outlier Rule is just a guideline and may fail to catch cases that should be considered outliers or may nominate cases that are not really all that unusual. It doesn’t relieve us of the need to think about the context of the data in deciding whether to consider a point an outlier.

It’s a good idea to point out to students that these are deliberately vague concepts. That insight can help students who might otherwise think that they don’t understand them. Don’t underestimate the importance of these vague concepts. Although methods we deal with later for inference require precise distributional assumptions for their theoretical background, in practice the best we can do is look at the shape of histograms to judge whether they are roughly unimodal and symmetric. And we will insist that students do just that before using these advanced methods.

A word about accuracy. Some math-oriented students may show signs of frustration with what they see as lack of precision here. There are alternative methods for finding quartiles that can result in different answers. We don’t report calculations of means and standard deviations to several decimal places. Many questions don’t have one indisputably correct answer. We may not know at first which “average” is meant when we hear an average reported for some data—and we see in this chapter that it can matter to how we think about that statistic. Some students wish for immutable rules and precise “answers.” The fact that Statistics has a lot of grey areas may frustrate them.

It may help to point out to your students that Statistics deals with the real world, not “MathWorld.”

They are used to dwelling in MathWorld in all of their other math courses. In MathWorld the bicyclist rides in a straight line along a level road at a constant speed without encountering any air resistance past a tree that stands perpendicular to the ground. For this scenario, there is a “right .

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Part I Exploring and Collecting Data

answer”. In Statistics we liberate ourselves from these unrealistic constraints, but that liberty comes at a price. Uncertainty, caution, and approximation will accompany everything we do. The business world is more complicated – and more interesting – than MathWorld.

It is precisely the demands of dealing with the real world that generate the ambiguities in Statistics.

So we deal with them by maintaining our connection to the real world. The answer is almost never just a number. Instead, it must come with interpretation in terms of context of the data and the questions we asked of the data—the things we considered at the Plan step and now must report in the Report step. We emphasize the value of identifying the business question to be addressed and show reports in a business memo format.

We introduce centre as the first thing we want to know about a variable. Spread is the second thing we want to know. These two numbers, centre and spread, should always be reported together.

Emphasize the importance of spread to your students, as well as the different ways it can be measured, and why sometimes one is more appropriate than another.

In Chapter 10, we’ll make it clear that the central concept in all of Statistics is that distances and differences are measured with a ruler based on spread. The practical approach we take in this book makes the vague concepts of this chapter central to even the most advanced methods found in the concluding chapters. That may not be what you’ve seen in older texts where one could fly by these ideas and never look back, but it is the way that practicing statisticians think and work.

Why y? This is the first chapter in which it becomes clear that we use the letter y to denote the variable we are summarizing, about which we would like to know something, or that we are modelling. This won’t bother your students, who are not likely to have any preconceived ideas about what letter to use. But it might take you by surprise if you studied Statistics from a text that used x by default or are reading other texts for background.

We made this choice with much thought. Most Statistics methods model the world with equations.

Once we get beyond one-variable summaries, the Statistics convention is to denote the variable that is being modelled, described, or predicted as y, and to plot it on the vertical axis of graphs. We consistently use x (and even x1, x2,...) to denote the variables that describe, predict, or model y.

Why should we do otherwise when we consider only one variable? We want to emphasize that we are modelling the world throughout the course. The consistency of naming the variable that we model y whenever we need a generic letter helps to convey that message.

As a purely practical matter, if we had used x in early chapters, we’d have had to switch to modelling y for Chapters 7 and 8, then back to x for the early chapters on inference, and back again to y for regression inference and beyond. That seemed unnecessarily confusing.

Nevertheless, it may be worth pointing out the choice to your students. Other textbooks do use x, as do some functions on calculators, in statistics software, and in some Excel statistics functions.

Looking Ahead Almost everything we do in Statistics starts by making a picture of the data. We’ll be making histograms and other displays to check out our data in almost every chapter of the book. Displays can reveal things that the data itself and numerical summaries may miss or hide. You may want to invite your students to thumb through the book to see how often we show displays of data even in the later chapters and exercises. They should be encouraged to understand that these skills and concepts are fundamental and will be used throughout the course.

The notion that we need to look at the data (a histogram, for example) before deciding which .

Chapter 5 Displaying and Describing Quantitative Data

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descriptive statistics are appropriate is important in a broader sense. Every statistical method we encounter will require some underlying assumptions. We should not forge ahead blindly until we think about whether those assumptions are reasonable in our situation. That’s an essential part of our Planning. Often we need to check some conditions before proceeding. Making a histogram of the data to rule out skewness, multiple modes, and outliers before finding the mean and standard deviation sets the stage for other such checks that we’ll require later on.

Class Do’s Continue to build students’ vocabularies. There are many terms that are critical to sound understanding and clear communication. We can’t Plan, Do or Report without understanding and using the proper terminology.

Some students may think that skewness is backwards – that the body rather than the tail should determine whether we say a distribution is skewed left or right. If you liken “skewed” to “strewn” it can help. The bulk of people’s incomes are low, but a histogram looks as though a few have been strewn further to the right of the main body – hence “skewed to the right.” (Think Bill Gates and Oprah Winfrey.) It might be a good time to talk about the difference between skewed and biased. News media and others sometimes use “skewed” to mean “biased.”

Ask students when to use a bar graph and when to use a histogram. They have to think about the type of variable first. And be sure they understand that bars represent piles of data. We often see ill-conceived displays in which a bar is used for each datum.

Get students to think about why one might prefer one type of graph to another. In what ways are stem-and-leaf displays, dotplots, and histograms all similar? What information is apparent in some but not others? What are the advantages and disadvantages of using each type of graph?

Be sure that students learn to look for all the important aspects of a distribution. The shape, centre, and spread describe different features. No description is complete without talking about all three, and adding comments about any unusual features such as outliers, gaps, or clusters.

Present the summary statistics as ways of describing centre and spread. Group them that way: median and IQR; mean and standard deviation. Median and IQR are appropriate for any data.

Mean and standard deviation are a bit more powerful, as they have some important mathematical properties we can exploit. But we cannot use mean and standard deviation effectively when the data are strongly skewed or outliers are present.

When reporting measures of centre and spread, use statistics that “go together”. Median and IQR are the same type of measurement, based on positions in the data. Likewise, mean and standard deviation are based on the size of the data, so they should be paired. We would not talk about, say, median and standard deviation together.

Focus on the meaning of the quartiles rather than on the details of calculation. There are alternative methods for finding the quartiles. Some sources advise including the median in each half of the data; others leave it out entirely. In large data sets this is rarely an issue. Tell students that they should clearly state their choice of method if there can be any doubt about the answer.

Students often misuse the terms “range” and “interquartile range,” mistakenly thinking these are intervals rather than numbers. They will say things like “The company’s profits were in the IQR” when they mean they were between the first and third quartiles (or within the middle 50% of all company profits). As always, clear and correct use of the vocabulary is critical. You may want to use examples regarding population incomes when it comes to the IQR. Students may realise that a .

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Part I Exploring and Collecting Data

larger IQR from one period to another means that there has been an increase in income disparity between the top earners in the bottom 25% of all income earners and the bottom earners in the top 25% of all income earners is growing.

Be sure students understand the effect of outliers on summary statistics. When there is an outlier in the data, the mean, range, and standard deviation can change a lot. The median and IQR are more stable, and therefore better descriptors in the presence of outliers.

Students should understand the relationship between the median and the mean in skewed distributions. Once they see that a high outlier can raise the mean without having much impact on the median, it’s easy to talk about skewed distributions; maybe there’s no outlier, but there are several unusually large values. When they see that the median divides the area in half but does not provide a balance point for the distribution, it becomes clearer why we use different summaries for skewed distributions. We have also included the geometric mean to show students that for exponential changes, the arithmetic mean is not appropriate. You may want to show them examples of this as we have done in the text.

Students who check assumptions and conditions before proceeding are more likely to deal properly with outliers, skewness, and other issues. Keep requiring this key step.

It may help some students to calculate a few standard deviations by hand—but don’t do many.

This is not an essential skill. It’s more important that they know how to have a calculator or statistics package find SDs, and even more important that they grasp what is being measured.

Remind students that summarizing distributions, or comparing them, still requires discussion of shape, centre, spread, and unusual features. As always, answers are sentences, not numbers.

As usual, it’s more interesting (and more difficult) to compare two distributions. Insist that graphical comparisons always use graphs drawn on the same scale. Verbal comparisons should include shape, centre, spread, and unusual features.

If you are comparing two distributions with different shapes (one skewed and the other symmetric), use the median and IQR to describe both. Although the mean and standard deviation would work fine for the symmetric distribution, you couldn’t compare them to the median and IQR of the other data.

The Importance of What You Don’t Say Don’t say “Measure of central tendency.” At this point, without a Law of Large Numbers, there’s no “tending” involved. The mean, median, and geometric mean are measures of centre; the IQR and standard deviation are measures of spread. Simpler, shorter phrases encourage correct use.

Many students will have heard about three centres: mean, median, and mode. Some, if not most, will not have heard about the geometric mean. Emphasize why this is an appropriate measure for geometric growth averages. Don’t make a big deal out of trying to attach a number to the mode.

This can be problematic, especially in a small data set. A single data value that is repeated can be declared the “mode” even when it is nowhere near the middle hump of a unimodal distribution.

We care more about the overall shape of a histogram, and especially if it tells us there are two commonly occurring clusters of data rather than just one. It’s the number of modes that matters, not the numerical value of a mode.

Don’t fret over the different ways to find quartiles. There are many different ways to find the quartiles; in fact, the various software packages employ 5 or 6 different algorithms! Some include the median in both halves (as we recommend in the text); some exclude it. Still others use various .

Chapter 5 Displaying and Describing Quantitative Data

5-5

methods of interpolation. This means two things: (1) different methods don’t always agree with each other about the precise values of the quartiles, and (2) it really doesn’t matter. The values will usually differ very little, especially for large data sets. Our goal is to get a feel for the story the data tell. For example, knowing that the upper quartile of incomes is around $45,000 paints the picture we want. It’s really irrelevant that the book’s method finds Q3 to be $45,103 and a calculator says it’s $45,104.50. Close enough.

Don’t get hung up over the divisor of (n – 1) in the formula for standard deviation. It’s an issue we’ll discuss in more detail later in the course. Some students accept it willingly, others grudgingly. You can appeal to the fact that if the mean is used in the calculation, then only n – 1 of the n scores are actually free to vary. Or you can discuss the fact that using n in finding standard deviations of small groups would underestimate the variability actually present in the larger groups they represent—and this is perfectly compensated for by dividing by n – 1.

Don’t get hung up on calculating the standard deviation by hand. That’s what the technology is for. It’s far more important that students understand what the standard deviation says about the distribution.

Don’t worry about always reporting answers to a fixed number of decimal places. This is the real world, not MathWorld. We simply can’t find an average height to the nearest thousandth of an inch. Just keep a reasonable number of significant digits when rounding off, and be careful about using rounded values in future calculations.

Don’t ignore or “throw away” outliers. Outliers can affect the summaries that we use to describe our data, and we can’t pretend they don’t exist; we can’t toss them aside unless they are obvious errors. Outliers are often the most interesting values in a data set because they don’t follow the same pattern as the other values. We want to call attention to them, not conceal them. The best policy when we identify an outlier is to make note of it and gather more information about it – if possible. (Is it an error? Was it collected under different conditions than the rest of the data?) If it’s not an error, do your calculations both with and without the outlier to see how much effect it has on your results.

Don’t use the word “trend” if time is not a variable. Try to be careful about the use of the word “trend”. There is a trend of increasing numbers of women involved in sports since it is changing over time. Nonetheless, in general men tend to be stronger physically than women. That’s a tendency, not a trend, because there’s no time element involved.

Class Examples and Activities 1. Use the data you collected the first day. There are many distributions you can compare: •

Make a histogram of the number of siblings. It will be skewed to the right.

•

Make a histogram of the number of provinces visited.

•

Compare heights for males and females. (Or use shoe sizes.) Point out the importance of having the same scale for both graphs.

•

Describe the centre, spread, and any unusual features in each of these variables.

•

Compare two histograms or two dotplots, asking which has a larger SD.

.

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Part I Exploring and Collecting Data

2. Imagine we were to examine data for the number of inches of snowfall in December in Ottawa, for the past 100 years. What information could we learn from looking at a histogram that would be difficult to see in a time series plot? What information could we see in a time series plot that would not show up in a histogram?

Solution: Using a histogram, we could learn about the shape, centre, spread, and unusual features of the distribution of snowfall in December in Ottawa. We would be able to see the average snowfall over the 100-year period, how spread out the distribution of snowfalls was, and whether the distribution was unimodal, bimodal, skewed, or uniform. Using a time series plot, we could learn whether December snowfall totals are increasing, decreasing, or staying the same year to year over the 100-year period.

3. Collect some data and plot them.

•

A look at home sales records in your city or a scan of the real estate ads can give you data about home values in your area. These data will most likely be skewed, with many low and moderately priced homes and fewer very expensive homes. Such data can often be made more symmetric with a logarithmic re-expression.

4. The mini case studies at the ends of the chapters are another source of material for classroom examples and activities.

5. If you haven’t done so yet, it’s time to get students working with statistical software. By now they can create pie charts, histograms, and boxplots, and they can calculate summary statistics.

Be aware that Excel doesn’t make boxplots; for that students will have to use the DDXL addin.

Resources ActivStats*

• Lessons teach students to construct dotplots and histograms, and to describe distributions.

• The activity on standard deviations is a nice introduction, explaining the process and saving you from having to explain the formula. Both the activities on centres and spreads let students drag points to see the influence of outliers and individual points on alternative measures of centre and spread. A few minutes with these activities can be much more effective than any abstract discussion of the effects of outliers on these statistics.

Web Links

• At the U.S. Census Bureau’s International Data Base (CIDB), you can create population pyramids for any country and any year (and even animate the changes) http://www.census.gov/

• For still more data sets, try the Data and Story Library (http://lib.stat.cmu.edu/DASL/).

• This is Statcan’s weblink for population pyramids along with some lesson plans and suggested activities that may be useful: http://www12.statcan.ca/english/census01/teacher's_kit/activity4.cfm * ActivStats is available www.pearsonmylabandmastering.com/northamerica/mystatlab/

.

Chapter 5 Displaying and Describing Quantitative Data

5-7

Basic Exercises 1.

Consider the data given below on weights of trucks in tons at a weighbridge on highway 401 in Ontario.

62.5 64.0 59.5 62.5 63.0

52.0 58.5 66.0 62.5 64.0

70.5 67.5 53.0 59.5 63.5

63.0 64.5 74.5 69.5 56.5

57.0 61.0 66.0 52.0 74.0

67.0 63.5 55.5 70.0 64.5

a.

Develop a frequency distribution using the following bins: 50-55, 55-60, 60-65, 65-70, and 70-75. b.

Develop a relative frequency distribution using the bins from part a. c.

Draw a histogram based on the relative frequency distribution from part b. d.

Is this distribution symmetric or skewed?

2.

Consider the data given below on the average number of products returned per day at a retail store in Fredericton, NB, during 20 consecutive months.

2.9 2.8

2.1 2.2

3.5 3.3

a. b. c. d.

Construct a stem-and-leaf display.

Is the distribution unimodal, bimodal or multimodal?

Is the distribution symmetric or skewed?

What would be a better measure of the centre for this distribution, the mean or the median?

3.

Consider the data given below on monthly average gas prices in Belleville, ON.

110

125

a. b. c. d. e.

Compute the mean.

Compute the median.

What is the mode?

What is the lower quartile, Q1?

What is the upper quartile, Q3?

99

4.2 3.4

115

5.1 4.4

119

6.5 5.5

95

7.8 6.1

110

8.1 4.2

132

4.5 2.5

3.8 2.1

85

4. A population increased by 3% at the end of year 1, by 5% at the end of year 2, and by 1% at the end of year 3. What was the compound annual growth rate? a. b. c. d. e.

3% 2.99% 2.98% 3.1% 28.97% .

5-8

Part I Exploring and Collecting Data

5.

Using the data set in exercise 3,

a. b. c. d.

Compute the range.

Compute the interquartile range (IQR).

Compute the variance.

Compute the standard deviation.

6.

The data given below shows the average number of customers per day purchasing a promotional product in a health food store in Guelph, Ontario, over 13 consecutive weeks.

10.5 22.5

27.0 18.5

21.5 19.0

15.0 17.0

19.0 20.0

20.0 21.0

12.5

a. b. c. d.

Provide the five-number summary.

Compute the value of the upper fence.

Compute the value of the lower fence.

Show the boxplot.

7.

The number of orders received per day at a clothing web site in British Columbia has a mean of 1175 and a standard deviation of 250. a.

One day the number of clothing orders received is 1500. Calculate its standardized value (its z-score). b.

What does its z-score tell us? c.

Another day the number of clothing orders received is 950. Calculate its standardized value (its z-score). d.

What does its z-score tell us?

8.

The number of orders received per day at a jewellery web site in Nova Scotia has a mean of 52 and a standard deviation of 15. a.

One day the number of jewellery orders received is 97. Calculate its standardized value (its z-score). b.

What does the z-score tell us? c.

Which value is more extreme, 97 jewellery orders or 1500 clothing orders (from #7a above)? Explain. d.

What number of jewellery orders would be the same number of standard deviations below its mean as 950 clothing orders (from #7c above)?

9.

The following data represent quarterly profits (in thousands of dollars) for a small cleantech start up business in Ottawa over a three year period.

-20

-10

-15

10

12

a. b.

Construct a stem-and-leaf display.

Construct a time series plot. .

15

12

22

22

28

25

38

Chapter 5 Displaying and Describing Quantitative Data c. d.

Are the data stationary? Explain.

Which display (stem and leaf or time series plot) is preferable for these data? Why?

ANSWERS 1.

a. and b.

Note that 70.0 is referred to the last bin.

Bins Frequency 50 – 55 3 55 – 60 6 60 – 65 12 65 – 70 5 70 – 75 4

c.

d.

Fairly symmetric

2.

a.

3.

b. c. d.

Unimodal Skewed to the right Median, as the distribution is skewed.

a. b. c. d.

110 110 110 99 .

Relative Frequency .10 .20 .40 .17 .13

5-9

5-10

Part I Exploring and Collecting Data e.

119

4.

b.

2.99%

5.

a. b. c. d.

47 20 223.25 14.94

6.

a.

Max Upper Quartile, Q3 Median Lower Quartile, Q1 Min

27.0 21.0 19.0 17.0 10.5

b. c. d.

27.0 11.0

7.

a. b. c. d.

z = 1.3 This observation is 1.3 standard deviations above the mean. z = -0.9 This observation is .9 standard deviations below the mean.

8.

a. b. c.

z=3 This observation is 3 standard deviations above the mean.

The 97 jewellery orders is more extreme because it is more standard deviations from its mean.

38.5

d.

.

Chapter 5 Displaying and Describing Quantitative Data a.

b.

Time Series Plot of Profit 40 30 20 Profit

9.

10 0 -10 -20 1

c. d.

2

3

4

5

6 7 Index

8

9

10

11

12

No. There is an increasing trend.

The time series plot is preferable because the stem-and-leaf display cannot capture the trend.

.

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6-1

Chapter 6 Scatterplots, Association, and Correlation What’s It About?

We turn our attention to looking at associations between quantitative variables. We draw scatterplots and look for patterns. We describe the association in terms of direction, form, and strength. When the association appears to be linear, we find the correlation. We use these concepts and some new vocabulary to write descriptions of associations, while warning students not to jump to conclusions about cause and effect.

We teach students to create scatterplots and find correlations using technology.

Comments Our emphasis on the importance of standard deviations as a kind of universal yardstick and on standardized scores as a unit-less way to talk about and compare data begins to pay off here. We use these ideas to motivate and explain the idea of correlation as a numerical description of the underlying association between two variables. We’ve found that it helps build an intuitive understanding of both correlation and regression. Thinking in z-scores and writing the formula in z-scores helps the students understand that the correlation is unit-less and why a change of units in x or y doesn’t change the correlation.

We continue to require that students Plan before rushing into their analysis. Before calculating a correlation, students must look for straightness in a scatterplot. Correlation requires the assumption of linearity. We check the condition by looking at the scatterplot. If it’s not linear, they should not calculate a correlation.

Looking Ahead Thinking of associations of quantitative variables in terms of standardized scores prepares students to deal with regression, which will be coming next.

We emphasize that statistical procedures are valid under certain assumptions, and we require students to think about those assumptions by checking corresponding conditions. This process continues throughout the course, and will be especially important when we get to inference.

Class Do’s Help students with the vocabulary: association, correlation, explanatory (or predictor) and response variables.

Emphasize the importance of the first rule of data analysis: make a picture. A scatterplot can reveal a lot about the relationship between two variables, including whether the relationship is linear and whether there are any outliers.

Don’t rush right to correlation. Have students describe scatterplots verbally. For histograms the mantra was shape, center, spread, and unusual features. Now it’s direction, form, strength, and unusual features. Make them Report what they see in a scatterplot, clearly, concisely, completely, and in context. (They’ve heard that before, too.) We’ll quantify the strength of the association after they know what they are looking at.

It’s especially important to point out the precise use of the word “correlation” . In everyday usage, correlation refers to any kind of association between any two issues. Proper usage in Statistics is much more limited. Correlation is a number that measures the direction and strength of a linear association between two quantitative variables. There can’t be a correlation between the model of .

6-2

Part I Exploring and Collecting Data

a car and how likely it is to be stolen. There can be a strong association between the speed at which you drive and the fuel economy you attain, yet the correlation is probably low, since the relationship is probably not linear. (A curved relationship with terrible fuel economy at 5 and 100 kilometres per hour and highest fuel economy between 40 and 60 kph may produce a correlation near 0, even though there is certainly an important relationship present here.)

Use the idea of z-scores to develop the formula. We don’t emphasize actually doing the calculation—that’s why we have computers— but students need to understand why correlation is unitless, and why it’s immune to changes of scale, units, or roles of x and y. If the correlation between highway speed (in mph) and traffic density (in cars per mile) is -0.54, ask them how this changes if we measure speed in kilometers per hour and density in cars per 100 meters.

Be sure to emphasize the conditions for correlation (linearity, no outliers, quantitative data).

Checking conditions will be a major issue throughout much of the course—continue to make that process a fundamental part of doing Statistics.

Spend some time exploring how a single point can greatly affect the value of r. If you have a computer projector, the interactive tool in Chapter 6 of ActivStats* provides a vivid animated demonstration and offers students the “game” of placing points on a scatterplot with the goal of attaining a specified correlation—a task that even some statisticians find challenging.

A common misunderstanding of Statistics involves confusing correlation with cause and effect. A scatterplot of the amount of damage caused by different building fires plotted against the number of firefighters sent to those fires would show a positive correlation. Does this mean that we should send fewer firefighters in order to decrease fire damage? Of course not! The size of the fire is a lurking variable that affects both the amount of damage and the number of firefighters.

The Importance of What You Don’t Say Don’t emphasize calculating the correlation by hand. That’s not an important skill, and it distracts from the conceptual understanding we hope to foster. And it takes a long time.

Don’t use the computational formulas. They look much scarier and reveal nothing. Once upon a time, when introductory Statistics students used pencil and paper to calculate a correlation, these other formulas were shortcuts. That day is long gone, though. And if you remember that day (as we do), don’t regale students with it; they really don’t care.

Don’t say “r is 66%.” Talk about correlations as decimals. A correlation is not a percent of anything. In chapter 18, we’ll introduce R2, which is a percent of variance.

Don’t permit casual use of “correlation.” In Statistics, correlation is a formal term with a very specific meaning. Use it then, and only then. Require students to say “association” when that’s what they mean. You’ll find them misusing the term if they write up any of the mini case studies— call them on it.

Class Examples 1. Propose two variables and have students imagine and describe what they think a scatterplot might look like. Some examples: • Drug dosage and degree of pain relief Answer: The association is likely to be strong, positive, and curved. Assuming, of course, that the drug is an effective pain reliever, as the dosage increases, the degree of pain relief * ActivStats is available at www.pearsonmylabandmastering.com/northamerica/mystatlab/.

.

Chapter 6 Scatterplots, Association, and Correlation

•

•

• •

•

6-3

will increase. Eventually, the association is likely to level off, until no further pain relief is possible because the pain will be gone. This assumes we can quantify “pain relief”.

Calories consumed and weight loss Answer: The association is likely to be moderate, negative, and linear. As fewer calories are consumed, more weight is likely to be lost. The association will not be strong because some people lose weight easier than others, and there are other variables involved like overall health, exercise, and beginning weight.

Hours of sleep and score on a test Answer: The association is likely weak, positive, and possibly linear. Generally, a wellrested person is expected to score higher on a test. The relationship is weak because there are other variables involved. Maybe a person got less sleep because they were up studying.

Shoe size and grade point average Answer: There is no association between shoe size and GPA. The scatterplot is likely to be randomly scattered.

Time for a kilometre run and age Answer: The association between time for a kilometre run and age is likely to be moderate and curved, with no dominant direction. The very young will likely have high run times.

Run times are likely to be the lowest for people in their late teens or early twenties. Older people are likely to have high run times.

Age of car and cost of repairs Answer: The association between age of car and cost of repairs is positive, moderate, and linear. As cars get older, they usually require more repairs.

2. Look at actual scatterplots, and elicit good descriptions. There are many examples in our text, and much data is available on the web.

3. Collect data on heights and weights. Use your data to make scatterplots and find correlations.

You can re-use these data to look at regression, linear modeling, residuals, and R 2 later on, too.

If some students object to listing their weights anonymously, you could alternatively collect data on height/shoe size or height/hand-span (the width of a person’s hand from the tip of the thumb to the tip of the little finger, when the fingers and thumb are fully spread).

4. Help students develop a feel for how the number calculated as correlation relates to the pattern they see in the scatterplot. ActivStats has several appropriate activities.

5. Show students that one point can change the correlation dramatically. We don’t discuss influential points formally yet, but you can begin to develop some ideas. Points near the center of the scatterplot have little effect, points that fit the pattern increase the strength of the correlation (and more so the farther the point is from the center), and points that don’t fit the pattern decrease (and can even reverse the sign of) the correlation. Again, ActivStats has very effective tools.

Resources ActivStats Lesson 6—”Scatterplots” explores scatterplots, including the associated vocabulary and descriptions. It also offers work on re-expressing data that you may do now or postpone until later. The section on correlation introduces students to the calculation and meaning of .

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Part I Exploring and Collecting Data correlation. It also provides two interactive activities you might use as classroom demonstrations. You can move a correlation slider between –1 and +1 and watch the scatterplot change. ActivStats asks students to construct scatterplots having a specified target correlation. You can use it to place several points in a moderately strong pattern, and then drag one more point around the screen to show how the correlation changes in the presence of outliers.

Web Links • Statistics Canada’s CANSIM tables can be accessed for a tremendous amount of data that students and instructors alike can access. http://www5.statcan.gc.ca/cansim/a01?lang=eng • Many sources will have bivariate data. The Data and Story Library’s (lib.stat.cmu.edu/DASL) organization makes it easy to find examples of data suitable for discussing correlation.

Assignments This is a short chapter of important basic concepts that lay the groundwork for the bigger issues coming next. Assign some reading and spiral the assignments through the important topics. First, do general descriptions of scatterplots, next correlation, and then the properties of r and the What Can Go Wrong issues, and finally a quick look at re-expression.

Basic Exercises 1.

Consider the following data on the average share price of a Canadian mining company and tons of copper extracted per day over 10 quarters.

Share price ($) 2 3 7 9 10 10 12 15 16 20 a. b. c. d.

2.

Copper (in tons) 10 11 13 14 18 20 20 22 22 26

Prepare a scatterplot.

What can you say about the direction of the association?

What can you say about the form of the relationship?

What can you say about the strength of the relationship?

Consider the following scatterplot. .

Chapter 6 Scatterplots, Association, and Correlation

6-5

Scatterplot of y vs x 60

50

y

40

30

20

10 10

a. c. d. e.

3.

30 x

40

50

What can you say about the direction of the association?

What can you say about the form of the relationship?

What can you say about the strength of the relationship?

What else does the scatterplot reveal about the data?

Data on salary (in $1000’s) and years of experience were collected for a sample of employees at a large multinational corporation. a. b. c. d.

4.

20

Which variable is the explanatory or predictor variable?

Which variable is the response variable?

Suppose the correlation coefficient is r = + .66. Before interpreting this value, what would you like to see? Why? Suppose that all correlation conditions are met, what does r of .66 tell us?

The profits and costs for a health food restaurant in Edmonton are given in the table over the last 6 years.

Profits Costs ($million) ($million) 7 20 3 14 9 15 10 30 15 32 15 40 a. b. c. d.

Compute the means for profits and costs.

Compute the standard deviations for profits and costs.

Compute the correlation coefficient.

Interpret the correlation coefficient. .

6-6

Part I Exploring and Collecting Data e.

5.

The following table gives data on profits and costs of a coffee shop franchise in Kingston Ontario over 5 years.

Profits Costs ($million) ($million) 10 3 6 4 6 5 3 9 1 8 a. b. c. d. e.

6.

What would be the correlation coefficient if profits and costs were both measured in thousands of dollars instead of millions?

Compute the means for profits and costs.

Compute the standard deviations for profits and costs.

Compute the correlation coefficient.

Interpret the correlation coefficient.

What would be the correlation coefficient if profits were measured in thousands of dollars and costs were measured in millions of dollars?

The scatterplots below represent four pairs of variables with correlation coefficients (in ascending order) -0.55, +0.03, +0.55, and +0.86.

.

Chapter 6 Scatterplots, Association, and Correlation a. b. c. d.

6-7

Which scatterplot has a correlation coefficient of + .86?

Which scatterplot has a correlation coefficient of +.03?

Which scatterplot has a correlation coefficient of +.55?

Which scatterplot has a correlation coefficient of -.55?

ANSWERS 1.

a.

b. c. d.

positive linear strong

2.

a. b. c. d.

negative linear strong an outlier

3.

a. b. c.

years of experience salary We need to see the scatterplot in order to check for outliers and linearity condition.

It indicates a moderately strong positive linear association between years of experience and salary.

d.

4.

a. b. c. d. e.

9.83 for Profits and 25.17 for Costs 4.67 for Profits and 10.44 for Costs + .87 The correlation coefficient of +0.87 indicates a strong positive linear association. +.87, no change.

5.

a.

5.20 for Profits and 5.80 for Costs .

6-8

6.

Part I Exploring and Collecting Data b. c. d. e.

3.42 for Profits and 2.59 for Costs -.898 The correlation coefficient of – 0.898 indicates a strong negative linear association.

-.898, no change.

a. b. c. d.

II. IV. I. III.

.

7-1

Chapter 7 Introduction to Linear Regression What’s It About?

In Chapter 6 we saw how to use correlation to quantify the strength of a linear association; now we see how to use it to create a line that models the association. We build on the concepts from Chapter 6, first showing students that if we standardize both variables, the “best fit” line has slope equal to r. This makes several things clear, among them the concept of “regression to the mean.”

We also emphasize that regression is different from correlation. The regression line depends on which of our variables is the “explanatory” one, whereas in correlation each variable is treated identically.

We then show how to find the equation of the regression line first from summary statistics and correlation, by reading the output of software-generated regression analyses.

We help students interpret the slope, the y-intercept, and the value of R2 in context. As always, we caution students about possible misuses and misinterpretations of these ideas.

We also show students that not all relationships are linear, but that we can transform them so that they are.

Topic Order Why do we introduce regression now? Many other texts save it for later, bundling together all of the regression and inference discussions. We have several reasons and many years of experience teaching topics in this order. Throughout the text we use the concept of a model. Models are ways that we describe the world so that we can understand it. More important for Statistics, models have parameters, which we will soon be estimating and using as targets of statistical inference. The first part of this text introduces the principal models: distributions of individual variables (parameterized by their means and standard deviations) and linear models for association between two quantitative variables. We find that it helps for students to understand the concept of modeling and be comfortable with the most common models. The last section shows how, by using techniques such as taking the logarithms of variables, nonlinear relationships can be transformed to linear ones.

We’ll return to each of the models later to apply inference methods. Then we’ll push beyond the basic models to see generalizations (multiple regression), extensions (ANOVA), and other applications (time series). This order gathers similar concepts together. It also, frankly, guarantees enough time to introduce the fundamentally important topic of regression and time for students to absorb and be comfortable with it.

If you are more comfortable delaying the material in Chapter 7 until later in the course, the chapters are written to accommodate that. But we urge you to try this order.

Comments The simple fact that r is the slope of the line that best fits the scatterplot of z-scores is the key to a myriad of important insights. The concept of regression, the equation of the regression line, the meaning of R2, and even our previous assertion that the correlation coefficient is never greater than 1 (or less than –1) all grow out of this basic fact. .

7-2

Part I Exploring and Collecting Data

Looking Ahead You may find it worthwhile to glance far ahead, to Chapter 18, where we return to regression to do inference. Although you won’t be discussing those issues with your students until later in the course, understanding the theoretical regression model will help you know what to emphasize in the current chapter and provide you with more insight as you handle some of the peskier questions likely to arise now. Notice that in Chapter 18 we still check the same conditions and assumptions, we still examine the scatterplot for linearity and the residuals for lack of evident pattern, we still create the same models, and we still interpret the model the same ways. Chapters 19, 20 and 21 also cover more advanced topics in regression, including interpreting the residuals and multiple regression. You can safely promise your students that the Plan and Report habits they learn in Chapters 6 and 7 will serve them well throughout the course.

Class Do’s Be sure to emphasize the model theme again. Open the discussion with a question about what might constitute a “good” line through a scatterplot. At first, talk in general about the model using examples involving means and standard deviations. Help students understand the general nature of a linear association before trying to create a model in terms of the original data and units. Build several ideas here.

• The first important idea is that the mean of one variable should be associated with the mean of the other. In other words, the line should go through the mean-mean point.

• The model line may not pass through any of the points, but it should be close to a lot of them.

Minor discrepancies between the line and the points are of less concern to us than huge gaps.

Define the residual of a point. See that we want to somehow minimize these residuals. Note that adding them all up should produce a sum of zero, as points above and below the line should balance each other out (in terms of residuals, not counts.) The idea that squaring the residuals addresses all these issues should make sense (and students may suggest it themselves, given their experience with finding standard deviation).

• Attack the problem of finding the “best” line (as defined by least squares) by looking at the standardized scatterplot. You need not actually calculate z–scores, just think about them.

Spend some time discussing the idea that the slope of the regression line for z-scores is the correlation coefficient. Concentrate on getting students to understand the concept of regression.

Consider as an example a mythical college where the correlation between the high school marks of freshmen and the GPAs they earn is 0.50. What should the college predict about the freshman GPA of an applicant who has an average high school mark? Or someone who has an aboveaverage GPA?

• It’s common sense that students with average high school marks should be expected to achieve average GPAs. And remember, the line goes through the mean–mean point.

• When there’s a positive association, we’d predict y to be above the mean when x is above the mean. Students with above-average high school marks should be expected to achieve aboveaverage GPAs. • From there we ask how far above the mean we should predict a student’s GPA will be if he is two standard deviations above the mean high school mark. Initially students will say that if x is at +2 SD, then y should also be at +2 SD. Further discussion should convince them that a .

Chapter 7 Introduction to Linear Regression

7-3

model’s prediction must depend on the strength of the association. It’s unusual for anything to be two standard deviations above the mean. Predicting that will happen is risky and we statisticians are a cautious lot. We’ll venture that far out on a limb only if we have pretty strong assurance that it’s safe. In fact, only if the correlation coefficient is 1. If the correlation is near 0, we stick close to the safety of the trunk—the mean. We would not feel comfortable predicting GPAs far above average for students whose shoe sizes are two standard deviations above the mean (assuming we think those two variables are unrelated . . .). It turns out that the correlation coefficient is our guide. With a correlation coefficient of 0.5, we are willing to go halfway, predicting GPAs will be 0.5  2 = 1 standard deviation above the mean for students who entered that college 2 standard deviations above the mean in high school mark. This caution, this backing off, is called “regression to the mean.”

To write an equation of a best-fit line in original units (rather than z-scores) we need only know two things – and we have both. A point on the line? The mean-mean point! And the slope? Well, we know that in the standardized plane r tells us how many standard deviations to move in the y direction for each standard deviation in the x direction. And that’s still true in the original scatterplot; the slope is r SDs in y per SD in x. Armed with the slope we can use the known point to produce the equation.

Always insist that students use meaningful variable names in writing the equations for the models.

In the interest of understanding and clear communication, we want Shipfee = 2.33 + 0.82Weight , not y = 2.33 + 0.82 x . You can probably guess what the first equation means even without knowing anything about the data that led to it. The second equation provides no such insight.

Spend time on the interpretations of slope, intercept, and R2– always in context. Insisting on real variable names will help students be clear that with regression we are no longer dealing with standardized variables. We are now working with the data in their natural units. The regression slope is in “y-units per x-unit”. The intercept is in y-units. Encourage students to write the phrase “y-units per x-unit” and then fill in the name of the y-units and the name of the x-units. The resulting phrase usually describes well what the regression is about. (We see phrases like “dollars per year of employment” or “grams of protein per ounce.”)

Often the best (indeed, the only reasonable) interpretation of the intercept is as a starting value.

Caution students that if x = 0 makes no sense (a baby with zero weight, the economy in the year “0”), then the interpretation of the intercept as a predicted value for x = 0 makes no sense.

Remind students frequently that the line only models the relationship—it isn’t the “truth.” As George Box said, “All models are wrong, but some are useful.” Be sure that the interpretations students make show they understand the distinction between models and reality. “Weight increases 3 kilograms for every inch of height” is not a correct interpretation of slope. We always need some weasel-words: “The model suggests that weight increases about 3 kilograms, on average, for every additional inch of height”. The same is true of estimates made with the model. Students should say something like, “Using the model, we would predict that this person weighs about 68 kilograms,” not “The person weighs . . .”.

Students have trouble with the distinction between assessing the appropriateness of the model and the strength of the model. These are very different issues. “Appropriate” refers to whether the underlying relationship is linear. Look at the scatterplot or (better) the residuals plot. If there is no sign of a pattern (especially curvature), then a linear model is appropriate. But how strong is the .

7-4

Part I Exploring and Collecting Data

association? Look at R2. See how much of the variability the model explains. These two issues are not the same. An appropriate linear model can have a low R2. The underlying relationship appears to be linear, but weak. And for a different relationship the wrong model can have a high R2.

Imagine a scatterplot showing a series of points lying along a slight but distinct curve. A line may fit them pretty well (high R2) but not be appropriate (because of the curve). Also look at se to see how well the model fits the data. Compare se with sy to see how much the model improves our ability to predict values of y.

We need to be sure students understand that the line for predicting x from y is not the inverse of the line for predicting y from x. Especially tricky is that if we regress standardized y on standardized x we’ll get a slope of r, but if we reverse the roles and regress standardized x on standardized y we don’t get a slope of 1/r, but r again. Ironically, it’s usually the more mathematically oriented students who have trouble with this. They want to back-solve the equation for the new slope rather than refit the line. Have them write the equations of both lines. They should see, for example, that the weight predicted from a height of 190 cm does not itself predict a height of 190 cm. This provides another opportunity to deal with the concept of regression toward the mean.

Here’s another time that we want students to check on an assumption. Among other things, linear regression assumes that there is an underlying linear association between the two variables. As is always the case with assumptions we cannot be sure that is true, but we can check to see if it is plausible. We eyeball the scatterplot or, even better, the residuals plot to see if there is a sign of curvature. The Linearity Condition affirms the assumption and allows us to proceed. It is worth noting that we make this check of residuals after the fact. First we create the model by doing the regression, and then we look at the residuals to decide whether we should use that model.

For nonlinear relationships, show students how taking the square root, logarithm or reciprocal of variables can transform them into linear relationships. You should then ask them to explain this relationship.

The Importance of What You Don’t Say Don’t say “dependent variable”. Though still used in some disciplines, the terms independent and dependent are confusing for two reasons. First, so-called “independent” variables aren’t statistically independent of anything. Second, saying that one variable depends on another sounds like causation – one of the major misinterpretations we are trying to help students avoid.

Unfortunately, these terms are standard in some statistical software, so students should get used to seeing them. That’s why we’ve sometimes used the term “dependent variable” in the generic computer output we show in the text and exercises.

Don’t talk about populations, inference, and so on. Now that we are building models for data, there's a temptation to take the next step to inference. But there's no need to go there yet. We are modeling the world, and still exploring. That's sufficient motivation at this point in the course.

There’s currently no need to talk about populations, and surely no need to wonder whether our model is “true”. That's easy: All models are wrong. We hope that our regression models are useful, but they certainly – and obviously – simplify the relationship and fail to fit every point exactly.

Don’t say that we “throw out the unusual points.” When talking about outliers, try to resist suggesting that we would ever “throw them out.” It's vivid and tempting language, but we should always keep them around to examine and think about. If we find they are genuinely errors, then, of course we are justified in correcting them or deleting them. But otherwise, we “set outliers aside,” comparing the analysis with and without the unusual points. .

Chapter 7 Introduction to Linear Regression

7-5

Class Examples 1. Use ActivStats* to give students an opportunity to play with the least squares idea. (You may also find some effective demos or applets online).

2. We can use cricket chirps to estimate temperature. Just count the number of chirps per minute, divide by 7, and add 4.5. This will give you the temperature (in degrees Celsius) of the cricket's location. How could we discover such a relationship? We collect data on temperature and chirps, and make a scatterplot. If there’s evidence of a strong linear association, then the next question is about the line.

3. Interpreting R2 can be problematic. Students are eager to say “Height explains 46% of a person’s weight” instead of “Differences in height explain 46% of the variability in weight,” or some similar statement about variability. They have trouble both hearing and understanding the difference. An easier-to-understand example can help. Suppose we open a pizza place, selling our plain pizzas for $8 and adding $1.50 per topping. Create a scatterplot graphing cost of pizza against the number of toppings. Because this association is perfectly linear we know that r = 1.00. Ask students which statement correctly describes this situation: A: The number of toppings explains 100% of the cost of a pizza.

B: Differences in the number of toppings explain 100% of the variation in the cost of pizzas.

Students should now see that the first statement is false. It attributes all of the cost to the toppings, ignoring the base price of a plain pizza. The second correctly explains the association.

4. The final requirement is that students be able to create linear models by reading regression output.

5. If you have access to computer labs and software, use a data set to teach students how to perform and interpret a regression analysis.

6. Tell your class that a vocabulary achievement test given to elementary school children discovered a strong association between weight and vocabulary score. Look for explanations that make causality mistakes. Ignoring political correctness, students may suggest that children who are overweight probably stay indoors and read instead of playing sports (weight indirectly causing a better vocabulary). Certainly that chain of causality could be reversed – children who read a lot spend less time in fitness-enhancing activities. Neither of those explanations considers age as a lurking variable: Older children are both bigger and have better vocabularies, so there may be a clear association between weight and scores on a vocabulary test given across many grade levels.

Resources ActivStats • Lesson 7 – “Least Squares Regression”. The tool that encourages students to position a line on a scatterplot and view the sum of the squared residuals offers an intuitive demonstration that there really is a unique line that minimizes the sum of squared residuals. * ActivStats is available at www.pearsonmylabandmastering.com/northamerica/mystatlab/

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7-6

Part I Exploring and Collecting Data

Web Links

• Statistics Canada’s CANSIM data tables can provide a multitude of data from which relationships can be analysed: http://www5.statcan.gc.ca/cansim/a01?lang=eng

• Four lessons at the NCTM’s Illuminations website investigate linear relationships, correlation, and regression lines. http://illuminations.nctm.org Basic Exercises 1.

Data from a sample of employees from a large Canadian mining company were used to estimate the following least squares regression equation: Salary = 36775 + 1590 Years of Experience a. What is the explanatory variable? b. What is the response variable? c. What does the slope mean in this context? d. What does the y-intercept mean in this context? Is it meaningful?

2.

Based on the regression equation from the previous exercise, Salary = 36775 + 1590 Years of Experience , answer the following questions: a. What is the predicted salary for an employee with 10 years of experience? b. If the salary for an employee with 10 years of experience turned out to be $56,500, what is the residual? c. What is the predicted salary for an employee with 20 years of experience? d. If the salary for an employee with 20 years of experience turned out to be $64,550, what is the residual?

3.

Advertisements for used cars in Winnipeg gave the following data and summary statistics: Car price Age of car ($K) (years) 6 20 9 18 12 10 15 8 18 9 25 4 Mean car price = 14.17 ($K) Mean car age = 11.50 (years)

Standard deviation of car price = 6.79 ($K) Standard deviation of car age = 6.19 (years)

Correlation r = -0.920. a. What is the slope of the regression equation? b. What is the intercept of the regression equation? .

Chapter 7 Introduction to Linear Regression

7-7

c. What is the predicted price of a 20 year old car? d. What is its residual?

4.

A real estate agency fit a regression equation to determine the length of time a property is on the market (number of months) before it sells based on asking price (in thousands of dollars).

The following results were obtained.

Time on Market = − 0.64 + .041 Asking Price

R 2 = 50.5% a. Interpret the meaning of R2. b. Is the correlation between Time on Market and Asking Price positive or negative? How do you know? c. What is the coefficient of correlation between Time on Market and Asking Price? d. What percent of the variability in Time on Market is not accounted for by Asking Price?

5.

For the regression model predicting Time on Market using Asking Price in the previous exercise, se = 7.459. a. What does se tell us? b. Using the model to predict number of months on the market for a home priced at $550,000, the residual is 13.82. How many standard deviations away is the prediction from the actual value? c. The residual plot is shown below. Based on this plot, is the Equal Spread Condition satisfied? Explain.

Versus Fits

(response is Time on Market) 15

Residual

10 5

0 -5

-10 5

6.

10

15 Fitted Value

20

25

Below are residual plots (residuals plotted against predicted values) for three linear regression models. Indicate which condition appears to be violated (Linearity, Outlier or Equal Spread) in each case.

a. .

7-8

Part I Exploring and Collecting Data Versus Fits

(response is y) 15

Residual

10

5

0

-5

-10 -10

0

10

20

30 Fitted Value

40

50

60

70

27.5

30.0

32.5

b.

Versus Fits

(response is y) 50 40

Residual

30 20 10 0 -10 15.0

17.5

20.0

22.5 25.0 Fitted Value

c.

Versus Fits

(response is y) 200

Residual

100

0

-100

-200 100

200

300

400 Fitted Value

.

500

600

700

Chapter 7 Introduction to Linear Regression ANSWERS 1. a. b. c. d.

Years of Experience Salary For each additional year of experience, salary is predicted to increase by $1590.

It is a predicted salary of employees with no experience (0 years of experience). It is meaningful as a starting salary.

2. a. b. c. d.

$52,675 $3825 $68,575 -4025 or $4025 less than the model predicts

3. a. b. c. d.

-1.01 25.79 5.59 ($K) or $5590 -0.41 ($K) or $410 less than the model predicts

4. a. 50.5% of the variation in time on market can be explained by variation in the asking price. b. Positive, because the slope is positive. c. .71 d. 49.5% 5. a. It is the standard deviation of the residuals. So it tells us how scattered the points are around the fitted line. b. 1.85 c. No. The spread increases (cone or fan shape evident).

6. a. Linearity b. Outlier c. Equal Spread

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7-9

8-1

Chapter 8 Randomness and Probability What’s It About?

We begin our discussion of probability with randomness and the Law of Large Numbers. We then look at basic probability rules for complementary events, unions of disjoint events, and intersections of independent events. Also, we introduce conditional probability; discuss probability tree and Bayes’s Rule.

Comments We’ve introduced the concept of relative frequency in Chapter 4. Chapter 8 formalizes that concept with the basic definitions and rules of probability. Some students will be pleased to have these ideas properly defined. Others may find the mathematics challenging. That might alert them that the interplay between math and data analysis lies at the heart of business decision making.

Probability is a deceptively difficult topic. The arithmetic, especially with a calculator in hand, is pretty easy. The questions sound straightforward. The words – and, or, not, of, if, independent, mutually exclusive, “given that” – can be unexpectedly confusing.

The essential concepts here are independence and the way that conditional probabilities work. We will require independence for all of the inference methods we discuss later in the course.

Misunderstandings of conditional probabilities are known to drive bad business decisions. Students need a sound understanding of the calculations because they may not be able to rely on their intuitions.

Looking Ahead Both independence and conditioning will continue to be important. We discuss random variables more generally in Chapter 9.

Class Do’s As always, emphasize the importance of vocabulary and notation for clear understanding and effective communication.

Note that probability is first defined as long-run relative frequency of occurrence, aka empirical probability. Resist the temptation to try to create some more formal-sounding “mathematical” definition. We believe in the Law of Large Numbers. That is, we believe that over the long run (the very long run), events will occur with a certain relative frequency, and we call that the probability of the event. This is consistent with common understanding of what we mean when we say a coin has a 50% chance of landing heads, aka theoretical probability. We are not saying that it must land heads one of every two times you toss it, or exactly 50 times in 100. We are saying that over many, many tosses we will see about 50% heads. Sometimes we need a third definition of probability to answer questions like “What is the probability that the price of gold will exceed $1500 an ounce next year?” aka subjective probability.

For calculating probabilities, just build three simple ideas, based on three words:

• “not” means subtract from 1. If 30% of people have blue eyes, then the other 70% of people do not.

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8-2

Part II Understanding Probability Distributions and Statistical Inference

• “or” means add. The probability that your eyes are green or brown is the sum of the probability of having green eyes and the probability of having brown eyes. That’s true, of course, only if you can’t have both green and brown eyes. If events are not disjoint, we must adjust the rule.

• “and” means multiply. The probability that two students sitting near each other in class both have blue eyes is the product of the probabilities that each has blue eyes. That’s true, of course, only if their eye colors are independent. Adjusting for events that are not independent leads us to conditional probabilities.

A two-way table is a good way to develop the formula for conditional probability when the joint probabilities are known. Students saw conditional probabilities in Chapter 4. The probability of an outcome on one variable given a certain result for the other variable is easily seen to be the number of ways that both happen out of the total number of ways the given event happened.

Define independence in terms of conditional probability. Conceptually we know that independent events don’t affect each other. More specifically, events are independent if the outcome on one of them does not change the probability of the other. That leads us directly to say that when given knowledge that A has occurred, the probability of B remains the same. Hence, in formula form, two events are independent if P(B | A) = P(B). Make students interpret that requirement in context every time they face a question about independence.

Continue to discuss the difference between independent events and disjoint events. This will be confusing for some students. Have ready examples. Ask them if events can be both disjoint and independent. We can hope that they will explain (with some coaxing, perhaps) that if the events are disjoint then they cannot both happen at the same time. That means that as soon as one of them happens the other cannot – the probability of the other changes to 0. When the occurrence of one event changes the probability of the other, the events are not independent. Ever. Perhaps what is most confusing is the fact that while possible events that are disjoint cannot be independent, events that are not disjoint may or may not be independent. Football games are sometimes played in the rain, so “football” and “rain” are not mutually exclusive. But they’re not independent either because games are less likely to be played when it rains.

Finally, emphasize again the value of the guided examples. Students can see how the rules are implemented and how to check for independence. They will find these examples a great help when they work on the exercises.

The Importance of What You Don’t Say Don’t mention combinatorics, permutations, and so on. Many of us studied probability, including counting theory, formal definitions, theorems, and proofs. Good stuff, different course. Students need to understand enough probability to be able to work with expected values and sampling models. Don’t try to turn them into math majors – this is a Statistics course.

Don’t speak kindly of the “Law of Averages”. Actually, do talk about the Law of Averages, as an example of What Can Go Wrong? Here’s another place where common usage misapplies a fundamental principle – this time, the Law of Large Numbers.

Simply put: “Law of Averages” is false, “Law of Large Numbers” is true. The so-called Law of Averages assumes that the more something has not happened, the more likely it becomes. We hear such reasoning frequently. Many people believe that if a coin has landed heads 5 times in a row they should now bet on tails. Such reasoning overlooks the fact that unusual outcomes are just that .

Chapter 8 Randomness and Probability

8-3

– unusual, not impossible. Over a long season a baseball player will go 12 at-bats without a hit, just as he’ll have a game when he gets 4 hits in a row. And if you toss a coin enough times, you will see some long runs of heads or of tails.

The “Law of Averages” assumes that some kind of compensatory mechanism exists – that the coin knows it is “over” in heads and hence will tend toward tails to balance things out. That’s false.

(Ask students how they know that the coin didn’t come up heads 10 times in a row before it arrived in their hands, and thus that the first time they toss it they should bet on tails.)

The Law of Large Numbers says that short-run anomalies get drowned out in the long run. Ten heads in a row seems weird, but after another 500 tosses (expecting 50% heads) the resulting 260 to 250 imbalance is not at all unusual. The LLN says long-term behaviour settles down ever closer to what we would expect (on a percentage basis), but that this happens because of the large number of trials and not because there is any intent on the part of the event (or the coin) itself.

There are at least three reasons why people believe in the “Law of Averages.”

1. They misunderstand the Law of Large Numbers, thinking it suggests that randomness compensates for anomalies instead of just ignoring and overwhelming them.

2. They confuse situations involving independent events with what they know about events that aren’t independent, like drawing a card from a deck without replacement. The longer we go without seeing an ace, the more likely it is that the next card will be one. This is not the same situation as the coin or the baseball player.

3. They sense that things actually do happen that way. They see a coin land heads five times in a row and think it will probably be tails next. When it’s heads again, they become even more sure it will be tails the next time. And so on. When the coin finally does land tails, their reaction is “I KNEW IT!,” thus confirming the “Law of Averages.”

Don’t endorse the “Hot Hand” either. The flip side of the Law of Averages is a belief in the socalled “hot hand”. Caution students about this line of reasoning, too. We often hear about a basketball player being “in the zone”, implying that he’s currently more likely to make shots so his teammates should pass him the ball while he’s hot. Fans and players alike believe this is true, yet when Cornell professor Tom Gilovich and others looked at player data supplied by the Philadelphia ‘76ers, they found that the frequencies with which a player hit a shot were the same whether he had missed or made the previous shot. In fact, even if the player had made several shots in a row, he made or missed the next shot at about the same rate as his season-long shooting percentage. There’s simply no evidence to support the belief in a “hot hand”. Students will argue this one, and passionately! Ask them if it’s good strategy at the end of a basketball game to foul the player who has missed all six previous foul shots. Does he have the cold hand, or is he due to make one? We can’t have it both ways – although people seem to try when they’re at casinos (or making investments!). At the roulette wheel gamblers tend to bet on the Law of Averages, increasing wagers on red when the ball has landed on black several spins in a row. Meanwhile at the craps table they’ll bet with the player who has won several times in a row – the hot hand!

Class Examples 1. Suppose that 40% of cars in your area are manufactured in the United States, 30% in Japan, 10% in Germany, and 20% in Canada. If cars are selected at random, find the probability that: .

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Part II Understanding Probability Distributions and Statistical Inference

• A car is not U.S.-made.

Answer: P(not U.S.-made) = 1 − P(U.S.-made) = 1− 0.40 = 0.60

• It is made in Japan or Germany.

Answer: P(Japanese or German) = P(Japanese) + P(German) = 0.30 + 0.10 = 0.40

• You see two in a row from Japan.

Answer: P(two Japanese in a row) = P(J and J) = P(J)P(J) = (0.30)(0.30) = 0.09

• None of three cars came from Germany.

Answer : P(Not German in three) = P(Not German and Not German and Not German) = P(Not German) P(Not German) P(Not German) = 0.30  0.30  0.30 = 0.729

• At least one of three cars is U.S.-made.

Answer: P(at least one U.S. in three) = 1 − P(No U.S. in three)

= 1 − (0.60)(0.60)(0.60) = 0.784

• The first Japanese car is the fourth one you choose.

Answer: P(First J is the fourth car) = P(not J and not J and not J and J) =  P(Not J)3 P(J) = (0.70)3 (0.30) = 0.1029

2. Estimate the probability of something by having the class do it many times. Plot the cumulative results to demonstrate the Law of Large Numbers. Avoid tossing a coin because students have a preconceived idea of what should happen. Spinning a coin is better. Still better is tossing a thumbtack to see if it lands point up or down.

3. Use your data from tossing coins or thumbtacks – or do a simulation on the computer* – to create a list of 100 or more outcomes. Examine the list for runs to dispel faith in the “Law of Averages”. Notice that a “fair” coin or tack will exhibit several long runs of heads or point-ups.

Identify every place where there were, say, 4 heads in a row to see what happened the fifth time.

Lots of good “Law of Averages” examples can be found in Activity-Based Statistics† and in the media. Students will contribute others. Here’s a beauty from the Salt Lake City Tribune of Friday, October 11, 2002.

Bureaucrat’s Math Makes Dizzy Dozen The menu at the Coffee Garden at 900 East and 900 South in Salt Lake City has included a scrumptious selection of quiche for about 10 years. The recipe calls for four fresh eggs for each quiche. A Salt Lake County Health Department inspector paid a visit recently and pointed out that research by the Food and Drug Administration indicates that one in four eggs * The site www.random.org will provide any number of truly random numbers. And students know

that anything—including randomness—can be found on the internet. † Activity-Based Statistics, 2nd Ed., Schaeffer, Gnanadesekan, Watkins, and Witmer; Key College Press, 2003. .

Chapter 8 Randomness and Probability

8-5

carries Salmonella bacterium, so restaurants should never use more than three eggs when preparing quiche. The manager on duty wondered if simply throwing out three eggs from each dozen and using the remaining nine in four-egg-quiches would serve the same purpose.

Resources ActivStats‡

• Lesson 14 – Simulations let students experiment with random events and discover the LLN for themselves.

Basic Exercises 1.

Consider the random phenomenon of selecting a playing card from a deck of 52. a. b. c. d.

What is the probability of selecting the King of Hearts?

What is the probability of selecting the King of Hearts or the King of Clubs?

What is the probability of selecting a King or Queen?

What method did you use in assigning these probabilities?

2.

The following contingency table shows opinion about global warming (non issue vs. serious concern) broken down by political party affiliation (Liberal, Conservative, NDP).

Political Party Liberal Conservative NDP Total a. b. c. d.

Opinion on Global Warming Non Issue Serious Concern 60 440 290 210 90 110 440 760

Total 500 500 200 1200

What is the probability that an individual selected at random believes that global warming is a serious issue?

What type of probability did you find in part a.?

What is the probability that an individual selected at random is a Conservative and believes that global warming is a serious issue?

What type of probability did you find in part c.?

3.

The following contingency table shows opinion about global warming (non issue vs. serious concern) broken down by political party affiliation (Liberal, Conservative, NDP ).

Opinion on Global Warming Political Party Non Issue Serious Concern Total Liberal 60 440 500 Conservative 290 210 500 NDP 90 110 200 Total 440 760 1200 a.

What is the probability that an individual selected at random believes that global ‡ ActivStats is available at www.pearsonmylabandmastering.com/northamerica/mystatlab/

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Part II Understanding Probability Distributions and Statistical Inference b. c. d.

warming is a non issue given that the individual is a Liberal?

What is the probability that an individual selected at random is a Liberal given that the individual believes that global warming is a serious concern?

What type of probability did you find in parts a. and b.?

Are political party affiliation and opinion on global warming independent? Explain your answer.

4.

A national survey indicated that 30% of adults conduct their banking online. It also found that 40% are under the age of 50, and that 25% are under the age of 50 and conduct their banking online. a. b. c. d.

What percentage of adults do not conduct their banking online?

What type of probability is 25%?

Construct a contingency table showing all joint and marginal probabilities.

What is the probability that an individual conducts banking online given that the individual is under the age of 50?

ANSWERS 1.

a. b. c. d.

1/52 2/52 8/52 Model-based (theoretical)

2.

a. b. c. d.

760/1200 = .63 Marginal 210/1200 = .175 Joint

3.

a. b. c. d.

60/500 = .12 440/760 = .58 Conditional No. The marginal probability of “Non Issue” = .37 and is not equal to the conditional probability found in part a. (.12).

4.

a. .70 b.

Joint c.

Age Yes Under 50 .25 50 or Older .05 Total .30 d. .25/.40 = .625

.

Online Banking No .15 .55 .70

Total .40 .60 1.00

9-1

Chapter 9 Random Variables and Probability Distributions What’s It About?

Chapter 9 introduces random variables and describes them with probability models. We look at expected values and standard deviations, and examine the effects of shifting and scaling on mean and variance. We introduce the all-important concept that when adding or subtracting independent random variables, we add their variances. Then we use these concepts to examine three discrete probability models: the Geometric, the Binomial, and the Poisson. We then consider three continuous probability models, the Uniform, the Normal, and the Exponential, and find that the Normal can be used to approximate the Binomial. We have also included a section on normal probability plots to help in deciding whether the normal model is appropriate.

Comments We’ve already dealt with many of these concepts earlier in the book, but there we were describing data—the distribution, center, and spread. Here we turn our attention to probability models. The ideas are the same, but rather than examining actual data we examine what might happen in theory.

Instead of a distribution of observed results, we have the probabilities of the possible outcomes.

Instead of the mean of observed values, we have the mean of the probability model – the expected value. The standard deviation of the data measured the spread of the data. Now it measures the spread of the probability distribution. And the effects of shifting and scaling are the same: both affect the center, but only scaling affects the spread.

The notion that we add the variances of sums or differences of independent random variables may seem complicated and counterintuitive at first (especially differences), but there’s nothing in this chapter that’s more fundamental. Be sure students leave this chapter with that idea well established. Calling this the Pythagorean Theorem of Statistics helps them understand and remember it.

Now, with the concept of random variables in hand, we can examine more special probability models. Students can find it difficult to distinguish between situations that are Geometric and those that are Binomial. Some of the confusion stems from the fact that all involve Bernoulli trials, so going through the checklist of those criteria fails to make the distinction. The difference lies in the definition of the random variable – are we counting the number of trials before the first success or the number of successes in a fixed number of trials, or are we working with a small probability of success and a large number of trials?

Teachers differ in which of the probability models they choose to discuss and in when they introduce these concepts. We’ve offered a wide variety of models. We’ve place the chapter relatively late in the book to reflect the fact that it is not really needed for any of the other work.

This chapter is background information for those students and teachers who find that helpful.

Class Do’s Emphasize the importance of vocabulary and notation. Random variables have a probability model; actual sample data have a distribution. The model’s mean is the expected value of the variable – the long-run average – and can be written µ or E(X), but not x. The model’s variance is 2 Var(X) or 2, not s . .

9-2 Part II Understanding Probability Distributions and Statistical Inference Help students understand that the expected value may very well be impossible on any given trial.

The chapter’s opening example about death and disability insurance, for instance, results in an expected value of $200. The policy could pay off $100,000, or $50,000, or nothing at all. No individual policy will pay off $200. That’s just the expected average payout per policy if there are a very large number of policies. This is a good time to revisit the Law of Large Numbers.

The LLN also explains how folks who run gambling games and lotteries can make such juicy profits despite offering a multi-million dollar prize. They would be foolish to offer such a payoff if there were only a few players. Even though the probability is very low, someone might win – a large downside risk. But the huge prize generates headlines (especially when someone does win) and entices a large number of people to play. Sure, one or two may win megabucks, but the LLN ensures that so many others will lose that running the game will be a very lucrative venture.

You can make sense of the formula for the variance by looking at the formula for mean. In a sample, to “average” something we add individual values and divide by a count. For random variables the “add-and-divide” routine is simply replaced by “multiply by the probabilities and add.” We find the mean by multiplying each possible value by the corresponding probability and then summing. We do the same to find the variance, this time multiplying squared deviations from the mean by the probabilities and summing. (Ultimately, a calculator or computer will do the heavy lifting.)

Students find it difficult to tell when they’re multiplying by a constant (2X) and when they are summing separate random variables ( X1 + X 2 ) . These aren’t the same. Doubling the roll of a single die produces more variability in the result than summing the two dice. Help them think about this issue as they attack each problem.

Spend time with the “variances add” idea. There are three important issues:

• First, notice that it’s the variance that actually deals with deviations from the mean, though they are squared. We can add on the variability from another random variable only at this stage – not after we have taken the square root. If we want the total variability, we need to add variances. Only then can we take the square root to get our units in line with the original units (“square dollars” aren’t helpful). Students know from the Pythagorean Theorem that a right triangle with legs of 3 and 4 does not have a hypotenuse of 7. Just as Geometry’s Pythagorean Theorem is about areas, the Pythagorean Theorem of Statistics is about variances – squared deviations from the mean.

• Most confusing to students is the fact that we still add the variances even if we are interested in the difference of the random variables. Think of variance as a measure of uncertainty. In any situation, the more variability there is in the possible outcomes, the less certain we can be of the results. Every time another random outcome is introduced, another layer of uncertainty is added to the final outcome.

• All of this discussion is true only if the variables are independent. If one outcome affects what will happen next, then the uncertainty (read “variance”) is diminished, so we should not add the variances. It is easy to find business examples in which the assumption of independence turned out to be false—and mattered. Consider, for example, home mortgage defaults, which were independent only so long as the housing market was stable and growing. .

Chapter 9 Random Variables and Probability Distributions 9-3 When you get to distributions, talk about Bernoulli trials first. Get students to recognize what characterizes Bernoulli trials: two outcomes, fixed probability of success, independence from trial to trial. They should see that such situations are common. Coins, dice, foul shots in basketball, guessing at true-false or multiple choice questions, and even sampling voter opinions seem to have the right characteristics – or are at least close enough to make the models useful.

Once students know what Bernoulli trials are, then we can define a random variable. It is the definition of that variable that determines whether a Geometric or Binomial model will prove useful. The Geometric model applies to waiting time situations—that is, counting the number of trials to achieve our first success. If instead we want to count the number of successes we achieve in a given number of trials, then we want a Binomial model. Students will make that distinction more readily when they see that it depends on what we choose to count as we conduct our Bernoulli trials.

Your decision on which (if any) of the Math Box derivations you choose to show students will depend on the interest and background of your class.

• The expected value of the Geometric model requires that students understand infinite series.

Instead you can simply make the plausible argument that when rolling a die we’d expect to see our first 4 after an average of 6 rolls, since the probability is 1/6. (The formula for the standard deviation of a Geometric model is rarely used.)

• It’s similarly obvious that we’d expect to get about 100 4’s in 600 rolls of a die, so the mean of the Binomial model is np.

• The standard deviation of the Binomial model is not obvious. The derivation is convincing, not very difficult, and uses the revelation that we add the variances of independent random variables. This is a great opportunity to apply that important concept.

• Insist that students check assumptions and conditions before using probability models.

Sometimes there is no condition to test. For example, the question of whether the trials are independent when we toss a coin repeatedly is left to us to think about. We can’t think of any reason why not, so we simply assume that they are.

• We can confirm the Large Sample Assumption with the Success/Failure Condition (np and nq at least 10).

• The Independence Assumption is violated whenever we sample without replacement, but is overridden by the 10% Condition (as long as we don’t drain off more than 10% of the population, the probabilities don’t change enough to matter).

Make sure students understand that they are required to check, not just state the conditions. In other words, it is not sufficient to simply write down np ≥ 10 and put a little check mark there.

Students must show that they checked this condition using the values given in the question, for example, np = 220(0.75) = 165 ≥ 10.

Use software with various sets of data to show normal probability plots to students. From this, you can ask them to tell you if approximations using the normal model would be appropriate.

The Importance of What You Don’t Say Don’t get lost in combinatorics. There’s no need to delve deeply into factorials, permutations, and combinations. If students understand the issue, that they need to know how many different ways they can get 5 successes in 7 trials, know how to write out the formula, and how to evaluate it on a .

9-4 Part II Understanding Probability Distributions and Statistical Inference calculator, that’s enough. This is not a formal probability course. What’s important is recognizing when and how to apply the Binomial model.

Don’t worry about the continuity correction. While it is technically true that some errors in the Normal approximation to the Binomial model come from the fact that the Normal is continuous and the Binomial discrete, these are generally negligible.

Class Examples 1. Play Greedy Pig. All the students stand up. You start rolling a die. Each student gets the number of points on the die. Students then may keep their points by sitting down, or remain standing for another roll. With each roll, points accumulate for those who choose to remain standing. BUT . . . if you roll a 5, all those still standing lose their points, ending the round with a score of 0. Play a couple of practice rounds to be sure they understand the game, and then play 5 rounds to see who can amass the highest total score. (Play with your door closed – this gets loud . . . .) Afterward, ask what strategies they used. Some will have a points-based rule: Sit down when you have 12 points or more. Others will count rolls: Sit down after 4 rolls.

Some will have a combination strategy: Stand for 6 rolls, but sit early if you get a lot of points.

Tell them there is an optimal strategy, and let this sit for a couple of days. Some will play (or simulate) the game, and all will be interested to get back to it later.

After students understand expected value, they can determine the strategy by noting that if they have P points now and decide to remain standing, there is 1 chance in 6 that the next roll will give them P + 1, P + 2, P + 3, P + 4, 0, or P + 6 points. Their expected score if they remain standing is therefore:

(P +1) + (P + 2) + (P + 3) + (P + 4) + (0) + (P + 6) = 5P 6+16 1 6

1 6

1 6

1 6

1 6

1 6

That’s greater than their current score P if P < 16, so in the long run the optimal strategy is to sit down whenever your point total hits 16 (or more).

2. To introduce the concept of random variable and expected value, propose a game. A player pays you $5 and draws a card from a deck. If he draws the ace of hearts, you pay him $100.

For any other ace, you pay $10, and for any other heart, you pay $5. If he draws anything else, he loses. See if anyone would be willing to play. Would they play for a top prize of $200?

Point out that such “games” are common. Obviously those include casino games and lotteries, but sports betting, the stock market, and the entire insurance industry also work this way.

List the values of the random variable, payoff: 95, 5, 0, and –5. List the corresponding probabilities to create the probability model. Appeal to the idea that this is a model (i.e., theoretical perfection) to find the expected value (or average) outcome. In a perfect world we’d expect 52 outcomes of this game to yield one $100 winner, three $5 winners, twelve $0 winners, and 36 people who lose their $5. Write the average as:

E ( X )=

95(1) + 5(3) + 0(12) + (−5)(36) = −$1.35 52

Discuss the meaning of the result. Obviously no one ever loses a $1.35 playing this game. Get the students to explain what it really means, and be sure to bring in the Law of Large Numbers. .

Chapter 9 Random Variables and Probability Distributions 9-5 3. Define Bernoulli trials and look at many examples and counterexamples.

4. A new sales gimmick has 30% of the M&M’s covered with speckles. These “groovy” candies are mixed randomly with the normal candies as they are put into the bags for distribution and sale. You buy a bag and remove candies one at a time looking for the speckles.

• Note that this situation involves Bernoulli trials. There are two outcomes: success = speckles, failure = ordinary. The probability of success, based on the information from the candy company, is 30%. Trials can be assumed independent – there is no reason to believe that finding a speckled candy reveals anything about whether the next one out of the bag will have speckles.

• What’s the probability that the first speckled one we see is the fourth candy we get? Note that the skills to answer this question come from the very first day of the probability unit. The only new issue here is formally naming the model that describes waiting time probabilities: Geometric. Remind students that they already know one useful probability model – the Normal. Now they have a second. The Normal model had two parameters: mean and standard deviation. The Geometric model has but one: probability of success.

Answer: P(First speckled is fourth candy) = (0.7)3 (0.3) = 0.1029

• Find other Geometric probabilities. What’s the probability that the first speckled one is the tenth one? Write a general formula. What’s the probability the first speckled one is one of the first three we look at? How many do we expect to have to check, on average, to find a speckled one?

Answers: P(First speckled is tenth candy) = (0.7)9 (0.3)  0.0121 General Formula: P( X = x) = qx −1 p

P(First speckled is among first three) = (0.3) + (0.7)(0.3) + (0.7)2(0.3)  0.657 How many do we expect to check, on average to find a speckled one?

1 1 E( X ) = = = 3 13 candies p 0.3 5. Return again to the speckled M&M’s, but change the question.

• What’s the probability that we’ll find two speckled ones in a handful of five 2

3

candies? Students will quickly come up with (0.3) (0.7) . You can use a tree diagram (you won’t have to draw the whole thing!) to convince them that that’s the correct probability for every branch that ends up with 2 of 5 speckled M&M’s. They need to count the number of orders in which it can happen. Let them make a list to see that there are 10. Explain that mathematicians call such counts “combinations” and show the notation. (If your students have TI calculators, you can also show them the nCr operation on the TI’s MATH PRB menu.)

.

9-6 Part II Understanding Probability Distributions and Statistical Inference

  Answer: P(Two speckled candies in first five) =  5  (0.3)2 (0.7)3 = 0.3087  2 

• Calculate a variety of other probabilities for practice in setting them up.

Train students to show the set-up in their answers.

• Discuss this new model, the Binomial; it has two parameters, the number of trials and the probability of success. Students need to see that the underlying situation is the same – Bernoulli trials – whether the question asks about waiting time (Geometric) or number of successes (Binomial).

• Note that we are actually drawing M&M’s without replacement from a finite population, so the Independence Condition is violated; the probability that the next one is speckled depends on how many speckled ones we have removed from the pool so far. Obviously our sample (the small bag we bought) is such a tiny subset of the vast number of M&Ms in the world that it doesn’t really matter much, and that’s the essence of the 10% Condition.

Now rewrite the calculation by distributing the common denominator: 36 E( X ) = 95( 521 ) + 5( 523 ) + 0( 12 52 ) + (−5)( 52 ) = −$1.35

Written in this form, it is easy to see how the expected value is computed directly from the probability model. Have students check to see if they would play for a top prize of $200.

The variance is the expected value of squared deviations from the mean, so:

Var ( X ) = ( 95 − (−1.35) ) ( 521 ) + ( 5 − (−1.35) ) ( 523 ) 2

2

36 + ( 0 − (−1.35) ) ( 12 52 ) + ( ( −5) − ( −1.35) ) ( 52 ) 2

2

 = SD( X ) = $13.80 Be sure students can explain why the standard deviation is so much larger than the mean in this situation.

6. Talk about combining independent random variables.

Consider a dice game: no points for rolling a 1, 2, or 3; 5 points for a 4 or 5; 50 points for a 6. Find the expected value and the standard deviation.

Solution: Plan I want to find the mean and expected value of a dice game where 1, 2, and 3 are worth 0 points, 4 or 5 are worth 5 points, and 6 is worth 50 points. Let Y = the number of points.

.

Chapter 9 Random Variables and Probability Distributions 9-7 Do

Report I expect to win 10 points per roll in the long run. The standard deviation is 18.03 points, suggesting a highly variable game.

• Imagine doubling the points awarded. What are the new mean and standard deviation?

Plan I want to find the mean and expected value of a dice game where 1, 2, and 3 are worth 0 points, 4 or 5 are worth 10 points, and 6 is worth 100 points. This is double the payout of the original dice game.

Do

E(2Y ) = 2E(Y ) = 20

Var(2Y ) = 22Var(Y ) = 4(325) = 1300 , so SD(2Y ) = 36.06

Report To calculate the mean and standard deviation of a game in which the rewards are doubled, just double the mean and standard deviation of the original game. I expect to win 20 points per roll in the long run. The standard deviation is 36.06 points, suggesting a highly variable game.

• Now imagine just playing the game twice. What are the mean and the standard deviation of your total points?

Plan I want to find the mean and expected value of the total of two plays of a dice game where 1, 2, and 3 are worth 0 points, 4 or 5 are worth 10 points, and 6 is worth 100 points.

Do and Report To calculate the mean of the sum of two random variables, add the means.

E(Y1 + Y2) = E(Y1) + E(Y2) = 10 + 10 = 20 I expect to win 20 points total for every two rolls in the long run.

To find the standard deviation of the sum of independent random variables, use the Pythagorean Theorem of Statistics: add the variances, and then take the square root.

 = SD(Y1 + Y2 ) = Var(Y1 + Y2 ) = Var(Y1 ) +Var(Y2 ) = 325+ 325 = 650  25.50 .

9-8 Part II Understanding Probability Distributions and Statistical Inference Notice that this is not the same situation as playing once for twice as much. Think about tossing a coin once—heads you win $50. That’s not the same as tossing a coin five times at $10 a head. Each outcome is a potentially different value of the random variable, and they vary independently. For independent random variables, the variances add.

• Suppose you and a friend both play the dice game. What are the mean and standard deviation of the difference in your winnings?

Plan I want to find the mean and expected value of the difference of two plays of a dice game where 1, 2, and 3 are worth 0 points, 4 or 5 are worth 5 points, and 6 is worth 50 points.

Do and Report To calculate the mean of the difference of two random variables, subtract the means.

E(Y1 – Y2) = E(Y1) – E(Y2) = 10 – 10 = 0 The difference in points won is expected to be 0 points in the long run.

To find the standard deviation of the difference of independent random variables, use the Pythagorean Theorem of Statistics: add the variances, and then take the square root.

 = SD(Y1 − Y2 ) = Var (Y1 − Y2 ) = Var (Y1 + Y2 ) = Var (Y1 ) + Var (Y2 ) = 325 + 325 = 650  25.50 Point out that there is as much potential variability (looking at the range of values) in the difference (–50 to 50) as in the sum (0 to 100). Variances add, even when you are subtracting random values.

7. Some justifications of the “variances add” idea:

• You mix a quart of lemonade. There should be 32 fluid ounces, but your measuring process isn’t perfectly accurate so, while 32 ounces is the expected amount, there remains some variability. You pour a 12-ounce glass. Of course, that measurement is not perfect either, so it’s actually somewhere around 12, give or take a little. How much is left in the pitcher?

Should be around 20 ounces, of course, but given the uncertainty about the initial amount and the variability in how far you filled the glass, you can’t say exactly how much is left.

Indeed, you are less sure about the amount left because you removed an unknown amount.

Subtracting some lemonade has increased the variability in the amount remaining.

Variances add.

• We have two piles of fruit: grapefruits weighing from 16 to 24 ounces and oranges weighing from 9 to 13 ounces. We pick one of each at random. The sum of the weights could be as high as 24 + 13 = 37 ounces or as low as 9 + 16 = 25 ounces, a range of 12 ounces. The difference in the weights has a max of 24 – 9 = 15 ounces and a min of 16 – 13 = 3 ounces, again a range of 12 ounces. Okay, this is range, not variances, but both measure spread – and it’s the same for sums as for differences (in both cases the sum of the ranges of the two variables, 8 + 4).

.

Chapter 9 Random Variables and Probability Distributions 9-9

• Imagine a survey of the class that asked how many hours a student slept between noon yesterday and noon today. The second question is how many hours the student was awake during that period of time. We could find the mean and standard deviation for each set of answers. What are the mean and standard deviation of the total? Certainly the mean = 24 hours and standard deviation = 0. We don’t add the variances this time because the answers are not independent.

8. Suppose a used car dealer runs autos through a two-stage process to get them ready to sell. The mechanical checkup costs $50 per hour and takes an average of 90 minutes, with a standard deviation of 15 minutes. The appearance prep (wash, polish, etc.) costs $6 per hour and takes an average of 60 minutes, with a standard deviation of 5 minutes.

• What are the mean and standard deviation of the total time spent preparing a car? (Note that we cannot find the standard deviation if we do not believe that the two phases of the process are independent, an important assumption to check.)

Solution: Plan – We want to know the mean and standard deviation of the total time spent preparing a car. We believe that the two phases of the process are independent, so variances add.

Let M = mechanical checkup time, A = appearance prep time, T = total time Do –

T=M+A

E(T ) = E( M ) + E( A) = 90 + 60 = 150 minutes

Var(T ) = Var( M ) +Var( A) = 152 + 52 = 250 SD(T ) = 250 = 15.8 minutes Report – We expect the total preparation time to take an average of 150 minutes (2.5 hours), with a standard deviation of 15.8 minutes.

• What are the mean and standard deviation of the total expense to prepare a car?

Solution: Plan – We want to know the mean and standard deviation of the total expense of preparing a car. We can find mean and standard deviation of expenses for each part of the preparation by multiplying the mean and standard deviation of the times (in hours) by the rate of $50/hr. We believe that the two phases of the process are independent, so variances add.

Let

M = mechanical checkup hours A = appearance prep hours TE = total expenses

E ( M ) = 1.5 E ( A) = 1.0

SD( M ) = 0.25 SD( A) = 5 / 60 = 1/12 = 0.0833

Do – TE = 50 M + 6 A E (TE ) = E (50M + 6 A) = E (50M ) + E (6 A) = 50 E (M ) + 6E ( A) = 50(1.5) + 6(1.0) = $81

Var (TE ) = Var (50M + 6 A) = Var (50M ) + Var (6 A) = 502Var ( M ) + 62Var ( A) = 2500(.252 ) + 36(1/12) 2 = 156.50 .

9-10 Part II Understanding Probability Distributions and Statistical Inference

SD(T ) = 156.50 = $12.51 Report – We expect the total preparation expense to average $81, with a standard deviation of $12.51.

• What are the mean and standard deviation of the difference in costs for the two phases of the operation?

Solution: Plan – We want to know the mean and standard deviation of the difference in expenses of preparing a car. We can find mean and standard deviation of expenses for each part of the preparation by multiplying the mean and standard deviation of the times (in hours) by the rate of $50/hr. We believe that the two phases of the process are independent, so variances add.

Do – D = 50 M − 6 A E(D) = E(50 M − 6 A) = E(50 M ) − E(6 A) = 50E( M ) − 6E( A) = 50(1.5) − 6(1.0) = $69 Var ( D) = Var (50M − 6 A) = Var (50M + 6 A) = Var (50M ) + Var (6 A)

= 502 Var ( M ) + 62Var ( A) = 2500(.252 ) + 36(1/12) 2 = 156.50 SD(D) = 156.50 = $12.51

Report – We expect the difference in preparation expenses to average $69, with a standard deviation of $12.51.

• What is the probability that it will take longer to do the appearance prep than the mechanical checkup? (Note that we cannot answer this question unless we believe each phase of the process can be described by a Normal model.)

Solution: Plan – To calculate the probability that it will take longer to do the appearance prep than the mechanical checkup, we need to know the mean and standard deviation of the difference in time spent on each. If the difference (mechanical checkup time – appearance prep time) is less than 0, then the appearance prep takes longer. The two phases of the process are independent, so variances add. We will assume that each of the processes varies according to a Normal model.

Let D = M − A .

Do –

E(D) = E( M − A) = E( M ) − E( A) = 90 − 60 = 30 minutes SD( D) = SD( M − A) = Var ( M − A) = Var ( M + A ) = Var ( M ) + Var ( A) = 152 + 52 = 250  15.8 minutes

The difference in times varies according to the Normal model with  = 30 and  = 15.8 .

The next step is to find the z-score associated with y = 0.

.

Chapter 9 Random Variables and Probability Distributions 9-11

z=

y−



0 − 30 15.8 z  −1.899

z=

Using software, we find Area ( y  0 ) = Area ( z  −1.899 ) = 0.028782  0.0288 , which means that 2.88% of the scores fall below –1.899. So, the Normal model estimates that about 2.88% of differences fall below zero.

Report – We expect that approximately 2.88% of the appearance prep works will take longer than the mechanical checkups.

9. Have students design their own games of chance. They need to describe the rules, the payoffs, and the cost of playing. Games could be based on cards, dice, coins, spinners, random numbers, and so on. The objective is to create a game that looks appealing, so many people would be eager to play, but with an expected value and standard deviation that would make it likely that the person running the game would realize a profit.

Resources ActivStats*

• Chapter 16—Simulations in this chapter generalize the Randomness Tool to continuous random variables. Students will use this version of the tool later to understand the Central Limit Theorem (CLT) and inference, so it is worthwhile to introduce it here.

Activity-Based Statistics†

• The Lazy Student Assignments Split the reading into two sections, assigning about five problems a day as you cover the chapter in three days. Continue to assign a couple of these problems for a few extra days along with work in the next chapter to be sure students understand the concepts.

Basic Exercises 1.

Consider the following probability distribution for the discrete random variable x.

* ActivStats is available at www.pearsonmylabandmastering.com/northamerica/mystatlab/ † Activity-Based Statistics, 2nd Ed., Schaeffer, Gnanadesekan, Watkins, and Witmer; Key College Press, 2003.

.

9-12 Part II Understanding Probability Distributions and Statistical Inference X 0 1 2 3 4 a. b. c. d.

2.

What is P (X = 1)?

What is P (4 ≤ X ≤ 8)?

What is P (X ≥ 4)?

Find E(X) and SD(X).

Suppose that X is normally distributed with µ = 50 and σ = 2. a. b. c. d.

7.

What is P (X = 0)?

What is P (X = 2)?

What is P (X ≤ 2)?

Find E(X) and SD(X).

Suppose that X is uniformly distributed between a = 1 and b = 9. a. b. c. d.

6.

What is P (X = 0)?

What is P (X = 2)?

What is P (X ≤ 2)?

Find E(X) and SD(X).

Suppose that X follows the Poisson probability model with λ = 2 for some specified time interval. a. b. c. d.

5.

E (X + 10) and SD (X + 10) E (5Y) and SD (5Y) E (X + Y) and SD (X + Y) What assumption must you make in part c.?

Suppose that X follows the Binomial probability model with n = 10 and p = .20. a. b. c. d.

4.

Is this a valid probability distribution? Explain.

Find the expected value of X. Find the variance of X. Find the standard deviation of X.

Consider the discrete random variables X and Y. Suppose that E(X) = 100, E(Y) = 90, SD(X) = 12, and SD(Y) = 8. Find each of the following. a. b. c. d.

3.

P (X = x) .35 .30 .22 .10 .03

What is P (X ≥ 50)?

What is P (X ≥ 55)?

What is P (X ≤ 55)?

What is P (45 ≤ X ≤ 55)?

Suppose that X follows the Binomial distribution with n = 500 and p = .60.

.

Chapter 9 Random Variables and Probability Distributions 9-13 a. b. c. d.

8.

Can the Normal probability model be used to approximate this distribution? Explain.

Find E(X) and SD(X).

What is P (X ≥ 320)?

What is P (X ≥ 280)?

Suppose that X is the time between arrivals and that arrivals follow the Poisson distribution with λ = 6 per hour. a. b. c. d.

What distribution can be used to model the time between arrivals?

What is the mean of this distribution?

What is P (X ≤ 10)?

What is P (X ≥ 10)?

ANSWERS 1.

a. b. c. d.

Yes, the probabilities sum to 1.0.

1.16 1.2144 1.102

2.

a. b. c. d.

110 and 12 450 and 40 190 and 14.422 X and Y must be independent.

3.

a. b. c. d.

.1074 .3020 .6778 2 and 1.265

4.

a. b. c. d.

.1353 .2707 .6767 2 and 1.414

5.

a. b. c. d.

0 .50 .625 5 and 2.309

6.

a. b. c. d.

.50 .0062 .9938 .9876

7.

a. b. c. d.

Yes, the success/failure condition is satisfied.

300 and 10.95 .0375 .9693

8.

a.

The Exponential distribution

.

9-14 Part II Understanding Probability Distributions and Statistical Inference b. c. d.

1/6 hour or 10 minutes .6321 .3679

.

10-1

Chapter 10 Sampling Distributions What’s It About?

We develop the concept of a sampling distribution by simulation. We see that for sample proportions, this sampling distribution is unimodal and symmetric. We learn that the distribution can be modeled with a specific distribution model—the Normal. We use the properties of the Normal distribution to be more specific about the sampling distribution of a proportion. Finally, we turn our attention to means and the Central Limit Theorem.

Comments At its heart, inference involves looking at sample data and trying to guess what’s actually going on in a population. It is vitally important that students be able to see that such comparisons involve known, observed data on the one hand and unknown—indeed, unknowable—facts about the world on the other. To help make that distinction we have adopted the term “model” to represent these images of theoretical perfection, as distinguished from what we observe. In class, begin now to draw the lines of distinction between these two worlds. The difference between what we see in actual data and conjectures we make about the world is a central issue throughout this course, and something students often confuse. Emphasize the idea that statisticians talk about models, and that these models differ from real data. We will build on this understanding over and over. It is one of the central unifying themes of the book.

When we look at proportions, we must think about the true proportion in the population ( p ) and the proportion we might actually see in any sample, ( p̂ ). That proportion ( p̂ ) will vary from sample to sample. We draw (or imagine that we draw) many samples and see that the distribution of sample proportions centers around the true population value but shows some variability – socalled sampling error. We note that the distribution of sample proportions appears to be roughly unimodal and symmetric, and see that under certain conditions a Normal model is useful. That Normal model is the sampling distribution model for sample proportions.

When we look at sample means, the plot thickens. First, we draw a sample from a distribution with some underlying shape – uniform, unimodal, bimodal, skewed, or whatever – but we may not know what the correct model for that is. We can make a histogram to examine the distribution of the data in this sample. We find the sample mean ( y ), and wonder how it differs from the population mean (  ). If we imagine drawing repeated samples, we could make a histogram showing the distribution of the various means. Surprisingly, this histogram appears to be roughly unimodal and symmetric. Again, under certain conditions a Normal model proves to be useful.

That Normal model is the sampling distribution model for sample means. By calling it a “sampling model” we help students distinguish it from a model for the population and the histogram of the sample.

Looking Ahead Most of the rest of this book depends on the ideas in this chapter. If sampling models for proportions and means did not turn out to be easily describable (indeed, Normal) then statistical inference as we know it would not exist.

The sampling model for sample proportions leads us directly to inference for proportions, found in Chapters 11 and 12. .

10-2 Part II Understanding Probability Distributions and Statistical Inference

The sampling model for sample means supports (with minor adjustment) inference for means, coming in Chapters 13 and14.

The idea of a sampling model returns in Chapter 18, where we examine the variability in slopes of regression lines based on different samples drawn from the same population. That sampling model will allow us to do inference for regression and carry us on to multiple regressions in Chapter 20 Class Do’s Help students understand the transition we are making here. We have talked about samples and created descriptive statistics like the sample proportion (for categorical variables) and the sample mean (for quantitative variables). We understand that such statistics vary from sample to sample and that each is an estimate of a population model parameter. We have discussed the notion of sampling error and understand that it is unavoidable. The truly remarkable thing is that sampling error is understandable and predictable. We hope it is small, but, remarkably enough, we can know how large it is likely to be. Armed with that knowledge, we can look at sample results and reach important decisions. We may decide that the results are within a reasonable margin of variation from what we expected. Or we may deem our results to be so unlikely that we don’t believe they could simply be sampling error. Then we call the outcome “statistically significant.” We are about to learn how to make that distinction.

This transition involves stepping back a level in our analysis. We used to focus on the data and derive a statistic from it. Now we focus on the statistic itself. The sample proportion (or mean) becomes our datum, and in our imaginations we compare it to all the other values we might have obtained from all the other samples we might have taken. (Or we might pretend that we’re about to draw many more random samples or perform many more randomized trials, but we’re just procrastinating and thinking about what’s likely to happen when we finally get around to it – if ever.)

Remind students repeatedly that the population is whatever the population is – it remains constant.

Only the samples vary. This key insight returns again and again in the chapters ahead.

Take time to construct sampling distributions in class to be sure students know what is happening here. The differences among the population, the sample distribution, a distribution of sample statistics from several samples, and the sampling model that describes such a distribution are potentially confusing, yet profound.

Help students to appreciate the stunning results in this chapter. It is not obvious that Normal models would prove to be so useful. This is one of the most startling and powerful insights in mathematics. We present the Central Limit Theorem as the Fundamental Theorem of Statistics. No one is surprised to think that larger samples look more like the population, or that the means of large samples probably vary less from the mean of the population. But who would guess that the variability among sample means would have a Normal model even when the population does not?

The Importance of What You Don’t Say This chapter touches on a number of topics that you will probably recognize as central to Statistics.

But it isn’t time to discuss them yet. Please don’t distract students with advanced concepts that they are not yet ready for. We have important principles to establish now. We’ll get to the more advanced application of these principles in due time. .

Chapter 10 Sampling Distributions 10-3 Don’t talk about probability density. We have in fact introduced a Normal density function, but that formal concept isn’t actually needed in this course. By contrast, the concept of a model is fundamental. Push that term instead. Discussing the Normal model reminds students of its role as an idealized model.

Don’t say “Empirical Rule.” The 68-95-99.7 Rule name is more useful because it carries the important numbers. The rule is “empirical” only for historical reasons; that name carries no useful information.

Don’t say “critical value.” That’s an idea that shows up in inference, but it doesn’t belong here.

Don’t say “The data are Normal.” No data are ever or can ever be Normal. The Normal model can serve as a useful way of thinking about data, but the model is not the data. Always make this distinction clear.

Avoid saying “sampling distribution of sample means (or proportions).” Students have a hard time discerning whether you are talking about the actual distribution of data in the sample or the theoretical variability in means from sample to sample. The distinction is critical, and made clearer by talking about “sample distributions” and “sampling models.”

Finally, as we said earlier, try not to get hung up making students learn to use the Normal table. If they have a calculator that computes Normal probabilities or easy access to a statistics program or spreadsheet, encourage them to use it. The Normal table is a backup for students without these calculators. That said, don’t let students use all the calculator shortcuts. We want them to develop an understanding of z-scores.

Class Examples 1. Draw some Normal models for sample proportions based on the 68-95-99.7 Rule. Always check the conditions first!

• Of all cars on the highway, 80% exceed the speed limit. What proportions of speeders might we see among the next 50 cars?

Solution: Plan – We want to find the distribution of the proportion of the next 50 cars that may be speeding on the highway. 80% of all cars on the highway are speeding.

10% Condition: The 50 cars can be considered a representative sample of all cars on the highway, and 50 is less than 10% of all cars on the highway.

Success/Failure Condition: np = 50(0.80) = 40, nq = 50(0.20) = 10 are both at least 10.

Do – Therefore, the sampling model for the proportion of speeders in 50 cars has mean 0.80 and

pq (0.80)(0.20) =  0.057. n 50 The model for p̂ is N(0.80, 0.057). standard deviation of SD( pˆ ) =

.

10-4 Part II Understanding Probability Distributions and Statistical Inference

Report – According to the Normal model, we expect 68% of the samples of 50 cars to have proportions of speeders between 0.743 and 0.857, 95% of the samples to have proportions between 0.686 and 0.914, and 99.7% of the samples to have proportions between 0.629 and 0.971.

2. Suppose that 52% of voters plan to vote “Yes” on a referendum on increasing school funding.

We poll a random sample of 300 voters. What might the percentage of yes-voters appear to be in our poll?

Solution: Plan – We want to find the distribution of the proportion of 300 voters that may vote “Yes” on the referendum. 52% of voters plan to vote “Yes.” 10% Condition: The 300 voters can be considered a representative sample of all voters, and 300 is less than 10% of all voters.

Success/Failure Condition: np = 300(0.52) = 156, nq = 300(0.48) = 144 are at least 10.

Do – Therefore, the sampling model for the proportion of “Yes”-voters in a sample of 300 has mean 0.52 and standard deviation of SD( pˆ ) =

pq = n

(0.52)(0.48)  0.029.

300

The model for p̂ is N(0.52, 0.029).

Report – According to the Normal model, we expect 68% of the samples of 300 voters to have proportions of “Yes” voters between 0.491 and 0.549, 95% of the samples to have .

Chapter 10 Sampling Distributions 10-5 proportions between 0.462 and 0.578, and 99.7% of the samples to have proportions between 0.433 and 0.607.

3. Use the sampling model to calculate some probabilities.

• A new sales gimmick has 30% of the M&M’s covered with speckles. These “groovy” candies are mixed randomly with the normal candies as they are put into the bags for distribution and sale. “Groovy” M&M’s are supposed to make up 30% of the candies sold.

In a large bag of 250 M&M’s, what is the probability that we get at least 25% “groovy” candies?

Solution: Plan – We want to find the probability of getting a bag of 250 M&M’s that has at least 25% “groovy” candies.

10% Condition: The 250 M&M’s can be considered a representative sample of all M&M’s, and 250 is less than 10% of all M&M’s.

Success/Failure Condition: np = 250(0.30) = 75, nq = 250(0.70) = 175 are at least 10.

Do – Therefore, the sampling model for the proportion of “groovy” M&M’s in a sample of 250 has mean 0.30 and standard deviation of SD( pˆ ) =

pq = n

(0.30)(0.70)  0.029.

250

The model for p̂ is N(0.30, 0.029). z=

z=

pˆ − p pq n 0.25 − 0.30

(0.30)(0.70) 250 z  −1.73

Report – According to the Normal model, the probability that our bag contains at least 25% “groovy” M&M’s is approximately 95.8%.

4. You can discover the Central Limit Theorem using the simulation activity from ActivStats* in class. You can also find Web applets and downloadable programs that do this simulation in a variety of ways. Alternatively, if you have time, draw random samples from some manageable population and create a sampling distribution of the sample means for various sample sizes.

5. As you did with the sampling model for proportions, use the Central Limit Theorem together with the 68-95-99.7 Rule and Normal percentiles to see what samples drawn from various populations might look like.

* ActivStats is available at www.pearsonmylabandmastering.com/northamerica/mystatlab/

.

10-6 Part II Understanding Probability Distributions and Statistical Inference 6. At birth, babies average 7.8 pounds, with a standard deviation of 2.1 pounds. A random sample of 34 babies born to mothers living near a large factory that may be polluting the air and water shows a mean birth weight of only 7.2 pounds. Is that unusually low?

Solution: Plan – We are interested in the probability that a sample of babies has mean birth weight below 7.2 pounds. At birth, babies average 7.8 pounds, with a standard deviation of 2.1 pounds. The model for birth weights should be roughly unimodal and symmetric, if not Normal.

Random Sampling Condition: The 34 babies were sampled randomly.

Independence Assumption: It’s reasonable to think that the weights of the randomly sampled babies will be mutually independent.

10% Condition: As long as more than 340 babies were born to mothers living in the vicinity of the factory, 34 babies represent less than 10% of all babies.

Do –

As the model for all babies is unimodal and symmetric, the Central Limit Theorem applies, especially because the sample size, 34 babies, is comparatively large. Under these conditions, the sampling distribution of y is Normal with mean  = 7.8 pounds and standard deviation SD( y) =

 n

=

2.1

 0.36 pounds.

34

Report – According to the Normal model, only about 4.75% of samples of 34 babies are expected to have mean birth weights below 7.2 pounds. The samples of babies near the factory appear to have a usually low mean birth weight.

Resources ActivStats

• Chapter 10 contains activities regarding sampling distributions for proportions and means, and presents the Central Limit Theorem. It is a great idea to assign the simulations before covering this material in class. Students can discover the Central Limit Theorem for themselves experimentally, which makes it much easier to believe and to remember. These .

Chapter 10 Sampling Distributions 10-7 simulations also make excellent classroom demonstrations, if you have suitable computer projection technology.

Web Links

• There are several websites and applets that explore the Central Limit Theorem. (Search for “central limit theorem applet”.)

Basic Exercises 1. Assume that the true proportion of college students who have taken at least one online course is .22. Suppose a sample of 500 college students is selected. a. What would you expect the shape of the sampling distribution for the sample proportion to look like? b. What would be the mean of this sampling distribution? c. What would be the standard deviation of this sampling distribution? d. If the sample size is increased to 1000, how would the sampling distribution change?

2. Assume that the true proportion of college students who do not have dental insurance is .85. Suppose that a sample of 200 college students is selected. a. b. c. d.

Is the success/failure condition satisfied? Explain.

What is the probability that the sample proportion will be at least .85?

What is the probability that the sample proportion will be less than .80?

What is the probability that the sample proportion will be between .80 and .90?

3. The net worth of families in New Brunswick in thousand dollars has a mean of 220 with a standard deviation of 40. A sample of families in New Brunswick is interviewed and the sample mean net worth is computed. a. Assuming a large enough sample size, what would you expect the shape of the sampling distribution for the sample mean to look like? b. What would be the mean of this sampling distribution? c. If the sample size is 50, what would be the standard deviation of this sampling distribution? d. If the sample size is 100, what would be the standard deviation of this sampling distribution?

4. Assume that new eco-friendly plastic bags have a mean tensile strength of 90 MPa with a standard deviation of 10 MPa. Suppose that a sample of 64 bags is selected. a. What is the probability that the sample mean will be at least 90 MPa? b. What is the probability that the sample mean will be between 87.5 MPA and 92.5 MPa? c. What is the probability that the sample mean will be less than 91MPa? d. What is the probability that the sample mean will be more than 93.75 MPa? .

10-8 Part II Understanding Probability Distributions and Statistical Inference

ANSWERS 1. a. b. c. d.

2.

3.

4.

a.

Normal .22 .0185 The mean would not change, and the standard deviation would decrease to .0131.

Yes, the success/failure condition is satisfied: np = 200(0.85) = 170  10

b. c. d. a. b. c. d.

nq = 200(0.15) = 30  10 .5 .0239 .9522 Normal 220 5.66 4

a. b. c. d.

.5 .9544 .7881 .0013

.

11-1

Chapter 11 Confidence Intervals for Proportions What’s It About?

We use the familiar setting of opinion polls to develop the concept of a confidence interval for a proportion, and then explore wider applications. We pay careful attention to helping students understand what a confidence interval means and how to interpret it correctly. We explore the interconnections among interval width, level of confidence, margin of error, and sample size, and see how to calculate the size of sample required to achieve a desired margin of error at a specified level of confidence. Finally we look at confidence intervals for the difference between two proportions.

Comments The order in which we present the various types of inference differs from that found in many other texts. We hope you’ll follow the order of our textbook; we are confident you will discover many benefits in this approach. We explained our rationale and the pedagogical advantages offered by this path through inference in a special commentary before Chapter 10 in this guide. Please have a look if you skipped that.

Looking Ahead We’ll apply the concepts of the form and meaning of confidence intervals again to means (Chapter 13), the difference in means and paired differences (Chapter 14), and regression slopes (Chapter 18).

Class Do’s Build your first confidence intervals without a messy formula or a critical value for z. You can base the discussion on the sampling model for proportions from the previous chapter and the 6895-99.7 Rule. It seems quite natural to use the sample proportion as an estimate of the unknown p in creating the standard error, and by now students easily associate two standard deviations with 95%. After they grasp the general idea, you can deal with the finer points – like a critical value of 1.96 rather than 2.

Open the discussion with a current poll. For example, Nanos (http://www.nanosresearch.com) has frequent polls, and other polling organizations are almost as prolific. Look at the bottom of the poll report for the information on the sample size and margin of error. (Most professional pollsters can be counted on to report these correctly.) Students have all seen these margins of error. Ask what they mean and where they came from. (Caution: If you try to reproduce the reported value, you will find that it is almost always computed for an estimated proportion of 0.5, not for each particular observed proportion.)

Concentrate on getting the interpretation right. While students usually get the general idea that they have an estimate plus or minus a margin of error, it’s quite common that they will misunderstand some of the nuances of meaning.

•

Students will think 95% is a probability. It’s better to use the term confidence. Strictly speaking, the probability is not 95% that the parameter is in our interval. This is based on the misconception that our interval is special in some way, and sometimes the parameter is in there. Remind them that it’s the parameter that’s special. It’s fixed. It’s our target – it does not move. We shoot at the target by forming this interval – like throwing a horseshoe at a stake we .

11-2

Part II Understanding Probability Distributions and Statistical Inference

can’t see. Sometimes (usually), we catch the stake somewhere within the horseshoe.

Sometimes we miss. The probability that we found the stake is 100% when the horseshoe spans it and 0% when it does not. We just don’t know which. So, we prefer to say we have 95% confidence that we’ve captured the parameter. The 95% reflects our own uncertainty about the interval.

•

Students will think that other sample results should fall in our interval. Different samples produce different intervals; 95% of those intervals contain the actual proportion we wish we knew. Maybe ours is one of those intervals. Or maybe it’s one of the other 5% that miss the target. We don’t know. We never will. The other horseshoes need not land near ours – they are aiming at the stake, not at our horseshoe.

•

Some students will think they’re 95% sure they know the sample proportion. Actually, we’re 100% sure of that. The value p̂ isn’t just in the interval, it is (by construction) in the very center of the interval. Remind them that sample results are known, but populations are not. We’re trying to catch a glimpse of the unknown, and that’s what we’ll write conclusions about. We care about the stake, not the horseshoe.

Spend time thinking about how our level of confidence is related to margin of error, interval width, and sample size. Let students ponder and discuss these connections based on common sense.

Messing with the formulas won’t be nearly as convincing.

•

It’s common sense that very precise statements are risky. That’s why the weatherman predicts that we’ll get between 4cm and 8cm of snow. A prediction of 5cm to 6cm would be more helpful perhaps, but it’s far less reliable.

•

It’s common sense that the more slack you cut yourself, the more faith you can have that you’re right. We’re very sure that between 1% and 50% of all cars are red.

•

It’s common sense that estimates based on larger samples are usually better. And better means that you get higher precision in your estimate (a narrower interval), higher confidence, or both.

It is, however, remarkable and fortunate that with sample sizes that are small enough to obtain we can usually make statements that are precise enough to be useful with sufficient confidence. There is no obvious reason why this should be so – and if it weren’t, inference would be impractical.

When you do the calculations that determine the sample size necessary to achieve a desired level of accuracy and confidence, make two important points.

•

These calculations are actually part of the process of designing a study.

•

We did say when we discussed sampling that the size of the required sample depended on the level of accuracy desired, not on the size of the population. It was hard for some to imagine that then, but now we see that the calculation rests only on level of confidence and margin of error, not on the size of the population. The calculation finds the minimal sample size that will work.

And, of course, be sure students see why to check the assumptions and conditions before proceeding.

Class Examples 1. Your local newspaper polls a random sample of 330 voters, finding 144 who say they will vote “yes” on the upcoming school budget. Create a confidence interval for actual sentiment of all .

Chapter 11 Confidence Intervals for Proportions

11-3

voters. Just use 2 SDs and concentrate on the conditions and the interpretation.

Solution: Plan – We want a confidence interval for the proportion of all voters who will vote “yes” on the upcoming school budget, based on a random sample of 330 voters. In our sample, 144 respondents stated that they would vote “yes”. We want to be reasonably confident in our results, so we will construct a confidence interval of ±2 standard errors around our sample proportion, or approximately 95% confidence.

Independence Assumption: It is reasonable to think that the responses were mutually independent, provided good surveying techniques were used.

Randomization Condition: The voters were sampled randomly.

10% Condition: Provided there are more than 3300 eligible voters, 330 are less than 10% of the population of voters.

Success/Failure Condition: npˆ = 144 and nqˆ = 186 , which are both at least 10, so the sample is large enough.

The conditions are satisfied, so we can use the Normal model to find a one proportion z-interval.

Do – Given n = 330 and pˆ =

144  0.436, so SE ( pˆ ) = 330

ˆˆ pq = n

( 0.436 )( 0.564 )  0.027 330

The margin of error is ME = z * SE ( pˆ ) = 2(0.027)  0.054 The confidence interval is 0.436  0.054 or (0.382, 0.490) Report – We are (approximately) 95% confident that between 38.3% and 49.0% of voters will vote “yes” on the upcoming budget.

2. An experiment finds that 27% of 53 subjects report improvement after using a new medicine.

Create a 95% confidence interval for the actual cure rate. Use z *= 1.96 . Why is this interval so wide? Make it narrower – 90% confidence. What are the advantages and disadvantages? What sample size would we need in a follow-up study if we want a margin of error of 5% with 98% confidence?

Solution: Plan – We want a confidence interval for the proportion of all people who will improve after using a new medication, based on an experiment involving 53 subjects. In our sample, 27% of the subjects improved.

Independence Assumption: One patient responding to the medication shouldn’t have an influence on other patients responding to the medication.

Randomization Condition: The patients were part of an experiment, which hopefully included random assignment of volunteers to treatment groups.

10% Condition: The sample is certainly less than 10% of the population. Success/Failure ˆ 53(0.73) = 38.69 , which are both at least 10, so the Condition: npˆ = 53(0.27) = 14.31 and nq= sample is large enough. .

11-4

Part II Understanding Probability Distributions and Statistical Inference

The conditions are satisfied, so we can use the Normal model to find a one proportion z-interval.

Do – Given n = 53 and pˆ = 0.27 , so SE ( pˆ ) =

ˆˆ pq (0.27)(0.73) =  0.061 n 53

The margin of error is ME = z * SE ( pˆ ) = 1.96(0.061)  0.12 The confidence interval is 0.27  0.12 or (0.15, 0.39) Report – We are 95% confident that between 15% and 39% of people will improve after using the new medication.

This interval is quite wide for a couple of reasons. The sample size of 53 people doesn’t provide a great deal of accuracy in our estimate. The standard error is still quite large. Also, the more confident we are that we have succeeded in capturing the true proportion, the less precise our interval becomes.

90% Confidence Interval The margin of error is ME = z * SE ( pˆ ) = 1.645(0.061)  0.10 The confidence interval is 0.27  0.10 or (0.17, 0.37) We are 90% confident that between 17% and 37% of people will improve after using the new medication.

This interval has the advantage of being more precise, but we are less confident in our ability to capture the true proportion within our interval.

Finding the sample size required for a margin of error of 5% with 98% confidence Consider the formula for margin of error. We believe the improvement rate to be 0.27 from ˆˆ pq our preliminary study. The value of z* for 98% confidence is 2.326. Using ME = z * , we n get

0.05 = 2.326

(0.27)(0.73) n

 (0.27)(0.73)  0.052 = 2.3262     n 2.326 2 (0.27)(0.73) n= = 426.546 0.052 To be safe, always round the estimate up.

We need to run an experiment with at least 427 people receiving the new medication in order to have a margin of error of 5% with 98% confidence.

3. What sample size does it take to estimate the outcome of an election with a margin of error of 3%?

Solution: We’ll do what polling organizations usually do, using 95% confidence and choosing .

Chapter 11 Confidence Intervals for Proportions

11-5

the most cautious proportion, 50%. So, pˆ = 0.5, qˆ = 0.5, ME = 0.03 and z = 1.96. Using

ME = z *

ˆˆ pq , we get n

0.03 = 196 .

(0.5)(0.5) n

 (0.5)(0.5)  0.032 = 196 . 2    n n=

1.962 (0.5)(0.5) = 1067.1 0.032

To be safe, always round the estimate up. A sample size of at least 1068 voters is required.

4. Have students simulate (or perform) 50 tosses of a fair coin. Each student should calculate his or her own observed sample proportion and find a 90% confidence interval for the true proportion of heads. Ask how many had confidence intervals that missed the target—that is, confidence intervals that did not contain the expected value of 50%. Ask what those students did wrong. (Trick question.) Make sure that everyone understands that you don’t have to do anything wrong to have a confidence interval that misses the truth. And because we never actually know the truth, we never know whether any confidence interval worked. We just know they usually do—and “usually” mean 90% sure in this case.

5. Try the coin simulation again with 200 tosses. Note that the 90% success rate holds true, but the estimates are more precise.

6. Have students conduct a survey on fellow students based on gender, year of study, etc.. For example, the proportion of males versus females that are smokers. Ask them to then construct a confidence interval for the difference in proportions between male and female smokers.

7. You’ll find plenty of other examples among the exercises.

Resources ActivStats*

• Chapter 10—“Inference for Proportions.” Simulations in this chapter give students handson experimental experience with confidence intervals. Here again, it is a good idea to let students discover how confidence intervals work for themselves before you tell them about it.

Web Links •

There are several Web applets that explore confidence intervals. Unfortunately, many deal with confidence intervals for means, but they are still useful for demonstrating the key ideas that the population parameter is fixed while confidence intervals vary from sample to sample, capturing the population value most – but not all – of the time. You can find links at http://wps.aw.com/aw_sharpe_business_1/237/60788/15561822.cw/index.html.

* ActivStats is available at www.pearsonmylabandmastering.com/northamerica/mystatlab/

.

11-6 •

Part II Understanding Probability Distributions and Statistical Inference Polling firms, such as Ipsos, poll regularly. And they almost always report proportions.

There’s nothing more up-to-date than today’s poll results from www.ipsos.ca.

Basic Exercises 1.

Suppose a sample is selected from a population in order to estimate the population proportion p.

Based on the sample data, the proportion is calculated to be .22. a.

What is the standard error if the sample size is 50? b.

What is the standard error if the sample size is 100? c.

What is the standard error if the sample size is 1000? d.

How is the standard error affected by the sample size?

2.

Suppose that you are constructing a confidence interval for the population proportion p. Find the appropriate z* value for each of the following levels of confidence. a.

90%. b.

95% c.

99% d.

How is the confidence interval affected by the stated level of confidence?

3.

Suppose that you are constructing a confidence interval for the population proportion p. You have selected a sample and based on the data calculate the proportion as .45. a.

What is the margin of error if the sample size is 50 and you want to be 90% confident? b.

What is the margin of error if the sample size is 1000 and you want to be 90% confident? c.

What is the margin of error if the sample size is 50 and you want to be 99% confident? d.

What is the margin of error if the sample size is 1000 and you want to be 99% confident?

4.

A random sample of 800 individuals provided 544 “no” responses. a.

What is the sample proportion? b.

Is the success/failure condition satisfied? Explain. c.

Construct a 95% confidence interval for the true proportion of “no” responses. d.

Interpret this confidence interval.

5.

Suppose you want to estimate the proportion of college students who own their own car. a.

What sample size is needed if you wish to be 95% confident that your estimate is within .02 of the true proportion? b.

What sample size is needed if you wish to be 99% confident that your estimate is within .02 of the true mean? c.

What sample size is needed if you wish to be 95% confident that your estimate is within .05 of the true mean? d.

Suppose that based on past studies, you have an idea that the proportion is .25. What sample size is needed if you wish to be 95% confident that your estimate is within .02 of the true proportion?

6.

Suppose that it was found that 75 out of 300 students enrolled in the accounting program failed at least one course, and that a similar study found that 80 out of 400 students enrolled in finance courses failed at least one course. Find a 95% confidence interval for the difference in proportions of failures between accounting and finance students.

ANSWERS 1.

a. b.

.0586 .0414

.

Chapter 11 Confidence Intervals for Proportions

11-7

c. d.

.0131 As the sample size increases, the standard error decreases.

2.

a. b. c. d.

1.645 1.960 2.576 As the stated level of confidence increases, the width of the interval increases.

3.

a. b. c. d.

.1157 .0259 .1812 .0405

4.

a. b. c. d.

.68 Yes, npˆ = 544  10 and nqˆ = 256  10 .

The 95% confidence interval is 0.68  0.032, i.e. between 0.648 and 0.712.

We are 95% confident that the true proportion of “no” responses is between .648 and .712.

5.

a. b. c. d.

n = 2401 n = 4148 n = 385 n = 1801

6.

The 95% confidence interval is 0.05  0.0619, i.e. between − 0.012 and 0.112.

.

12-1

Chapter 12 Testing Hypotheses about Proportions What’s It About?

Chapter 12 introduces the formal inference process for testing a hypothesis. We first look at the one-proportion z-test, and emphasize the general strategy and logic of hypotheses tests by comparing the process to a jury trial. We lay out a four-step process – hypotheses, model, mechanics, and conclusion – and demonstrate its implementation with step-by-step examples. We discuss the meaning and proper interpretation of P-values. We then look at comparison of two proportions. Throughout the chapter, we pay careful attention to writing conclusions.

Comments Students already know how to use Normal models, and they understand the sampling model for sample proportions that lies at the heart of the mechanics of finding the P-value. What’s new here is the formal structure of a hypothesis test.

You may be familiar with traditional hypothesis testing procedures that make decisions based on pre-specified alpha values or critical values of z. We have opted for the more modern P-value approach. We think P-values encourage students to think more and, as a result, they understand the reasoning better and—we hope—make better decisions. We explain that reasoning in the text, and amplify it below.

Looking Ahead The step-by-step process will be the template for all the tests of hypotheses we’ll encounter for proportions (Chapter 12), means (Chapters 13 and 14), distributions (Chapter 16), non-parametrics (Chapter 17) and association (Chapters 18 - 21). Once they are comfortable with this template, students can adapt more easily to each of the tests as they encounter them.

Class Do’s Emphasize the four assumptions and conditions of hypothesis testing, and that these have to be met in order to conduct the tests. Our four-step process for testing hypotheses fits comfortably within the Plan-Do-Report template. Each of the four steps has several requirements. When students organize their hypothesis tests according to these steps, they will be less likely to omit some key component in their answers. Present this procedure as “The Way” to test hypotheses; it is the path to the elusive “Complete” solution.

Plan: 1. Hypotheses

• Write the null hypothesis. A hypothesis is a claim about an unknown population model parameter, so be sure that students use the proper statistical notation: for example, p = 0.50 – not p̂ . Students may also write the null hypothesis in words; here again it is important to talk about the model, not the observed results: “The proportion of all voters who favour the amendment is 50%.” Make it clear that the null hypothesis specifies a parameter of interest and a value for that parameter.

• Write the alternative hypothesis. At this stage we must decide whether we are doing a twotailed or a one-tailed test, and if one-tailed whether we are interested in the upper tail or .

12-2

Part II Understanding Probability Distributions and Statistical Inference lower tail. We base this decision on what we are initially interested in learning about the situation (the Why of the study), not on what we think the observed results might show. If we care about whether the percentage of our customers who are teenagers has changed in the past 10 years, then we are equally interested in spotting an increase or a decrease and we do a two-tailed test. If we wonder whether a costly ad campaign directed at teens has had an impact, then we only want to know if the percentage has gone up. That calls for a one-tailed upper tail test. Note that in either case the null hypothesis is that nothing has changed; that is, that the percentage of teen customers is the same as before.

• Remind students that the null hypothesis proposes a model for the population. As with all models, we know this one is wrong – the parameter probably does not exactly equal the hypothesized value. Unlike previous models, now we ask if it’s wrong enough to reject. A null hypothesis is “null” because it is sceptical about the world. It generally asserts that nothing interesting or unusual is happening, not that the parameter value is necessarily 0 (or, for proportions, 50%).

2. Model

• Decide what inference procedure is called for. Right now this is easy – we’re just looking at one-proportion z-tests. But as our repertoire of inference procedures expands, this will become a key issue. Means or proportions? One group or two? Independent or matched?

Numerical values or rankings? We’ll work hard later on helping students make this decision.

• List the assumptions and check the conditions. We can never invoke any statistical procedure without being sure it properly fits the circumstances, and inference methods are based on sampling distribution models that assume specific things about the population.

Some assumptions are checked by just thinking about the way the data were collected. (If randomization was correctly applied, then the individual responses should be independent.)

Some come with associated conditions that we can check with the data. This is one of the most frequently overlooked steps in student responses on exams. Note also that it is not enough to merely state the condition np 10 . Students should be expected to show that they have actually checked that the condition is satisfied: np = 120(0.2) = 24  10.

• Name the test. Indicate that because the conditions are satisfied it is okay to use a Normal model and that we’ll perform (for now) a one-proportion z-test. Each of the inference methods in the book comes with a clear name. Encourage students to learn and use them.

These names are not universal throughout Statistics, but they are clear and unambiguous.

Do: 3. Mechanics

• Write down the statistics, emphasizing use of proper notation. Here, those are the sample size, the observed number of successes, and the sample proportion. Later on, the statistics may involve more than one group, means and standard deviations, distributions, tables, slopes, and so on. Often they come from technology – for example, from a summary table displayed by a statistics package. .

Chapter 12 Testing Hypotheses About Proportions

12-3

• Draw a curve depicting the sampling model. Mark the hypothesized parameter and the observed statistic, and shade the appropriate tail (or tails) to show the region corresponding to the P-value. We mean this seriously. Even professional statisticians make this sketch.

Calculate the value of the test statistic – here z, later on t,  2 and F or the variety of nonparametric test statistics in Chapter 17. Show the formula, substitute all the proper values, and give the final result. Although technology will do this and it’s fine to copy the results without actually doing the calculations, it’s a good idea for students to show the formula and values that produce that result. (If nothing else, it lets you award partial credit – a reason students readily accept.)

• Find the P-value. Most often we’ll just find the P-value using technology. When we must, we can use tables, functions on a calculator, or a statistics package’s calculations functions.

Report: 4. Conclusion

• Link the P-value to the decision. We need to be clear how the calculated P-value led to our decision. These statements may take many forms, such as “Because the P-value is so low…”, “Since P > 0.05, . . .”, or “This P-value of 0.03 means that the observed results are unlikely to be just sampling error, so . . .”

• State the decision about the null hypothesis – either we reject it or we fail to reject it.

• Interpret that decision in the proper context. The final conclusion we write must clearly explain what we have learned about the original question under investigation. A complete conclusion might look like one of these: “Because P = 0.003 is very low, I reject the null hypothesis. There is strong evidence that more than 50% of all voters favour the amendment.”

“The high P-value indicated that these results could be reasonably explained by sampling error, so I fail to reject the null hypothesis. We do not have evidence of a change in the percentage of teens purchasing our product.”

Students need practice writing hypotheses. Remind them that “null” means “nothing.” In other words, the null hypothesis is that nothing interesting is happening, nothing has changed, or there’s nothing unusual about the results. In practice, that usually means that we posit the defendant is innocent, the medicine does not work, the water is not polluted, and the rivets can’t withstand the necessary stress. The alternative hypothesis represents the change or difference that we are interested in or concerned about. It is the position we will have to take if the results are so unlikely (were the null hypothesis true) as to make the null hypothesis untenable.

Keep emphasizing that hypotheses are about parameters, not statistics. We’ll never see p̂ , y or b1 in a hypothesis. We know those values – we calculated them from the data we collected. There’s no need to hypothesize about them. Hypotheses are about the unknown (unknowable). That is, they are about models. The null hypothesis proposes a parameter for a model – p (or, later,  or 1 ). The test checks to see if the results we observed cast doubt on the viability of that model.

Those results can never confirm that the model is correct, but they can convince us (beyond a reasonable doubt) that it is wrong.

Be sure that students understand the P-value. It is not the probability that the null hypothesis is .

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Part II Understanding Probability Distributions and Statistical Inference

true. It is not even the probability that the null hypothesis is true given that these results were observed. In fact, it’s the other way around. Either the hypothesis is true or it’s not; we just don’t happen to know which. We assume that the null hypothesis is true – that the model behaves as we think it does. We realize that we’ll never collect data in perfect agreement with our model. We know that observed results will vary, and have called that variability sampling error. By now we understand how much variability we should reasonably expect. We look at the statistics from the observed data and assess whether they are consistent with the sampling model or so bizarre that it is not reasonable to continue to believe in the model. We ask how likely we’d be to see results like these if our model were correct. That’s the P-value: the conditional probability of observing results at least as extreme as ours if the null hypothesis were true. A very low P-value means our results are very unusual, and we then reject the null model.

How low a P-value do we need? That’s a personal decision, based on the situation, the belief in the null hypothesis, and the costs of being wrong. Traditionally, statisticians have adopted “alphas,” or “levels of significance,” of 5%, or maybe 10%, 1%, or 0.1%. These are good rules of thumb, but remember that they are general guidelines, not ironclad laws. The modern P-value approach asks a statistician (a.k.a. student) to look at the P-value, think about what it says about the situation under consideration, and then make a decision. If we are checking on a sociological phenomenon of interest – say, a change in teen smoking – and we find a P-value of 0.08, we might very well say that we have found some evidence of a change. If we are testing whether a new drug has harmful side effects, we would want an extremely small P-value. There’s nothing magical about 0.05.

Landy Godbold has summarized these issues in statistical inference by posing three questions when looking at data: 1. Am I surprised? (Should I reject the null hypothesis?)

2. How surprised am I? (What’s the P-value?)

3. What would not surprise me? (Write a confidence interval for the parameter.)

Landy’s last question is important. A hypothesis poses a single value for a parameter. We can be almost 100% sure that value is incorrect (though perhaps only slightly). A confidence interval proposes a range of plausible values for that parameter, and should almost always accompany a hypothesis test.

The Importance of What You Don’t Say Don’t “accept” the null hypothesis. We can never accept the null hypothesis. For one thing, it’s probably never true precisely. Is the probability that a coin lands heads exactly 50%? Isn’t it possible that there is a slight bulge on one side that makes the bounce of a quarter ever so slightly biased, and that the probability of heads might therefore be 49.997%? We cannot accept the null hypothesis just because we don’t find compelling evidence against it. The best we can do is to not reject the hypothesis. This is exactly what happens when a jury fails to convict. Their verdict is “not guilty.” In other words, the jury says that they did not see evidence of guilt “beyond a reasonable doubt.” Some jurors may still suspect that the defendant is indeed guilty. The jury does not proclaim that the defendant has been proven innocent. We never “prove” the null hypothesis— or even the alternative for that matter.

Don’t use “Standard Error” inappropriately. This is a subtle, but important point. In this book we reserve the term standard error for sample-based estimates of the standard deviation of the .

Chapter 12 Testing Hypotheses About Proportions

12-5

sampling distribution model. Some books have used the term more generally for that standard deviation, whether estimated or specified. Proportions have the special property that the value of the proportion determines the value of the standard deviation of its sampling distribution model. A null hypothesis that specifies a population model proportion value has automatically also specified its standard deviation. That then is not a standard error in our terminology, and we’ll write SD( pˆ ) rather than SE ( pˆ ) . Keeping the terminology separate helps students remember this difference.

Class Examples 1. Propose an example of a hypothesis to test, and build your discussion of the four-step format around it. Here’s a possibility: Health Canada monitors the safety of smoke detectors in the marketplace and estimates that 90% of all Canadian homes have at least one smoke detector. Calgary’s fire department has been running a public safety campaign about smoke detectors consisting of posters, billboards, and ads on radio and TV and in the newspaper. The city wonders if this concerted effort has raised the local level above the 90% national rate. Building inspectors visit 400 randomly selected homes and find that 376 have smoke detectors. Is this strong evidence that the local rate is higher than the national rate?

Hypotheses. The null hypothesis is that Calgary is no different than anywhere else. The alternative is that the public safety campaign has had a positive effect, so the rate of compliance is higher. The hypotheses deal with the actual percentage of all homes that have smoke detectors in that city.

H0: p = 0.90, HA: p > 0.90 Model. We have a simple random sample of 400 homes, and assume that’s less than 10% of all homes in this city, making them essentially independent. We expect np0 = 400(0.9) = 360 successes and nq0 = 40 failures, both at least 10, so the sample is large enough to use a Normal model. We’ll do a one-proportion z-test.

Mechanics.

n = 400, x = 376, pˆ = 0.94 0.94 − 0.90 = 2.67 (0.90)(0.10) 400 P-value = P( z  2.67) = 0.004 z=

Conclusion. The P-value is very low, making it unlikely that the observed results can be explained by sampling error. We reject the null hypothesis. There is strong evidence that, in this city, more than 90% of homes have smoke detectors.

Further Comments. Interpret the P-value carefully. If in fact the rate in this city is 90% – the same as anywhere else – then only 0.4% of samples (about 1 in every 250) would find (as ours did) a proportion of 94% or more. While that could be what happened here, it’s not the most logical explanation. In fact, we are 95% confident that between 91.1% and 96.9% of the city’s homes have smoke detectors.

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Part II Understanding Probability Distributions and Statistical Inference

2. Do a second example where we don’t reject the null. There are supposed to be 20% orange M&M’s. Suppose a bag of 122 has only 21 orange ones. Does this contradict the company’s 20% claim? Note that this is a two-tailed test. Even though the observed result looks as though there were not enough orange ones, before we had the sample we’d have been equally interested to discover that there were too many. Now the P-value is 2  P(z  −0.77) = 0.44 (see solution below.) That means that even if the company is correct about the 20% orange claim, 44% of all bags this size will differ from perfection by at least as much as this one did. It does not seem that this result is unusual, so we fail to reject the null hypothesis. We don’t have convincing evidence that the 20% figure is wrong. (On the other hand, we certainly didn’t prove it was right, so we cannot accept the null hypothesis.)

Solution: Hypotheses. The null hypothesis is that the proportion of M&M’s that are orange is 20%. The alternative hypothesis is that the proportion of M&M’s that are orange is something other than 20%. In symbols: H0 : p = 0.20, H A : p  0.20 Model.

Independence Assumption: It is reasonable to think that M&M colors are mutually independent.

Randomization Condition: The 122 M&M’s we have can be considered representative of all M&M’s.

10% Condition: 122 M&M’s is certainly less than 10% of all M&M’s.

Success/Failure Condition: np0 = 122(0.20) = 24.4 and nq0 = 122(0.80) = 97.6 , which are both at least 10, so the sample is large enough.

The conditions are satisfied, so I can use the Normal model to perform a one proportion z-test.

Since we are testing the hypothesis that the proportion of orange M&M’s is different than 0.20, we will use a two-tailed test.

Mechanics.

n = 122 , x = 21 , pˆ =

21  0.172, 122

0.172 − 0.2 = −0.77 (0.2)(0.8) 122 P -value= 2  P( z  − 0.77) = 2 ( 0.2206 )  0.44 z=

Conclusion. The P-value is quite high, so we fail to reject the null hypothesis. There is no evidence to suggest that the proportion of orange M&M’s differs from the advertised 20%.

3. It’s instructive to try the M&M’s example again, this time assuming that the rate is 22%. You won’t reject that hypothesis either. Or 15%. Try having different groups of students test hypotheses ranging from 5% to 30%. Some will be rejected, others not. Some models are clearly inconsistent with the facts, but many remain plausible. What would not surprise me?

The 95% confidence interval says that p could probably be anywhere from about 11% to 23%. .

Chapter 12 Testing Hypotheses About Proportions

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That’s still more justification for not “accepting” any particular null hypothesis; we just can’t reject some.

4. Have students conduct either an in-class or out-of –class survey of their fellow students.

Possible proportions could be found by gender. For example, the proportion of girls versus boys who vote for a specific party, who work, who are enrolled in a certain program, or who support a certain issue, etc. Students could then conduct hypothesis tests for their results to see if a difference does exist based on gender.

5. You’ll get plenty of opportunity to discuss all the issues as you go over the many questions on the homework exercises. If you still need more in-class examples, there are always unassigned exercises in the text or on the ActivStats*.

Resources ActivStats

• Chapter 11 – “Testing Hypotheses.” A simulation exercise in this chapter asks students to test whether the proportion of a striped bar that is red is 50%. Students gather data and draw their own conclusions. In the process, they develop the reasoning of hypothesis testing for themselves. Let them do this before you present it in class, and you’ll find they have a much deeper appreciation for the reasoning. Remind them to record the number of trials they needed before they rejected the null and the true proportion (which they will be told.) With these, they can compute their own personal critical value. How unlikely did the data have to be before they threw in the towel and rejected the null?

Websites:

• Ekos, Nanos Research and Ipsos Reid regularly conduct polls that you can use for in class examples or to give students real-world material to work from. Their websites can be found, respectively, at http://www.ekos.com/ http://www.nanosresearch.com/ http://www.ipsos-na.com/news-polls/ Basic Exercises 1.

The four cities, Toronto, Montreal, Vancouver and Calgary accounted for 40% of Canadian household income in a recent year. You interview a random sample of households this year to see whether that proportion has changed. Suppose that all assumptions are met and you test the null hypothesis H0: p = .40. a. b. c. d.

What is the standard deviation of the sample proportion if the sample size is 50?

What is the standard deviation of the sample proportion if the sample size is 100?

What is the standard deviation of the sample proportion if the sample size is 1000?

Why are we calculating a standard deviation and not a standard error?

* ActivStats is available at www.pearsonmylabandmastering.com/northamerica/mystatlab/

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Part II Understanding Probability Distributions and Statistical Inference 2.

In 2009, the Financial Post report on Canadian Demographics classified 25% of Canadian households as “Quiet Neighbours”. You interview a random sample of households this year to see whether that proportion is still the same. Suppose that all assumptions are met and you test the hypotheses H0: p = .25. a. b. c. d.

3.

Toronto, Montreal and Vancouver accounted for 30% of Canadian retail sales in a recent year. You interview a random sample of Canadian residents this year to see whether that proportion has changed. Suppose that all assumptions are met and you test the hypotheses H0: p = .30 vs. HA: p ≠ .30. A sample of size 500 results in a sample proportion of .25. a. b. c. d.

4.

What is the standard deviation of the sample proportion?

Compute the value of the z statistic.

Find the P-value associated with this z statistic.

State your conclusion at α = .05.

In a recent year, Canadian households in the territories and provinces other than Quebec accounted for 75% of national households. You interview a random sample of households this year to see whether the proportion of non-Quebec based households has increased. Suppose that all assumptions are met and you test the hypotheses H0: p = .75 vs. HA: p > .75. A sample of size 100 results in a sample proportion of .80. a. b. c. d.

5.

What is the critical value if HA: p ≠ .25 and α = .05?

What is the critical value if HA: p > .25 and α = .05?

What is the critical value if HA: p < .25 and α = .05?

What is the critical value if HA: p ≠ .25 and α = .01?

What is the standard deviation of the sample proportion?

Compute the value of the z statistic.

Find the P-value associated with this z statistic.

State your conclusion at α = .05.

In a recent year 55% of TV viewing time in Canada was associated with foreign programming. The government has recently introduced additional grants to subsidize the production of Canadian programming. You interview a random sample of Canadians this year to see whether the proportion of Canadians viewing foreign programming has dropped. Suppose that all assumptions are met and you test the hypotheses H0: p = .55 vs. HA: p < .55. A sample of size 1000 results in a sample proportion of .51. a. b. c. d.

What is the standard deviation of the sample proportion?

Compute the value of the z statistic.

Find the P-value associated with this z statistic.

State your conclusion at α = .05.

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Chapter 12 Testing Hypotheses About Proportions 6.

In a survey, it was found that 20% of all Canadians aged 15 or older had a university degree. You interview a random sample of Canadians this year to see whether that proportion has changed. Suppose that all assumptions are met and you test the hypotheses H0: p = .20 vs. HA: p ≠ .20. A sample of size 250 results in a sample proportion of .25. a. b. c. d.

7.

12-9

Construct a 99% confidence interval for p.

Based on the confidence interval, at α = .01 can you reject H0? Explain.

What is the difference between the standard error and standard deviation of the sample proportion?

Which is used in computing the confidence interval?

In a recent survey, it was found that of 200 people interviewed in Ontario, 25% spoke a language other than English or French at home. In British Columbia, a similar study of an equally sized sample found that 20% of people spoke a language other than English or French at home. Provided that all assumptions and conditions are met, can we conclude that there is a significant difference between British Columbia and Ontario in the proportion of people who speak a language other than English or French at home? Test at α = .05.

ANSWERS 1.

a. b. c. d.

.069 .049 .0155 We are calculating the standard deviation because we assume that the null hypothesis is true and that p = .40.

2.

a. b. c. d.

± 1.96 1.645 - 1.645 ± 2.575

3.

a. b. c. d.

.0205 - 2.44 .0147 Because P-value = .0147 <.05, we reject the null hypothesis H0. There is strong evidence that the proportion has changed.

4.

a. b. c. d.

.0433 1.15 .1251 The high P-value = 0.1251 >.05 indicates that these results could be reasonably explained by sampling error, so we fail to reject the hull hypotheses H0. There is no evidence to suggest that the proportion of non-Quebec based households has increased. .

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Part II Understanding Probability Distributions and Statistical Inference 5.

a. b. c. d.

.0157 - 2.55 .0054 Because P-value = .0054 <.05, we reject the null hypothesis H0. There is strong evidence that the proportion of Canadians viewing foreign programming has dropped.

6.

a. b. c.

.25 ± .065, i.e. between 0.185 and 0.315 No, because .20 is included in the 99% confidence interval.

The term standard error is used for sample-based estimates of the standard deviation of the sampling distribution model. The proportion determines the value of the standard deviation of its sampling distribution model.

Thus, we use SD( pˆ ) rather than SE ( pˆ ) .

The standard error, not the standard deviation, should be used for the confidence interval.

d.

7.

Since z = 1.196 and the critical value is z = ±1.96, we cannot conclude that a difference exists.

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13-1

Chapter 13 Confidence Intervals and Hypothesis Tests for Means Now that we have the principles of inference under our belts, it’s an easy slide downhill to the finish line. The theme of these chapters is “just like the other ones, only a little different.”

Emphasize that, and students won’t feel overwhelmed thinking there are new formulas and methods raining down on them with each chapter.

If you haven’t done it yet, point your students to the inside back cover of this book. The table there organizes all of the inference methods and shows them that they are really all alike – only a bit different. And the list of assumptions and conditions should remind students of the things they were looking for in their displays and summaries back in chapters 4, 7, and 8.

What’s It About?

Chapter 13 examines confidence intervals and hypothesis tests for a population mean. We introduce the t-models to deal with the extra variation caused by not knowing the population standard deviation, and examine the assumptions and conditions necessary for inference. We use this new application of the concepts of inference to re-examine the correct interpretation of a confidence interval, the relationship between confidence intervals and hypothesis tests, and the difference between statistical significance and importance. We show how to determine the sample size necessary for a study about means, and how to interpret output from software packages.

Comments We almost never do inference pretending to know the population standard deviation. We have emphasized throughout this course that Statistics is about making sound decisions in the real world, and we won’t change that now. In the real world no one ever knows the population standard deviation. (Well, almost never. There are cases when experience suggests that the standard deviation is known.) W. Gosset discovered the t-distribution to resolve this dilemma. What worked for Gosset works for us, too.

And we never mention the out-dated pre-technology approach of using a Normal model for “large samples.” It may have seemed that there was a compelling mathematical justification for switching over to z when the sample size exceeds 30, but there wasn’t. Typesetters, not mathematicians, are the source of that rule of thumb. About 30 lines of a t-table fit conveniently on a page. After that, the old advice was, turn to the z-table because it’s pretty close. In the age of graphing calculators and statistics software, there’s no reason to shy away from t for any number of degrees of freedom, so we can do better than approximate it with z. We can do inference for means the right way, using t. Always.

The bottom line is that nothing else in this chapter is really new. We still create and interpret confidence intervals the same way. We still use the same reasoning and four-step procedure for testing a hypothesis. The only new issue is the need to scrap the Normal model in favour of a tmodel -the adjustment students will readily make.

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Chapter 13 Confidence Intervals and Hypothesis Tests for Means 13-2 We’ve been very careful about the term standard error, and here we get some of the benefit. The standard error SE (y ) = s is a standard error because it is estimated from the data. And, of n course, that fact is central to our use of Student’s t.

Remember also that t-models are just that – models. Students’ most common confusion at this stageis between the standard deviation of the data, s, and the standard error we estimate for the mean, SE (y ). They need to be able to distinguish between the actual distribution of the data on the one hand, and the model for the sampling distribution of the mean on the other. Talk about the “model” often and clearly to help them keep that distinction clear.

 With the t-models we can show a simpler, clearer relationship between confidence intervals and hypothesis tests. Students should see these as two ways of viewing essentially the same calculations. That reduces by half how much new stuff there is here to learn.

Looking Ahead In the next chapter we expand our abilities to do inference for means when there are two groups.

Chapter 14 explains the proper procedures if the two groups are independent, and then looks at inference for paired differences if the groups are matched. The chapter continues to rely on tmodels. Student’s t returns again in Chapter 18 where we discuss inference for regression.

Class Do’s Start right in with an example involving data – like our coffee machine suggestion in the Classroom Examples section. The chapter discusses confidence intervals first, and then moves on to hypothesis tests. You should look at the two types of inference in that order – a hypothesis test with a follow-up confidence interval. Students will be able to set up the outline of the hypothesis test. After all, they’re all the same.

• Hypotheses. This time they’re about  . (Never, y of course. We know y ; that value comes to us from the sample. It’s the mean of the population we wonder about.)

• Model. Some of the conditions are obvious. We need a simple random sample (and we won’t worry as much about the 10% Condition for quantitative data). We’ll add another condition after some more discussion.

• Mechanics. It seems pretty straightforward at first. Find y , and then try to find z. Trouble. The Central Limit Theorem tells us the sampling model for y has a standard deviation of  divided by the square root of the sample size. Strike One: we don’t know  . Students suggest using s, the sample’s standard deviation, as an estimate. Great idea. This shows they have caught on to the kind of statistical thinking we want them to develop in this course. But there’s a problem. Unlike proportions, where the null hypothesis specifies p, and that in turn tells us the SD, here there is no link to the SD. The hypothesized population mean tells us nothing about the population’s standard deviation. And the sample standard deviation, s, varies from sample to sample. This introduces variation in the denominator, something that wasn’t there before. How can we deal with this extra uncertainty? Some students may suggest cutting ourselves a little more slack to make up for the increased variability, and that’s exactly what a t-model does.

Spend some time briefly describing t (suggestions follow). Note that the mathematics underlying the use of t requires the assumption that the sample comes from a population that is .

Chapter 13 Confidence Intervals and Hypothesis Tests for Means 13-3 Normal. Strike Two: we don’t know the population’s center (our hypothesis is about that), we don’t know its spread (that’s why we need t), and we don’t (indeed can’t) know its shape, either. All we can do is take a look at the distribution of the data in the sample to see if it’s plausible that these data were chosen at random from a Normal population. We hope to see a roughly mound-shaped, symmetric distribution without skewness or outliers. That’s the Nearly Normal Condition, telling us it’s OK to use a one-sample t-test for means.

With the conditions now complete, get right on with the hypothesis test. Calculate t, draw the curve (indicating the degrees of freedom), shade the region representing the P-value, and find P (using technology or a t table for critical values).

• Conclusion. It’s the same old story: linkage, decision, and conclusion in context. Be sure the conclusion talks about the mean of the population.

It’s a great idea to follow the hypothesis test with a confidence interval, looking up the critical value in the table. Spend some time with the interpretation. There are all kinds of new ways to be wrong about what the interval means! Most common are that some students will think that 95% of the population or 95% of all sample means must lie in the interval. Help students understand the correct interpretation.

Because the t-models are new, it helps to discuss their relationship to the Normal model. Common sense says that the increased uncertainty caused by using s to estimate  makes this model more spread out, and that this uncertainty (and therefore the spread) must be greater for small samples.

These ideas tell us that there are different t-models for different n’s, pointing to the new issue of degrees of freedom. Greater spread tells us that confidence intervals must be wider, so critical values of t must be larger than the corresponding critical value of z. A quick glance down one of the columns of the table will show that, and also show that t-models look increasingly Normal as n increases.

The Nearly Normal Condition is intentionally vague. First, be sure students understand that the issue is not whether the sample is Normal. Samples aren’t; they can’t be. We want to have a sample drawn from a population that’s Normal, but we can never be sure about the population.

The t-test is fairly robust. What we really need is that the distribution is unimodal and roughly symmetric – i.e., nearly Normal. We’re particularly concerned about outliers and skewness.

However, the usefulness of a t-model also depends on the sample size. As the sample size increases, the Normality of the population becomes less critical. (We say that the t-models with larger degrees of freedom are increasingly robust with respect to the assumption of Normality.)

For n’s near 20, some skewness is okay. For n’s near 30, we can use t unless the sample is strongly skewed or there are far outliers. For larger samples, we often don’t even look at the histogram, justifying use of a t-model simply by saying the sample is large.

Talk about what to do with outliers. We recommend performing the analysis twice. The concern, of course, is that the outlier will distort the results. If the conclusions don’t change (if, for example, we reject a hypothesis when we include the outlier in the data, and also reject it after we remove the outlier), then it’s clear that the decision is not based on the impact of the outlier. Generally, outliers inflate the standard error more than they move the sample mean, so they make confidence intervals wider and P-values larger.

We resort to z when we do a sample size calculation, because not knowing n leaves us unable to pick a critical value for t. This calculation will always underestimate the sample size we should use. If it calls for a very large sample, that’s fine. But if it points to a small sample, then we should .

Chapter 13 Confidence Intervals and Hypothesis Tests for Means 13-4 use that first estimate for n to determine a critical t and repeat the calculation.

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Chapter 13 Confidence Intervals and Hypothesis Tests for Means 13-5 The Importance of What You Don’t Say Don’t suppose we know  , the standard deviation of the population. We (almost) never will.

Don’t suggest that we use z instead of t for large samples. For inference about means, use t.

Always. (Almost.)

Don’t confuse notation or terminology. We find it confusing to write  y = 

n . Instead, we use

SD( y ). Of course it’s the same thing, but students seem to get confused with  ’s all over the place. And the difference between SD( y ) and SE( y ) reminds them of the difference in terminology and meaning.

Class Examples 1. The coffee machine in your company “break room” dispenses coffee into paper cups. You’re supposed to get 10 ounces of coffee, but the amount varies slightly from cup to cup. Here are the amounts measured in a random sample of 20 cups. Is there evidence that the machine is short-changing users?

9.9 9.9 10.0 9.9 10.0

9.7 9.6 9.5 9.6 9.9

10.0 9.8 9.7 10.2 9.5

10.1 9.8 10.1 9.8 9.9

Hypotheses. H0 :  = 10.0 , HA :   10.0 Model. Random sample; 20 < 10% of all cups (no reason to doubt independence). The histogram of sample data to the right looks roughly unimodal and symmetric, so it’s reasonable to believe that the amount of coffee in all possible cups could be described by a Normal model. Okay to do a t-test for the mean.

Mechanics. n = 20 df = 19 y = 9.845 s = 0.1986 9.845 − 10.0 t= = −3.49 0.1986 20

P- value = P( y  10.0) = P(t19  −3.49) = 0.0012 Conclusion. Such a small P-value makes it unlikely that the low sample mean resulted from sampling error, so we reject the null hypothesis. There is strong evidence that the mean amount of coffee dispensed by this machine is less than the stated 10 fluid ounces.

2. A company has set a goal of developing a battery that lasts over 5 hours (300 minutes) in continuous use. A first test of 12 of these batteries measured the following lifespans (in minutes): 321, 295, 332, 351, 281, 336, 311, 253, 270, 326, 311, and 288.

• Is there evidence that the company has met its goal? .

Chapter 13 Confidence Intervals and Hypothesis Tests for Means 13-6

Solution: We want to know if the mean battery lifespan exceeds the 300-minute goal set by the manufacturer. We have 12 battery lifespans in our sample to test the claim.

Hypotheses. The null hypothesis is that the batteries have a mean lifespan of at most 300 minutes. The 300-minute goal has not been met. The alternative hypothesis is that the batteries have a mean lifespan greater than 300 minutes. The 300-minute goal has been met.

H0 :  = 300 , HA :   300 Model.

Randomization Condition: This is not a random sample of batteries, but merely 12 batteries produced for preliminary testing. However, it is reasonable to assume that these batteries are representative of all batteries.

Nearly Normal Condition: The distribution of battery lifespans is roughly unimodal and symmetric, so it’s reasonable to assume that the lifespans of all batteries could be described by a Normal model.

4 3 2 1

240

280

320

Battery Lifespans (min)

Because the conditions have been met, we can do a one sample t-test for the mean, with 11 degrees of freedom.

Mechanics.

n = 12

df = 11

y− s

P-value = P( y  306.25)

n 306.25 − 300 t= 29.31

= 0.238

t=

y = 306.25

s = 29.31

= P(t11  0.7387)

12 t  0.7387 Conclusion. Since the P-value is high, we fail to reject the null hypothesis. There is no evidence to suggest that the mean battery lifespan exceeds 300 minutes. It does not appear that the company has met its goal.

• Find a 90% confidence interval for the mean lifespan of this type of battery.

Solution: The conditions have been met, so we can create a one-sample t-interval, with 90% confidence.

 29.31 y  t11*  SE( y) = 306.251.796   = (291.05,321.45)  12  We are 90% confident that the true mean battery lifespan is between 291.05 and 321.45 .

Chapter 13 Confidence Intervals and Hypothesis Tests for Means 13-7 minutes.

• If we wish to conduct another trial, how many batteries must we test to be 95% sure of estimating the mean lifespan to within 15 minutes? To within 5 minutes?

Solution: We want to know how many batteries to test to be 95% sure of estimating the mean lifespan to within 15 minutes. First, do a preliminary estimate using z* = 1.96 as the critical value. Our first estimate is about 15 batteries, a small sample (see calculations below.)

 s  ME = z *    n  29.31 15 = 1.96    n  (1.96)(29.31) n= 15 n  14.67 Now, do a better estimate, using t14* = 2.145 as the critical value.

 s  ME = t *    n  29.31  15 = 2.145    n  (2.145)(29.31) n= 15 n  17.57

We would need to sample about 18 batteries in order to estimate the mean battery lifespan to within 15 minutes, with 95% confidence.

Finally, to estimate the mean battery lifespan to within 5 minutes, you could do the entire process again, perhaps using a critical value with much higher degrees of freedom. We know that it’s going to take lots more batteries to cut the margin of error to a third of what it was. Alternatively, we know it will take a sample about 9 times as large, 18(9) = 162 batteries, since the margin of error was decreased to a third of its size. (The standard error involves the square root of the sample size in its denominator.)

3. Resources ActivStats*

• Chapter 12. The interactive tool in ActivStats that shows confidence intervals and P-values dynamically is valuable in cementing student understanding. One activity in this chapter differentiates the effects of smaller sample size on degrees of freedom from the effects of sample size on the standard error (in which we divide by the sample size). The change in standard error has a far greater impact than the spreading of the t-distribution with smaller * ActivStats is available at www.pearsonmylabandmastering.com/northamerica/mystatlab/

.

Chapter 13 Confidence Intervals and Hypothesis Tests for Means 13-8 degrees of freedom.

Decisions Through Data† Video Unit 19-21: Introduction to Inference Web Links

• You’ll find several good data sets at the DASL website. ( http://lib.stat.cmu.edu/DASL/) • Several websites have confidence interval applets, useful for demonstrating the key ideas that the population parameter is fixed while confidence intervals vary from sample to sample, capturing the population value most – but not all – of the time.

Basic Exercises 1.

Suppose a sample is selected from a population with mean µ and standard deviation σ. The sample mean is calculated to estimate µ. The sample standard deviation is also computed and found to be 120. a. b. c. d.

What is the standard error if the sample size is 36?

What is the standard error if the sample size is 64?

What is the standard error if the sample size is 100?

How does the standard error differ from the standard deviation of the sampling distribution of the sample mean?

2.

Suppose that you are constructing a confidence interval for µ. A sample is taken from a population with a nearly normal distribution and the sample mean and sample standard deviation are calculated. Find the appropriate t-value for each of the following. a.

A 90% confidence interval based on n = 30. b.

A 95% confidence interval based on n = 30. c.

A 99% confidence interval based on n = 30. d.

As the level of confidence increases, how will the resulting confidence interval be affected?

3.

Suppose that you are constructing a confidence interval for µ. A sample is taken from a population with a nearly normal distribution and the sample mean and sample standard deviation are calculated. Find the appropriate t-value for each of the following. a.

A 95% confidence interval based on n = 9. b.

A 95% confidence interval based on n = 25. c.

A 95% confidence interval based on n = 101. d.

As the sample size increases, how will the resulting confidence interval be affected?

4.

Suppose that you are constructing a confidence interval for µ. A sample is taken from a population with a nearly normal distribution. a.

If the sample size is 36, and the sample standard deviation is 24, at 99% confidence what is the margin of error?

† http://www.comap.com/contact/index.html

.

Chapter 13 Confidence Intervals and Hypothesis Tests for Means 13-9 b.

If the sample size is 36, and the sample standard deviation is 24, at 95% confidence what is the margin of error? c.

If the sample mean is 250, construct the 95% and 99% confidence intervals for µ. d.

Interpret these confidence intervals.

5.

Suppose that you are testing the hypotheses H0: µ = 1800 vs. HA: µ > 1800 and a sample of size 25 yields the sample mean of 1895 and the sample standard deviation of 250. a.

Compute the value of the t statistic. b.

Find the P-value associated with this t statistic. c.

At α = .05, what is the critical value for t? d.

State your conclusion at α = .05.

6.

Suppose that you are testing the hypotheses H0: µ = 85 vs. HA: µ < 85 and a sample of size 76 yields the sample mean of 84 and the sample standard deviation of 6.5. a.

Compute the value of the t statistic. b.

Find the P-value associated with this t statistic. c.

At α = .05, what is the critical value for t? d.

State your conclusion at α = .05.

7.

Suppose that you are testing the hypotheses H0: µ = 15 vs. HA: µ ≠ 15 and a sample size of 16 yields the sample mean of 17.5 and the sample standard deviation of 6. a.

Compute the value of the t-statistic. b.

Find the P-value associated with this t statistic. c.

State your conclusion at α = .05. d.

Would you expect the 95% confidence interval for µ to contain 15? Explain.

ANSWERS 1.

a. b. c. d.

20 15 12 The standard error is used when σ is unknown and must be estimated using s.

2.

a. b. c. d.

1.699 2.045 2.756 As the level of confidence increases, the interval gets wider.

3.

a. b. c. d.

2.306 2.064 1.984 As the sample size increases, the interval gets shorter.

4.

a. b. c.

10.9 8.12 95% confidence interval is 241.88 to 258.12 99% confidence interval is 239.1 to 260.9 .

Chapter 13 Confidence Intervals and Hypothesis Tests for Means 13-10 d.

We are 95% confident that the true mean is contained in the interval 241.88 to 258.12 and 99% confident that the true mean is contained in the interval 239.1 to 260.9.

5.

a. b. c. d.

1.90 between .025 and .05 (by the table); 0.0348 (by technology) 1.711 Since P-value = .0348 < .05 (alternatively, t24 = 1.90 > 1.711), we reject H0. There is strong evidence that the true mean is higher than 1800.

6.

a. b. c. d.

–1.34 between .05 and .10 (by the table); 0.0920 (by technology) –1.665 Since P-value = .0920 > .05 (alternatively, t75 = –1.34 > –1.665), we fail to reject H0. There is no evidence to suggest that the true mean is less than 85.

7.

a. b. c.

1.67 between .10 and .20 (by the table); 0.1163 (by technology) Since P-value = .1163 > .05 (alternatively, –2.131< t15 = 1.67 < 2.131), we fail to reject H0. There is no evidence to suggest that the true mean differs from 15.

d.

Yes, because we are not able to reject the H0 at α = .05.

.

14-1

Chapter 14 Comparing Two Means What’s It About?

We extend inference for means to comparing the means of two groups. We look at both confidence intervals and hypothesis tests. We emphasize that these methods are appropriate only when the groups are independent. This allows us to determine the standard error for the sampling distribution by adding variances (again). We advise students against pooling for inference about means of independent groups. We then examine data collected from subjects that have been paired.

We want to know how the groups differ, but instead of being concerned with how the group means differ, we focus on how large the mean (paired) difference is. It wouldn’t be appropriate to use the independent two-sample t-test because it addresses the wrong question and, of course, because paired groups are not independent. We discuss the assumptions and conditions and present examples of paired-t procedures for both hypothesis tests and confidence intervals. Many exercises require students to distinguish between groups of data that are matched and those that are independent.

Comments Students are now familiar with one-sample t-tests and confidence intervals, and can adapt those procedures to investigate the difference in two means. They see that the key is in adding the variances, and that makes it necessary that the two groups be independent.

We emphasize the two-sample t methods rather than the “pooled t” methods. The latter have the advantage of generalizing to the Analysis of Variance, but they carry the rather strong disadvantage of requiring the additional assumption of equal variances without offering very much in return in the way of power. Statisticians have generally concluded that the risk of using a pooled t when the equal variance assumption fails isn’t worth the small gain when it is true. We’ll address the issue of pooling when we work with ANOVA models in chapter 24.

Paired-t procedures are mechanically identical to the one-sample t procedures we first saw in Chapter 13. We simply apply those methods to the differences observed between the two measurements for each pair.

The problem students face is less in learning how to perform this kind of inference than in deciding when these procedures are necessary. At the beginning of the chapter the data came from two different and independent groups. Later on in the chapter, the data are two different measurements for the same individuals or two groups in which individuals are paired. To use the procedures from the earlier parts of Chapter 14, the groups must be independent. Paired measurements are clearly not independent, so the two-sample procedures are not viable. But more importantly, the question we ask of paired data is a different one. We actually don’t care whether the group means differ; we want to know about the mean difference within the pairs. The book tries to emphasize this by noting that pairing or matching is a form of blocking.

The recognition of what situation we have must be based on the way the data were collected, requiring students to Plan before they Do and Report.

Even students who are very adept at performing each of the individual procedures may find it quite difficult to decide which one is appropriate. We offer suggestions here for helping them learn to make those important decisions. .

14-2

Part II Understanding Probability Distributions and Statistical Inference

Looking Ahead Chapter 14 looks at inference about means when there are two sets of data using t-models. In the start of Chapter 14 our two groups will be independent. Under that condition variances add. In the latter part of Chapter 14 our two groups are paired, hence not independent. That means we’re unable to add variances, so we cannot use the methods we see in the earlier part of this chapter. It is critical that students start thinking about the independence issue now, because that is the vital determining factor in the decision they’ll soon face – whether to use two-sample or paired-t methods. The decision is forced by the design of the study and therefore not a “choice” a student can make.

The versatility of Gosset’s t in serving as a useful model for a wide variety sampling distributions is highlighted again in Chapter 18, where we discuss inference for regression.

Class Do’s Start by presenting the class with a two-sample example to consider, and expect them to be able to do the inference. As we’ve done in the past few chapters, we propose testing a hypothesis and following it up with a confidence interval, allowing you to deal with both kinds of inference in one class.

Hypotheses. Students should be able to write the hypotheses correctly. Be sure the notation is clear and correct, using parameters with well-defined or mnemonic subscripts. In particular, the hypothesis is not that the means are equal—that doesn’t meet the requirement of specifying a parameter and a hypothesized value. Instead, we consider the difference of the means and hypothesize that it is zero. That will help students see that the needed standard error is the SE of the difference in the means.

Model. Students can also write the assumptions and conditions, needing only to apply those for the one-sample t methods to both groups. Note that this means checking each group for signs of Normality, so they need to do two histograms. Students may or may not foresee the need to add the assumption that the groups are independent, but you can return to that when the issue arises.

Mechanics. As always, write the sample statistics using proper notation. Sketch the curve, centered on the hypothesized difference (usually 0), mark the observed difference, and shade the proper tails. Then there are two problems to face: degrees of freedom and standard deviation. We’ll let technology tell us the degrees of freedom. And once again the secret for finding the standard error of the difference in sample means is in adding the variances. That requires believing that the groups are independent. There’s no condition to check for that; we just need to think carefully about how the data were collected. This is a key point—in the next chapter we’ll encounter situations where the groups are not independent and we’ll need a different technique.

Conclusion. By now, a familiar story: linkage, decision, and conclusion in context. Be sure that the conclusion clearly talks about the difference in the means of the populations.

Confidence Interval. As we often do, we follow the hypothesis test with a confidence interval to see just how big the difference in means might be.

Emphasize that we need to believe that Normal models are appropriate for both populations. We test the plausibility of that assumption by looking at the distributions of both groups. We can look at histograms, dot-plots, stem-and-leaf displays, or boxplots. We’re more worried about outliers and skewness when the groups are small; if the samples are large enough, we can simply rely on the robustness of the t-procedures and go ahead with inference. .

Chapter 14 Comparing Two Means

14-3

Some students will want to use two separate confidence intervals to see if there is a significant difference in the means. They will argue that if the two intervals overlap, then the means may be the same, but if the intervals do not overlap, the means are probably different. This analysis is not precise enough. We should instead look for a significant difference using a single confidence interval for the difference in means. If 0 is in the interval, the means may be the same; if not, then we can conclude they are probably different. This approach correctly adds variances. Because confidence intervals have margins of error based on standard deviations, to look for an overlap in two separate intervals is to incorrectly add the standard deviations. (It’s a Pythagorean issue: 3 + 4 = 7, but

32 + 42 = 5 .) When there are two groups, we must use two-sample methods.

Then move on to paired data section and show how this differs from independent groups.

As always, notation is critical. Proper notation enhances both understanding and communication.

Here we are examining differences, d, not the actual measurements themselves. Now our statistic is the mean of those differences, d , not the difference in means, y1 − y2 , even though they have the same value. Similarly, our parameter of interest is the mean of those differences,  d , not the difference of the means, 1 − 2 .

But it’s not just the notation and the formulas that must be different. Each stage of our procedure reflects the fact that we are looking at paired differences rather than independent measurements.

Hypotheses. Now the null hypothesis is about the mean of the differences,  d .

Model. We note that the data are paired, and check a histogram of the differences for plausible Normality – one histogram, not two as in the last chapter.

Mechanics. Our method looks like the one-sample t approach, but our statistic is d , and degrees of freedom are simply n – 1, the number of pairs – 1.

Conclusion. We talk about what these data reveal about the mean difference. Our statements say things like “There is strong evidence that the training results in an increased test score on average.”

Confidence Interval. Again, our interpretation is about mean difference, as in “We are 95% confident that training increases an individual’s test score by an average of between 8 and 11.5 points.”

When pairing is a form of blocking in a designed experiment, there is rarely any question about it.

But observational data may be matched for a variety of reasons, making the pairing less clear. We can’t assume that samples of 50 men and 50 women are paired just because there are 50 of each.

But we might learn subsequently that these are really 50 married couples and conclude that they should be paired. This potential ambiguity emphasizes the need to determine the details of the study design first and determine whether the data are paired at the Plan step, before performing any analysis.

After students have learned how to do inference for means, it’s important to help them decide when. At this point in the course, they can find confidence intervals and do hypothesis tests for situations involving one proportion, two proportions, one mean, two independent means, or matched pairs. They may (we hope) be very good at each individual procedure. That’s only part of the problem, however. Students also need to be able to read about the situation and the questions to be investigated, and then to decide which type of inference is called for. Some find this extremely difficult. Try to help them gain this key ability in several ways. .

14-4

Part II Understanding Probability Distributions and Statistical Inference

The Importance of What You Don’t Say Don’t say, “Suppose we know the standard deviations of the populations.” We (almost) never will.

Don’t use z instead of t if both groups are large. For inference about means, we always use t.

Always.

Don’t spend time on the pooled t-test. Don’t suggest that students pool to achieve an estimate for a common population variance. It’s rare that such an assumption is warranted, there’s no way to test it, and little is gained (perhaps a few degrees of freedom, but with additional risk). The book briefly discusses the pooled variance t-test, but we recommend against its use. We do note a special situation where pooling can be defended - in a properly randomized experiment. There the pooled t will generalize to the F-test for Analysis of Variance, a topic students will encounter in Chapter 24. At this point, though, it is better to encourage students to use the two-sample t procedure. It’s never wrong not to pool.

Discourage students from using the “df = smaller sample size – 1” guideline for degrees of freedom. That rule is sometimes offered, but it is overly cautious, and can make a sizeable difference in the results. If technology is available to calculate the full number of degrees of freedom, there’s no reason to short-change ourselves.

Don’t say “choose” two-sample or paired t. There is no choice. We are compelled to use the method that is appropriate based on our knowledge of how the data were collected. There is an appropriate method and an inappropriate method, not an option.

Don’t say we “lose” degrees of freedom when we use matched pairs. No, we don’t; they were never there. True, there are more degrees of freedom when we have two independent samples, but if the data are matched, those degrees of freedom don’t exist. Nothing was lost. And we gained the control of variability afforded by matched and blocked designs. Usually that makes for a more powerful analysis, even though there are fewer degrees of freedom.

Class Examples 1. Resting pulse rates for a random sample of 26 smokers had a mean of 80 beats per minute (bpm) and a standard deviation of 5 bpm. Among 32 randomly chosen non-smokers, the mean and standard deviation were 74 and 6 bpm. Both sets of data were roughly symmetric and had no outliers. Is there evidence of a difference in mean pulse rate between smokers and nonsmokers? How big?

Solution: Hypotheses. H 0 : S −  NS = 0 H A : S −  NS  0 Model. We have independent random samples, each less than 10% of the population, and are told that the data appear to be approximately Normal. OK to proceed with a 2-sample t-test.

Mechanics.

nS = 26

yS = 80 sS = 5

nNS = 32 y NS = 74 sNS = 6

t=

(80 − 74) − 0

= 4.15 52 62 + 26 32 P  0.0001 df  56 .

Chapter 14 Comparing Two Means

14-5

Conclusion. Because the P-value is so small, the observed difference is unlikely to be just sampling error. We reject the null hypothesis. We have strong evidence of a difference in mean pulse rates for smokers and non-smokers.

Follow-up. How big is that difference?

 52 62  * * y − y  t  SE ( y − y ) = (80 − 74)  t +  = (3.1, 8.9) ( S NS ) 56 S NS 56   26 32    We can be 95% confident that the average pulse rate for smokers is between 3.1 and 8.9 beats per minute higher than for non-smokers.

2. Here is the saturated fat content (in grams) for several pizzas sold by two national chains.

We want to know if there is a difference in mean saturated fat contents for brand D and brand PJ pizzas. Be sure that in checking the conditions, students plot both sets of data.

17 8 6 11

Brand D Brand PJ

12 12 7 3

10 15 11 4

8 7 9 5

8 11 4 8

10 11 4 5

10 13 7 5

5 13 9

16 11

16 12

Solution: Hypotheses. The null hypothesis is that there is no difference is mean saturated fat content.

The alternative hypothesis is that there is a difference in mean saturated fat contents.

H 0 :  D −  PJ = 0 H A :  D −  PJ  0 Model.

Independent Groups Assumption – The two samples of saturated fat contents were chosen independently of one another.

10 8 6

Randomization Condition: There is no mention of randomness, so we will assume that these pizzas are representative of all pizzas by these two chains.

4 2 5.0

Nearly Normal Condition: Both distributions of saturated fat content are roughly unimodal and symmetric.

Since the conditions have been met, we can do a two sample t-test for the difference of means, with 32.76 degrees of freedom (from the approximation formula).

Mechanics.

nD = 20

yD = 11.25

sD = 3.193

nPJ =15

yPJ = 6.53

sPJ = 2.588

yD − yPJ = 4.72

df = 32.76 (from technology) .

10.0

15.0

Brand D

6 4 2

2

6 Brand PJ

10

14-6

t=

Part II Understanding Probability Distributions and Statistical Inference

(11.25 − 6.53) − 0  4.83

3.1932 2.5882 + 20 15 P-value  0.00003

Conclusion. Because the P-value is very low, we reject the null hypothesis. There is strong evidence to suggest that the two pizza brands have different mean saturated fat content.

Follow-up. The conditions have been met, so we can create a two-sample t-interval for difference in means, with 95% confidence.

 3.1932 2.5882  * * yD − yPJ  t32.76  SE ( yD − yPJ ) = 4.72  t32.76 +   = (2.73, 6.71)   20 15   We are 95% confident that the average saturated fat content for Brand D is between 2.73 and 6.71 grams higher than the average saturated fat content for Brand PJ. 3. The previous exercises address the important issue of using one confidence interval for the difference in means (correct) rather than seeing if separate confidence intervals for the two means overlap (incorrect). Here are some easy (madeFeed up) statistics you can use to demonstrate the point. They deal with a hypothetical experiment to see if there is any Diet A Diet B difference in the amount of weight gained by two groups Number of pigs 36 36 of pigs fed different diets. Based on these data we Mean weight gain 55 lb.

53 lb. estimate (with 95% confidence) that Diet A would Standard deviation 3 lb.

4 lb. produce a mean weight gain between 53.99 and 56.01 pounds, and Diet B between 51.65 and 54.35 pounds.

These two intervals overlap, making it appear that the two diets might actually produce the same mean weight gain (maybe 54.2 pounds, for example). This analysis is incorrect. It rests on the fact that the 2-pound difference is smaller than the sum of the two separate margins of error. Each of those margins of error is based on the standard error of the individual means, here 3 = 0.67 . We must instead sum the variances to examine the = 0.5 and 4 36 36 2

2

 3   4  + = 0.83 , less than 0.5 + 0.67. The 95% difference between the two means:    36   36  confidence interval for the difference in mean weight gains is 0.34 to 3.66 pounds. It does not look likely that the two diets produce the same results, because 0 is not in the confidence interval. (Note also that the correct method has 65 degrees of freedom rather than only 35.)

4. We offer an example that allows you to start with a hypothesis test, and then immediately find .

Chapter 14 Comparing Two Means

14-7

a confidence interval as further analysis.

One indicator of physical fitness is resting pulse rate. Ten men volunteered to test an exercise device advertised on television by using it three times a week for 20 minutes. Their resting pulse rates (beats per minute) were measured before the test began, and then again after six weeks. Results are shown in the table. Is there evidence that this kind of exercise can reduce resting pulse rates? How much?

Your students should see that the methods they know aren’t viable here. The before and after measurements are not independent. Someone will likely suggest looking at the change. Seeing that they should apply our one-sample t-procedure to the paired differences should allow the class to lead you through the solution.

Hypotheses. H 0 :d = 0 H A : d  0 (because we’re interested in a decrease) Model. Although the results are independent from one subject to the next, the measurements are paired by subject. A histogram of the paired differences is approximately unimodal and symmetric. Okay to do a matched pairs t-test.

Mechanics. n = 10, df = 9, d = −2.8 , sd = 2.53

t=

(−2.8) − 0

= −3.5

2.53 10 P-value = P( t9  −3.5 ) = 0.0034

Conclusion. With a P-value this low, we reject the null hypothesis. There is strong evidence that this form of exercise can reduce resting pulse rates.

Confidence Interval. The interval looks just like those we saw in Chapter 13; only the notation changes to represent the differences. We choose 95% as our confidence level. d  t9*  SE(d ) = −2.8  2.262

2.53

= (−4.61,−0.99)

10

We are 95% confident that six weeks of this exercise program can produce a mean decrease in resting pulse rate of between 1 and 4.6 beats per minute.

Resources ActivStats* • Lesson 13: Comparing Two Means Web Links • The Fast Food Nutrition Fact Explorer offers a lot of data for comparing groups (http://fastfoodnutrition.org/). * ActivStats is available at www.pearsonmylabandmastering.com/northamerica/mystatlab/

.

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Part II Understanding Probability Distributions and Statistical Inference

• Statistics Canada also offers data that allows you to compare various cities, regions, and provinces (www.statcan.gc.ca) Basic Exercises 1.

Suppose that two independent random samples were selected from two populations that are normally distributed. The table shows the data for each sample.

Sample 1 30 36 29 25 40 20 a. b. c. d.

2.

Calculate the two sample means.

What is the estimated difference between means?

Calculate the two sample standard deviations.

What is the standard error of the estimated difference between means?

Suppose that you are testing the hypotheses H0: µ1 – µ2 = 0 vs. HA: µ1 – µ2 ≠ 0 and obtained the following results from independent samples selected from two populations.

Sample 1 n1 = 60 1 = 122 s1 = 22.5 a. b. c. d.

3.

Sample 2 26 32 34 38 30 36

Sample 2 n2 = 55 2 = 125 s2 = 19.9

Compute the value of the t statistic.

What are the degrees of freedom?

What is P-value associated with the t statistic?

State your conclusion at α = .05.

Suppose that you are testing the hypotheses H0: µ1 = µ2 vs. HA: µ1 > µ2 and obtained the following results from independent samples selected from two populations.

Sample 1 n1 = 35 1 = 13.8 s1 = 2.5 a. b. c. d.

Sample 2 n2 = 45 2 = 11.5 s2 = 3.2

Compute the value of the t statistic.

What are the degrees of freedom?

What is P-value associated with the t statistic?

State your conclusion at α = .01. .

Chapter 14 Comparing Two Means 4.

Suppose that two independent random samples were selected from two populations that are normally distributed. The following results are obtained.

a. b. c. d.

5.

14-9

Sample 1 Sample 2 n1 = 20 n2 = 25 1 = 55.2 2 = 56.9 s1 = 5.1 s2 = 4.8 What is the estimated difference between means?

What is the standard error of the estimated difference between means?

Construct a 95% confidence interval for the difference between means.

At α = .05, can you conclude that the true means are the same? Explain.

Suppose that two independent random samples were selected from two populations that are normally distributed. The table shows the data for each sample.

Sample 1 145 133 150 120 115

a. b. c. d.

6.

Sample 2 158 162 170 150 135 129 144 What is the estimated difference between means?

If we assume that the population variances are the same for both groups, calculate the pooled estimate of the variance?

What are the degrees of freedom for the t distribution?

Construct a 90% confidence interval for the difference between means.

Suppose that you are testing the hypotheses H0: µ1 = µ2 vs. HA: µ1 < µ2 and obtained the following results from independent samples selected from two populations. Assume that the population variances are equal.

Sample 1 n1 = 100 1 = 68.9 s1 = 12.5 a. b. c. d.

7.

Sample 2 n2 = 80 2 = 74.9 s2 = 12.9

Compute the value of the t statistic.

What are the degrees of freedom?

What is P-value associated with the t statistic?

State your conclusion at α = .05.

Endurance measurements (in minutes) were made on a sample of individuals both before and after they participated in a fitness program. .

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Part II Understanding Probability Distributions and Statistical Inference

a. b. c. d.

8.

Suppose that you are testing the hypotheses H0: µd = 0 vs. HA: µd ≠ 0 and obtained the following paired observations. Assume assumptions are met for the paired t test.

a. b. c. d.

9.

Before After 20 22 15 18 22 20 30 38 22 32 20 26 15 15 25 22 Compute the “After – Before” pairwise differences.

What is the mean of the pairwise differences?

What is the standard deviation of the pairwise differences?

What is the standard error of the mean difference?

Pre – Post Pre Post 140 136 4 –2 133 135 150 148 2 142 135 7 132 128 4 135 135 0 151 144 7 –6 133 139 148 140 8 147 141 6 Compute the value of the t statistic.

What is the number of degrees of freedom?

What is P-value associated with the t statistic?

State your conclusion at α = .05.

Suppose that you are testing the hypotheses H0: µd = 0 vs. HA: µd < 0 and obtained the following paired observations. Assume assumptions are met for the paired t test.

Pre – Post Pre Post –3 5 8 –4 6 10 4 4 0 –3 6 9 –4 2 6 8 6 2 –5 9 14 7 6 1 a.

Compute the value of the t statistic. .

Chapter 14 Comparing Two Means b. c. d.

10.

14-11

What is the number of degrees of freedom?

What is P-value associated with the t statistic?

State your conclusion at α = .05.

Suppose that the time it takes to complete a certain task on a shop floor was measured for 16 workers. After reconfiguring the workspace, the time it takes to complete the same task was measured again for the same 16 workers. The mean and standard deviation of the pairwise differences (defined as “Before – After” reconfiguring the workspace) is found to be 10 minutes and 6.8 minutes, respectively. a. b. c. d.

What is the standard error of the mean difference?

At 95% confidence, what is the margin of error?

What is the 95% confidence interval?

At α = .05, is there a significant difference in the time required to complete the task before and after reconfiguring the workspace?

ANSWERS 1.

a. b. c. d.

30 and 32.67 –2.67 7.24 and 4.32 3.44

2.

a. b. c. d.

–0.76 113 .449 At α = .05, we fail to reject the null hypothesis. There is no evidence that the difference between true means of group 1 and 2 is not equal to zero; the means are equal.

3.

a. b. c. d.

3.61 78 < .001 (by the table); 0.0003 (using technology) At α = .01, we reject the null hypothesis. There is strong evidence that the true mean of group 1 is larger than the true mean of group 2.

4.

a. b. c. d.

–1.70 1.49 (–4.71, 1.31) Yes, because zero falls into the 95% confidence interval.

5.

a. b. c. d.

–17.11 223.46 10 (–32.97, –1.25)

6.

a.

–3.15 .

14-12

Part II Understanding Probability Distributions and Statistical Inference b. c. d.

178 0.00096  0.001 At α = .05, we reject the null hypothesis. There is strong evidence that the true mean of group 1 is smaller than the true mean of group 2.

7.

a. b. c. d.

2, 3, –2, 8, 10, 6, 0, –3 3 4.69 1.66

8.

a. b. c. d.

2.10 9 .065 At α = .05, we fail to reject the null hypothesis. There is no difference between Pre and Post measurements.

9.

a. b. c. d.

–2.16 7 .034 At α = .05, we reject the null hypothesis. There is strong evidence that the true mean of group Pre is smaller than the true mean of group Post. Pre measurements are significantly lower than post measurements.

10.

a. b. c. d.

1.7 3.62 (6.38, 13.62) Yes, because the interval does not contain zero.

.

15-1

Chapter 15 Design of Experiments and Analysis of Variance (ANOVA) What’s It About?

We consider observational studies and experiments, emphasizing the importance of controlled randomized experiments in establishing cause-and-effect relationships. We discuss the principles of control, randomization, replication, and blocking, and introduce – informally – the notion of statistical significance. We look at completely randomized one- and two-factor designs, as well as blocked and matched designs. We discuss the importance of control treatments, the use of blinding and placebos, and the problem of confounding variables.

Then we consider analyses of data collected in randomized experiments. We looked at equality of means for two groups in Chapter 14. Here, we extend our concern to more than two groups. In Chapter 14, we saw that the ratio of the difference of two means to its standard error was the natural statistic to use to test the null hypothesis and that the t-model gave us its sampling distribution. Now we have the generalization of that ratio: the ratio of two mean squares, modeled by the F-distribution. We present step-by-step examples of hypothesis tests. In the case that the null hypothesis is rejected, we may want to go further and examine confidence intervals for pairwise differences. We do this using the Bonferroni adjustment.

Comments This chapter is packed with important ideas and new vocabulary. Students usually grasp the issues in general, but are easily confused about the differences between confounding and lurking variables or between completely randomized and blocked designs, or about the different reasons for using randomization, control groups, and blocking.

The concepts in this chapter are fundamental to practical statistical literacy. You can’t open a medicine without encountering a report on the (randomized, blinded) clinical trial in which the medicine was tested prior to being approved by the FDA. Many people find it hard to grasp the fundamental importance of deliberately applying randomization. Doctors prefer to treat their patients with the therapy they think best rather than assign one at random. House painters prefer to select the paint brand they think will be top-quality rather than get one at random. But without randomizing, we cannot generalize experimental conclusions or assign cause and effect.

ANOVA becomes difficult to teach when too much time is spent on the formulas themselves.

Rarely will students need to calculate these quantities by hand. More important is to emphasize that the F-statistic measures the ratio of the variation among the group means to the (average) variation within each group. Students intuitively know that large differences lead to rejecting the null hypothesis, and that differences are easier to discern when the variation is smaller. Students need to understand that the variation within the groups is based on an average of all the groups and so, that variation needs to be similar for the average to make sense. The assumption of equal variances is checked by the equal variance condition. They’ve used residual plots to check conditions for regression, so this check is natural.

Class Do’s Vocabulary is important again here. We have observational studies and experiments; confounding and lurking variables; treatments, levels, and factors; single and double blinding; complete randomization, matching and blocking; explanatory and response variables. Insist that students always use the terms properly. We offer many exercises that summarize research and ask students .

15-2 Part II Understanding Probability Distributions and Statistical Inference to analyze the designs. Not only does this help them sort out the differences among designs and the vocabulary to describe them, but it pays off now when they have to make their own designs and later on when they start making statistical inferences about the results.

Emphasize the differences among sampling, observational studies, and experimentation. When our objective is to learn something about a population, we must start with a random sample of that population. When our objective is to see if there is a difference in the effects of two treatments, we use available subjects and divide them randomly into groups. At other times, we just use an existing situation, neither choosing subjects at random nor imposing treatments. These are very different kinds of research. Only when we have a random sample of a population can we generalize the findings of our study to the entire population.

Experiments rarely allow generalization – nor are they intended to. Students often mistakenly believe that an experiment is flawed if the subjects were not chosen randomly. Not so. A welldesigned experiment can tell us that a certain treatment caused a certain response in our group of subjects, something a sample or an observational study can never do. Such experiments may require replication on several other groups of subjects before the results are generalized.

Observational studies can be very useful, but they are in some ways the worst of both worlds. They are neither based on random samples nor do they randomly impose treatments. The results then cannot be generalized across the population nor can they be attributed to cause and effect. While there are many situations where observational studies provide us with the best information we have, at best they can establish the existence of an association between two variables. That can be important. A recent medical study revealed that among men who have had a heart attack, those with a higher level of a certain protein in their blood are at greater risk to have a second attack. In situations like this, students should think of three possibilities.

1. X causes Y: The protein causes heart attacks.

2. Y causes X: The risky condition of the patient’s heart produces the protein.

3. X and Y both result from a third (lurking) variable: Some other unknown aspect of the patient’s physical condition or body chemistry causes the second heart attack and also produces the protein as a harmless by-product.

Which is it? We don’t know. The fact that an observational study can’t tell us doesn’t make the study worthless, though. Knowing that the mere presence of this protein signals greater risk is valuable information for doctors working with heart attack patients.

It’s necessary – but not sufficient – that students be able to design a good experiment. Be sure they also understand the purpose of each component of the design. If we are to infer causation, we must try to ensure that the only possible explanation for any difference in outcomes is the difference in treatments. We control all sources of variation we can think of. We use a control group to help us know that observed effects are due to the treatment itself, rather than some unnoticed factor. We randomize to equalize unknown sources of variation. If we cannot control something important (like gender), we can at least block to reduce the variability. And we replicate to have as much data as possible, leaving us less at the mercy of some unusual and unknown issue present in our only subject.

The issue of blocking is particularly thorny. Students should not think that every experiment has to be blocked. Blocking in an experiment is akin to stratifying in sampling. We stratify when we have reason to believe that certain subsets of our population may have a different response than others, and we want to be sure that all these groups are properly represented in the sample. A simple random sample might, by chance, over- or undercount some groups. In an experiment, we may .

Chapter 15 Design of Experiments and Analysis of Variance (ANOVA)

15-3

suspect that some issue we cannot control may introduce variability in the response. Suppose that the drug we are testing works effectively. That should show up as a difference in response between the experimental and control groups. However, if both groups are mixed sex and men and women respond differently to the drug, then the variability between the sexes can drown out the true effect of the drug in each sex. We cannot cope with this problem by randomization. (It’s hard to randomly assign people to a sex.) Instead, we block by sex to reduce this variability. We might block our subjects by age, by exercise habits, or any other pre-existing condition that we suspect might change the response. As an extreme form of blocking, we might pair our subjects by some variable, say cholesterol level, matching the two highest and flipping a coin to decide which is assigned to which treatment group, then repeating with the next two, and so on.

We must be alert to the possibility of lurking or confounding variables. These are not the same thing.

• Lurking variables are a common problem in observational studies, when an apparent association between two variables is really just common response to a third unseen variable. A commonly cited example involves a positive association between ice cream sales and drownings. What is explanatory and what is response? Should we conclude that when people are saddened to hear of drownings in the news they attempt to cheer up by eating ice cream?

Or, more likely, that it’s dangerous to eat ice cream before you go swimming? In fact, the explanatory variable is probably summer heat, leading to both more ice cream sales and more drownings. Another favourite example is the strong positive association between the number of firefighters at a fire and the amount of damage. Perhaps you shouldn't call the fire department… The lurking variable is the size of the blaze, which "causes" both damage and determines how many fire fighters show up. Beware: don’t let these examples mislead students into thinking that lurking variables are obvious or even silly. Potentially misleading relationships are often far more subtle. Lurking variable examples are cautionary tales warning us not to infer causation from association.

• Confounding is a different issue. Confounding arises when the response we see in an experiment is at least partially attributable to uncontrolled variables. Confounded variables vary together so that one cannot tease apart which is responsible for any observed effect.

Confounding can occur because of poor design in an experiment. If a bank offers both a low interest rate and low fee to one group of customers and a higher interest rate and higher fee to another, then they'll never be able to tell whether customers were more motivated by the difference in interest rate or the difference in fee. But confounding can also simply be structural. Any study that takes place over time risks that something else will happen during that time that will vary with the treatments and could be responsible for the observed effect. A store’s special promotion may appear to increase video rentals, but the marketing folks cannot be sure that’s really what did it if the weather was particularly bad during the trial period. Bad weather may have kept people indoors and induced them to rent more videos anyway. Any actual effect of the special promotion is confounded by the weather.

• A hint: lurking variables are most common in observational studies, confounding variables in experiments. In designed experiments, we avoid lurking variables by randomizing subjects and controlling treatments, hence an external variable couldn't affect our treatment variable.

Confounding, however, can show up in an experiment either through poor design or just because there is no reasonable way to avoid it. Things happen that are not under our control, but might be confounded with the factors, making it impossible to tell whether our treatment or the external variable was responsible for the response. .

15-4 Part II Understanding Probability Distributions and Statistical Inference

• Having said all that, what students call these variables is less important than the thoughtful recognition that variables other than those named in the study may mislead us about the results.

When we hear about a study, we should be sceptical and think about possible external variables. When we find (or think we may have found) some, we must take extra care in interpreting the results, and we should always be especially cautious about inferring causation.

If differentiating lurking from confounding helps students to understand this, then go for it. But it’s more important to focus on how to interpret results with care when other variables might affect the data.

Blinding is an important tool in experimentation, and students should understand its purpose. The response can be confounded when people involved in the experiment (subjects, technicians, etc.), knowing what is happening, and adjust some aspect of their behaviour. And the evaluation of the results can be similarly tainted by knowledge of the process. An experiment may not be blinded at all, may be single-blind, or double-blind, depending on how we deal with these issues.

Be sure that in any proposed design students clearly define their response variable. With all the other things to think about, this is commonly overlooked. And raise the issue of evaluating that response variable. After all, we are looking for a difference between the groups. We expect there will be some difference just because of random variation. The great unresolved issue is how big that difference has to be before we can attribute it to the effect of the treatment. This is the core issue of statistical significance. Plant that seed now—we will begin to seek a way to resolve the question next.

Introduce ANOVA with a review of the t-test for two means. Use box-plots of various situations to ask students what their intuition says about equality of means with more than two groups. (Watch out for extremely large or small group sample sizes where the intuition may not match well.)

Start right in with data. Show the box-plots and ask students whether they think there is evidence of differences in group means. Ask whether the assumptions and conditions of ANOVA are satisfied:

• We believe that all the groups have the same standard deviation. This is the same

“homoscedasticity” as regression. This is pretty easy to believe for randomized experiments because when we randomly assign subjects to treatments, the resulting treatments groups start out with the same underlying standard deviation. So we only need to assume that (under the null hypothesis of no difference in treatment effect) treatments that didn’t change the means also didn’t change the standard deviations. The assumption’s crucial here because the MSE needs to make sense. It is dangerous to average essentially different quantities (we learned that in the Simpson’s paradox examples of Chapter 4). To check a corresponding condition, we can look at box-plots by group and/or check the scatter-plot of residuals vs. predicted for uniform spread.

• We believe that the errors – the differences of the observations from the group means – are independent. (We need to think about how the data were collected. If they come from an experiment, were the treatments appropriately randomized? If from an observational study, were the respondents chosen at random?)

• We also believe that the errors are normally distributed. (Check a normal probability plot of the residuals—provided the standard deviations appear to be equal.)

.

Chapter 15 Design of Experiments and Analysis of Variance (ANOVA)

15-5

Examine the ANOVA table. Without getting too caught up in the sums of squares and degrees of freedom, look at the two mean squares. The error mean square estimates the (average) error variance across the groups. So, its square root (sp) should give a reasonable estimate of the spread of the box-plots. Does it? Now, if the group means are equal, the treatment mean square should be about the same size as the error mean square. Is it? If it’s a lot larger, the F-test will reject the null hypothesis, and the larger the ratio, the more evidence there is against the null hypothesis. That’s the F-test in a nutshell.

Be careful to say what rejecting the null hypothesis doesn’t say. We know that all the means are not equal, but we don’t know which means are different. For that we need to do multiple tests, and when we do many tests, we must beware of the Type I error rate, so we use special methods for multiple comparisons.

The Importance of What You Don’t Say Don’t talk about a random sample of subjects. Students readily confuse the idea of sampling at random from a population with the idea of randomly assigning subjects to treatments. There is no need for the subjects in an experiment to be selected at random from some population. Often that would make no sense. For example, if we want to know which of two “quick-start” guides our customers like best, it would make no sense to sample from the public at large; we’d only look at those who purchased our product and received (at random) one or the other guide.

Don’t dwell on the sums of squares formulas for getting to the mean squares. Students need to understand the concepts and be able to interpret ANOVA output from software packages or a calculator. Don’t distract them from these central issues with scary formulas. If they can conduct a hypothesis test using software or a calculator to do the mechanics, they will be fine.

Class Examples 1. You can build much of the discussion around a single example of an experiment. Here’s another one. A company that manufactures fertilizer designs a field study, hoping that they will be able to claim in advertising that tomatoes grown with their fertilizer taste better. They specify three treatment groups: no fertilizer, half the specified amount, and the amount specified in the package instructions.

• Start with the randomized single factor design. Note that we don’t need a random sample of all the tomatoes in the world. Randomly assigning tomatoes to the treatment groups should balance out any inherent differences in the plants themselves. We don’t want to make the choices ourselves, as we might (even subconsciously) put the better-looking plants in the group that will get the fertilizer. We control all the other factors to make everything equal: temperature, light, soil, and so on. And we blind our tasters so that their evaluations of the fruit will not be influenced by the knowledge that the plant received special fertilizer.

• We can then add the issue of watering the plants or allowing nature to take her course. That introduces a second factor with differing levels. Our treatments now involve combinations of levels of the two factors, but the experiment is still completely randomized because each plant is equally likely to end up in any of the treatment groups.

• When we recognize potential sources of variability that we cannot control (different sources of the tomato plants here, or variables like sex or age in medical studies), we can use a blocked design, essentially running parallel experiments on the different groups of subjects. .

15-6 Part II Understanding Probability Distributions and Statistical Inference 2. Causation is an interesting issue. The case against smoking is instructive. A retrospective study that finds most people with lung cancer were smokers is inconclusive. A prospective study that follows a group of smokers and non-smokers for several years is a little better. But even when it finds a greater frequency of lung cancer among the smokers, we cannot attribute that to the effects of smoking. Perhaps the lifestyles of smokers differ from non-smokers in other ways, and those factors cause cancer. Perhaps there is something about a person’s body chemistry that both makes them enjoy smoking and also contributes to the later development of cancer.

Or perhaps the earliest subtle symptoms of precancerous changes in the body are soothed by smoking—in other words, lung cancer could cause smoking! Observational studies alone cannot resolve this issue, and an experiment is out of the question. Why? All we need to do is get a large group of 10-year-olds, randomly assign half of them to be smokers and the other half never to be around even second-hand smoke, then wait about 60 years and see what happens...

3. Turn this conclusion, from Newsweek, over to your students and have them tear it apart.

“Of all pre-college curricula, the highest level of mathematics one studies in secondary school has the strongest continuing influence on bachelor's degree completion. Finishing a course beyond the level of Algebra 2 (for example, trigonometry or pre-calculus) more than doubles the odds that a student who enters postsecondary education will complete a bachelor's degree.”

Then ask your students to propose a study design that might enable them to able to draw this conclusion (at least the causal interpretation of “influence on” and “doubling the odds”).

Discuss why it would be difficult and probably unethical to perform that study.

4. Spend some time talking about the ethics of experimentation. Most of us are okay with withholding fertilizer from a tomato plant or offering a better deal to some (randomly selected) credit card customers, but we recoil at the knowledge that treatment for syphilis was withheld from human subjects during the infamous Tuskegee experiment. There are many gray areas in between.

• Is it okay to use animals for experiments? Does it matter whether we are trying to develop treatments that may potentially save human lives, or just to develop cosmetics that won’t damage people’s eyes?

• Under what conditions would people consent to try an unapproved experimental treatment?

What is “informed consent”? If you have been told that you have only months to live, would you volunteer for an experiment? Would you want to be part of the treatment group or get the placebo?

• Is it okay to determine whether mastectomies are an effective treatment for breast cancer by randomly deciding which patients have mastectomies and which a less disfiguring procedure?

• Is it acceptable to drill placebo holes in patients’ skulls so that they and their doctors won’t know whether or not they received an experimental implantation of nerve cells as a possible treatment for Parkinson’s disease? (Yes, that experiment was approved.)

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Chapter 15 Design of Experiments and Analysis of Variance (ANOVA)

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5. Discuss the Salk polio vaccine trials.

• Have your students ask their parents (or grandparents) about the polio epidemics of the early and mid-20th century. Get them to imagine a disease that strikes children, killing many and crippling others. Parents were afraid, with just cause.

• Parents took many desperate measures to try to prevent their children from contracting polio. Because the epidemics peaked in the summer, some blamed them on swimming, so children were forbidden to go swimming. One conjecture linked the disease to peach fuzz, so children were forbidden to eat fresh peaches.

• Amidst this climate of fear, someone seeks parental approval to try an experimental vaccine in the hope of preventing the disease. This vaccine involves injecting a small dose of polio virus into your child. If you were a parent, would you give permission?

• In 1955, one of the authors was a subject in this experiment. Randomly chosen school districts sought permission from parents to use their second graders as subjects. Those whose parents agreed were transported to the local vaccination site in the spring. Imagine a long line of kids, inching forward to get a shot. We stood there, watching and listening as each child in turn reached the front of the line and cried when the vaccine was administered. We waited for our turn to cry. About a month later, we made the trip again for the second dose. Another line, another turn to cry. After the summer polio season, the figures were checked; it was clear that the incidence of polio was dramatically reduced among those who got the vaccine compared to those who received a placebo injection of a saline solution. The federal government mounted a nationwide program to vaccinate all children, and in a few years the disease was conquered.

• Now imagine being 7 years old and being told in October that the shots you got last spring were fake, so now you have go get the real ones. Your interest in experimental design is further piqued (no pun intended) when you learn that your best friend doesn’t need any more shots because he got the real ones the first time. (So, eventually you write a Statistics textbook.)

• Think more about the ethics here. Some children who got the placebo shot died or were crippled by the disease when the vaccine might have saved them. But how could that be known in advance? And if some doses of the vaccine were a little too “live,” they may have actually caused the disease in children whose parents gave permission in hopes of protecting them.

• Perhaps one of the most interesting sidebars is that the experiment actually had a second design, approved by some schools, in which all the children whose parents approved got vaccinated and the others were used as controls. Surprisingly, the results of this design were not as convincing. It seems that parents with higher levels of education were more likely to give permission, and also likely to raise their children in more hygienic conditions—which put those children more at risk of polio! Children reared in less hygienic surroundings were more likely to contract a mild form of polio early in life, thus developing a natural immunity to the disease when they were older. This rendered the treatment and control groups not identical at the start of the experiment, confounding the results. Moral: There’s no substitute for double-blinded, placebo-controlled, randomized clinical trials. .

15-8 Part II Understanding Probability Distributions and Statistical Inference 6. Resting pulse rates for a random sample of 26 heavy smokers had a mean of 80 beats per minute (bpm) and a standard deviation of 5 bpm. Among 32 randomly chosen light smokers the mean and standard deviation were 77 and 6 bpm. Among 30 non-smokers the mean was 74 bpm with a standard deviation of 5bpm. All three sets of data were roughly symmetric and had no outliers. The ANOVA table shows: Analysis of Variance Table Source Sum of Squares DF Mean Square F-Ratio Smoking 502.364 2 251.1818 7.639 Error 2794.800 85 32.88 Total 3294.164 87

P-Value < 0.001

Is there evidence of a difference in mean pulse rate between these groups?

Hypotheses. H 0 :  HS =  LS =  NS vs. HA: At least one mean is different from the others.

Model. We have independent random samples, the spreads of the three groups appear to be about the same, and we are told that the data appear to be approximately Normal. OK to proceed with an ANOVA Mechanics. The F-test shows that there is strong evidence that at least one mean is different from the others. The observed differences are unlikely to be due to sampling error. We reject the null hypothesis, concluding we have strong evidence of a difference in mean pulse rates across the three groups. In other words, we reject the hypothesis that the mean plus is the same for all three groups.

To get a 95% confidence interval for the difference between two groups, say light smokers and non-smokers, we would start by creating a confidence interval for the difference as in Chapter 14 replacing the two-sided t* critical value at  = .05 with a t** value at  = .05/3 = .01667.

For 85 degrees of freedom, t** = 2.163. So the Bonferroni confidence interval for the difference is:

(77 − 74)  t **  s p

1 1 1 1 + = 3  2.163 5.39 + = (0.037 , 5.963) bpm nLS nNS 32 30

As far as zero does not fall into the confidence interval, we conclude that there is a significant difference in the mean pulse rates between light smokers and non-smokers.

7. Here is the saturated fat content (in grams) for pizzas sold by ten chains:

.

Chapter 15 Design of Experiments and Analysis of Variance (ANOVA) Pizza Number 8 A 8 B 8 C 8 D 8 E 8 F 8 G 8 H 8 I 8 J

Mean 12.13 11.75 10.13 11.25 13.75 11.25 12.50 13.25 13.00 13.50

SD 2.53 1.58 3.64 2.49 3.20 1.16 2.88 3.20 3.93 3.66

Analysis of Variance Table Source Sum of Squares DF Mean Square F Ratio P-value Pizza 97.57 9 10.841 1.2415 0.2846 Error 611.25 70 8.732 Total 708.82 79 Do you find any systematic differences in the fat content from one brand to another? What concerns do you have about the design of the study?

The solution of this problem is similar to solution of problem 6.

Resources ActivStats* • Lesson 23 – ANOVA Web Links •

The Fast Food Nutrition Fact Explorer: http://fastfoodnutrition.org/

•

Statistics Canada offers data that allows you to compare various groups (www.statcan.gc.ca)

Basic Exercises 1.

In a completely randomized design, ten subjects were assigned to each of four treatments of a factor. Below is the partially completed ANOVA table.

Source Treatment (Between) Error (Within) Total a. b. c. *

DF

Sum of Squares 856.07

Mean Square

1177.97

What are the degrees of freedom for treatment, error and total?

What is SSE? What is MST?

ActivStats is available at www.pearsonmylabandmastering.com/northamerica/mystatlab/

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F-Ratio

15-9

15-10 Part II Understanding Probability Distributions and Statistical Inference d.

2.

What is MSE?

Refer to exercise 1. a. b. c. d.

3.

State the null and alternative hypotheses.

Calculate the F-ratio.

What is the P-value?

State your conclusion at α = .05.

Refer to exercise 2. Below is a graph showing side-by-side box-plots for the original data.

Boxplot of A, B, C, D 25

20

Data

15

10

5

0 A

a. b. c.

4.

B

C

D

Is the similar variance condition satisfied? Explain.

Is the nearly normal condition satisfied? Explain.

In addition to a side-by-side box-plots, what else can you use to check that the assumptions /conditions for ANOVA are satisfied?

Consider the following ANOVA table for a completely randomized balanced design in one factor.

Source Treatment (Between) Error (Within) Total a. b. c. d.

DF 2

Sum of Squares 2009

35

7023

Mean Square

F-Ratio

How many treatments are there in this design?

How many experimental units are assigned to each treatment (a balanced design means that the number assigned to each is the same)?

Complete the ANOVA table.

At α = .05, is there a significant difference between the treatments?

Explain. .

Chapter 15 Design of Experiments and Analysis of Variance (ANOVA)

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5.

Refer to exercise 4. a.

Is it appropriate to use the Bonferroni method for comparing group means?

Explain. b.

How many pairwise comparisons of group means are possible? c.

Given α = .05. Find the critical value t**. d.

Suppose that the sample means for treatments A, B and C are 65.33, 80.08 and 63.33, respectively. Based on the MSD (minimum significant difference) = 12.306, is there a significant difference between the means for treatments A and B? For A and C? For B and C? Explain.

6.

The ANOVA table below shows results for a completely randomized design involving four treatments each with 5 experimental units. The sample means for treatments A, B, C and D are 306, 312.6, 512 and 362, respectively.

Source Treatment (Between) Error (Within) Total a. b. c.

d.

7.

DF 3

Sum of Squares 137895

Mean Square 45965

16 19

18875 156770

1180

P-Value < .001

Is it appropriate to use the Bonferroni method for comparing group means?

Explain.

How many pair-wise comparisons of group means are possible?

Use the Bonferroni method to construct a 95% confidence interval for the difference between the means of treatments A and B. Is the difference significant?

Explain.

Use the Bonferroni method to construct a 95% confidence interval for the difference between the means of treatments C and D. Is the difference significant?

Explain.

Consider the following partially completed ANOVA for a multifactor design involving three levels of factor A and two levels of factor B.

Source Factor A Factor B Interaction Error Total a. b. c. d.

8.

F-Ratio 38.95

DF 2 1 2

Sum of Squares 1684.08 5251.04 71.58

23

8085.96

Mean Square

How many treatments are there in this design?

How many experimental units in this design?

What is the error degrees of freedom (dfError)?

What is SSE?

Refer to exercise 7. .

F-Ratio

15-12 Part II Understanding Probability Distributions and Statistical Inference a. b. c. d.

Calculate all mean squares (MSA, MSB, MSAB, and MSE).

At α = .05, is there a significant main effect due to factor A? Explain.

At α = .05, is there a significant main effect due to factor B? Explain.

At α = .05, is there a significant interaction effect? Explain.

ANSWERS 1.

a. b. c. d.

3, 36 and 39 321.90 285.36 8.94

2.

a.

H0: µ1 = µ2 = µ3 = µ4 HA: At least one mean is different from the others 31.92 < .0001; 3.02867E-10 (using technology) At α = .05, we reject the null hypothesis and conclude that at least one mean is significantly different from the others.

b. c. d.

3.

a. b. c.

Yes, the boxes are of equal width.

The box-plots indicate that the distributions are symmetric.

Histogram or normal probability plot of all residuals together.

4.

a. b. c.

3 12 for each treatment; 36 in total

Source Treatment (Between) Error (Within) Total

DF 2 33 35

Sum of Squares 2009 5014 7023

Mean Square 1005 152

F-Ratio 6.61

d.

Yes, the P-value associated with the F-ratio is .004 so we reject the null hypothesis and conclude that at least one mean is significantly different from the others.

5.

a. b. c. d.

Yes, because we did reject the null hypothesis.

3 For df = 33 and α = .05/3 = .01667, we have t** = 2.221 (using technology).

Those pair-wise differences greater than 12.306 are significant. Therefore, A and B are significantly different; A and C are not significantly different; B and C are significantly different.

6.

a

Yes, based on the F-ratio and P-value we can reject the null hypothesis and conclude that at least one mean is significantly different from the others.

6 For df = 16 and α = .05/6 = .00833, we have t** = 2.673 (using technology).

From the given ANOVA table: s p = MSE = 1180  34.351

b. c.

.

Chapter 15 Design of Experiments and Analysis of Variance (ANOVA)

d.

The Bonferroni confidence interval for the difference between A and B is 1 1 ( 306 − 312.6 )  2.673  34.351 + = −6.6  58.07 or ( −64.67 , 51.47 ) 5 5 Since it contains zero, the difference between A and B is not significant.

For df = 16 and α = .05/6 = .00833, we have t** = 2.673 (using technology).

From the given ANOVA table: s p = MSE = 1180  34.351 The Bonferroni confidence interval for the difference between C and D is 1 1 ( 512 − 362 )  2.673  34.351 + = 150  58.07 or ( 91.93 , 208.07 ) 5 5 Since it does not contain zero, the difference between C and D is significant.

7.

a. b. c. d.

8.

a.

6 24 18 1079.26

Source Factor A Factor B Interaction Error Total b. c. d.

DF 2 1 2 18 23

Sum of Squares 1684.08 5251.04 71.58 1079.26 8085.96

Mean Square 842.04 5251.04 35.79 59.96

F-Ratio 14.04 87.58 0.60

Yes, F = 14.04 with P-value = 0.0002 so we can reject the null hypothesis.

Yes, F = 87.58 with P-value = 2.45311E-08 so we can reject the null hypothesis.

No, F = 0.60 with P-value = .559 so we cannot reject the null hypothesis.

.

15-13

16-1

Chapter 16 Inference for Counts: Chi-Square Tests What’s It About?

We introduce  2 models. We see how to use the  2 statistic to test goodness of fit and homogeneity for distributions of categorical variables, as well as independence of categorical variables. We discuss the assumptions and conditions, present step-by-step examples, and look at how to use residuals to better interpret the results of our tests.

Comments Having spent three chapters on inference for means, we now return to questions we first raised in Chapter 4 about categorical variables. In particular, are observed differences among several proportions large enough to be significant?

To answer such questions we need a new family of models:  2 . Students first learned to work with Normal models and then made the transition to t-models. While the t-models looked familiarly “normal,” the  2 models are different. It’s important to consider the family in general so that students develop a “feel” for what their calculations and results mean. Fortunately, we always calculate the test statistic in essentially the same way, squaring the differences between what we observed and what our hypothesis says we should have expected. We always perform a test, rather than estimating with an interval. And the test always asks only whether the computed statistic is sufficiently large. So, mechanically, this chapter is pretty straightforward. But we do need to Plan.

While the first two  2 procedures look at distributions, the same kind of calculation can allow us to decide whether we think two categorical variables are independent. That question has been hanging over our heads since Chapter 4.

Looking Ahead The  2 test of independence tests the significance of associations between categorical variables.

Inference about associations between quantitative variables requires a linear regression tprocedure, coming in Chapter 18.

Class Do’s Help students understand the family of  2 models. Unlike the now familiar Normal and t-models, these models do not measure distances in standard deviation (or standard error) units, cannot take on negative values, and are skewed to the right. It’s harder to develop a feel for what the calculated result means. With z or t, a calculated result of 5 is pretty clearly significant; we can tell even without knowing the P-value, as it’s rare for anything to be five standard deviations from what was anticipated. With the  2 statistic, the picture is murkier. Values that are highly significant when there are few degrees of freedom fade into insignificance as the curve shifts to the right for greater degrees of freedom. You can help your students understand this in three ways:

• The more degrees of freedom, the more terms in the summation. We’d naturally expect the sum to be bigger when there are 10 cells in the table than when there are only 4. It takes a much larger value to surprise us when there are more degrees of freedom. .

16-2 Part II Understanding Probability Distributions and Statistical Inference

• It’s helpful to look at the  2 table. Pick a column, say 0.05. Look down the column to see how the critical value increases with higher degrees of freedom. Note that it can take a pretty large number to attain statistical significance when there are many degrees of freedom. You can watch this effect with the interactive  2 table in the Appendix of ActivStats. Drag down the table into higher degrees of freedom and watch the illustrative density at the top of the table as it shifts to the right and its shape gradually becomes more like a Normal.

• A fit that’s almost perfect would be somewhat unusual, so we don’t really expect to calculate a  2 value near 0. In fact, the expected value for a  2 model is equal to the number of degrees of freedom. We should be a little surprised by a calculated value of  2 = 9 when there are 3 degrees of freedom, but not at all surprised if there are 12 degrees of freedom. Having that initial feel for whether or not the calculated value is unusually large helps make the P-value we’ll find next a little less mysterious.

How much of the calculations must students be able to do? Well, they should be able to do it all – but most of the time they’ll use a calculator or software. Be sure students can:

• determine the number of degrees of freedom for any relevant table; • determine the expected value for any cell; • calculate the component of  2 for any cell; • write out an expression showing how the complete calculation would be done.

Be sure students can distinguish among the three situations, and choose the correct type of test.

• A goodness-of-fit test compares the distribution of observed outcomes for a single categorical variable to the expected outcomes predicted by a probability model to see if the model is viable. We start with one variable, one sample.

• A test of homogeneity compares observed distributions for several groups to each other to see if there is evidence of differences among the respective populations. There are several groups but still only one variable.

• A test of independence cross-categorizes one group on two variables to see if there is an association between them. Now there’s just one sample, but two variables.

The homogeneity/independence decision can be a difficult one. The data, usually summarized in a two-way table, look the same. We find ourselves in a position similar to that we faced in making the two-samples/matched-pairs decision when doing inference about means. Now, as then, it is the design of the study that matters. How were the data collected? Suppose we wonder what customers consider to be the biggest advantages and disadvantages of our product. Did we purposefully select a sample of registered owners, a sample of customers who purchased a competing product, and a sample of potential customers who own neither? If so, that’s a question of homogeneity. We’ll examine whether the distribution of answers is the same for each of the three groups. Or did we select one large sample of voters, asking them to respond to questions about our product and tell us whether they own one, too. If so, we’ve categorized each customer with respect to two variables and will examine whether the two variables are independent. .

Chapter 16 Inference for Counts: Chi-Square Tests

16-3

Another way to make the distinction is to think about the inference we’d like to make. In a clinical trial, we might give patients three treatments: a placebo, an herbal remedy, and a pharmaceutical, and then measure the responses (say in three different categories – “much better,” “about the same,” “much worse”). Compare that to an observational study relating baldness in men to heart disease. We might have three categories of baldness (none/little, some, much) and three categories of heart disease (severe, little, none). The two situations look the same, but the inference is quite different. The first concerns three specific treatments: placebo, herbal remedy, and drug. We don’t generalize from there. We’re interested in whether the proportions are the same across those three (and only those three) groups. On the other hand, with baldness, we are interested in whether baldness (as a variable) and heart disease are related. The specific levels of baldness are measured – they’re not specific groups – and represent the entire range of baldness. This is the essential difference. Homogeneity is about specific groups. Independence is about two variables.

Although the degrees of freedom are determined differently, the distinctions among the three types of tests have no impact on the actual calculation of  2 . This makes the mechanics fairly straightforward, and students readily learn to find the P-value of the test. Writing the hypotheses and interpreting the results, however, require clear understanding of the differences among the tests.

• The null hypothesis for a goodness-of-fit test is that the probability model is correct. (All faces of the die are equally likely to occur.) Our test assesses whether the observed results are consistent with that model. (Do the actual results from tossing the die vary so much from a perfect 1:1:1:1:1:1 ratio as to cast doubt on the die’s fairness?) The conclusion either rejects the model (the die is unfair) or fails to reject it (a fair die might behave like this one did).

• The null hypothesis for a test of homogeneity is that the two groups are drawn from populations in which the variable has the same model. (The distribution of eye color among boys is the same as it is among girls.) The test assesses whether the differences we found in the two sample distributions are consistent with that belief. (Are the percentages of the various eye colors in the sample of boys so different from those in the sample of girls that we can no longer believe the distribution of eye color is the same for all boys as it is for all girls?) Our conclusion either decides it’s unlikely that the populations follow the same model (frequencies of various eye colors differ by gender) or finds the observed discrepancy isn’t great enough to make us think so (if boys’ and girls’ eye colors don’t really differ, results like ours could be reasonably expected to turn up in samples like these).

• The null hypothesis for a test of independence is that there is no association between the two variables. (Smoking is independent of level of education.) The test assesses whether the observed differences in conditional distributions are greater than sampling variability reasonably allows. (Is the percentage of smokers among the well-educated subjects so much lower than the percentage among those who had only been to high school that the difference can’t be attributed to sampling error?) Our conclusion claims either that there is an association (well-educated people are less likely to smoke cigarettes) or that we don’t have convincing proof of that (there’s no evidence of an association between smoking and education level).

As usual, pay attention to the assumptions and conditions. Be sure students recognize that  2 tests are appropriate only for categorical data. Beware of this when you search for other examples to use for discussion in class. There are textbooks out there (written by non-statisticians) that incorrectly apply these procedures to variables like weight. .

16-4 Part II Understanding Probability Distributions and Statistical Inference We check to see that our sample is large enough by looking at the expected counts in each cell.

• All expected cell frequencies must be at least 5.

• That’s expected counts, not observed counts.

• Note that while observed counts must be integers, expected counts usually are not.

• Students must demonstrate that they checked this condition by actually showing all the expected frequencies. They can either find them themselves or copy them off a calculator- or computer-generated table, but it is not sufficient to just claim that it’s true without showing the evidence.

• If a small number of cells have expected counts slightly below 5, it’s okay to proceed with caution. However, if you end up rejecting the null hypothesis, check the  2 components to be sure that it was not the offending cells that triggered the rejection.

• When it makes sense to do so, it’s better to combine cells to achieve expected counts that are sufficiently large.

When you reject the  2 null hypothesis, the initial conclusion you reach is rarely satisfying.

We’ve decided the die is not fair. That just makes us want to know what’s wrong with it. We can identify which cells contributed most to the high value of by examining the components of the calculation. A good rule of thumb is to think of each component as the square of a z-score. Cells with components below 3 or 4 (z within  2) usually don’t arouse our suspicion. Cells with larger components do. Often, when the calculated value of  2 is large, there are one or two cells with glaringly large components that dominate the sum. So after our initial conclusion that a very low P-value causes us to reject the null hypothesis and conclude that there is evidence that the die is not fair, we get to add that there’s an indication that 3’s are unusually likely to appear (or that boys are more likely to have brown eyes, or that the high-school dropouts are over-represented among smokers).

Are  2 tests one-tailed or two? The test itself is almost always one sided. We reject the null only if we calculate an unusually large test statistic, so the rejection region is in the upper tail of the curve we draw. But whether a cell shows results higher or lower than expected, we square the deviation, always making it positive. We are essentially checking discrepancies in both directions at once.

You can ask the students what it would mean if the  2 statistic (say for a goodness-of- fit) is too low? They might say that the data fit the model too well – and that’s right. There is some question, for example, as to whether the data in Mendel’s original experiments on pea breeding were too good. The statistic for the genetic model proportions is extremely low. Perhaps some of his helpers were overzealous, knowing what Brother Mendel was expecting to see in his peas.

The Importance of What You Don’t Say Don’t attempt to express hypotheses symbolically. The mathematical notation required to express these hypotheses is complicated and not very meaningful to most students. Just say it in words.

Don’t suggest that it doesn’t matter whether it’s homogeneity or independence. It does matter. The hypotheses and the conclusions must be stated differently. Only the mechanics are the same. Don’t let students forget to Plan and Report properly just because the Do steps are the same. .

Chapter 16 Inference for Counts: Chi-Square Tests

16-5

Class Examples 1. Buy enough small packages of M&M’s to have one per pair of students—and treat yourself to one, too. Everyone should count the number of candies of each color in his or her package. Tell them the proportions claimed by the candy company: 13% brown, 14% yellow, 13% red, 24% blue, 20% orange, and 16% green. (Note: M&M’s no longer posts color proportions, and other kinds of M&M’s have different color distributions.) Calculate how many of each color you should have found in your bag (based on the total number of candies), and have each group do the same for theirs. Lead them through the goodness-of-fit test.

Hypotheses. H0 : The distribution of candy colors is as the manufacturer claims.

H A : The distribution is not what the manufacturer claims it to be.

Model. We have categorical data (counts), a random sample of M&M’s (we think), fewer than 10% of all the candies in the world (unfortunately), and we expected at least five of each color to show up in each group’s total (if we buy bags with enough candies for each group).

Mechanics. Demonstrate your calculation of  2 , and let the students do their own as they follow along. Explain  2 models, and find your P-value, having them each find theirs.

Conclusion. Many will fail to reject the null hypothesis, concluding that the content of their bags does not cast doubt on the claimed color distribution. Others will have P-values that are small enough to reject the null hypothesis. Use one of those to show everyone how to examine the components to be able to say more about how the distribution seems to be wrong.

2. Medical researchers enlisted 90 subjects for Treatment an experiment comparing treatments for Diagnosis Placebo St. J W Posrex Depression depression. The subjects were randomly 24 22 14 returned divided into three groups and given pills to No sign of take for a period of three months. Unknown 6 8 16 depression to them, one group received a placebo, the second group the “natural” remedy St. John’s Wort, and the third group the prescription drug Posrex. After six months, psychologists and physicians (who did not know which treatment each person had received) evaluated the subjects to see if their depression had returned.

Hypotheses. H 0 : The rate of recurrence is the same for all three treatments.

HA : The rate of recurrence differs for some treatments.

Model. These are counts of categorical data. Subjects were randomly assigned to treatments.

But now we run into an issue we need to resolve: expected counts.

• Add row and column totals to the table, and challenge students to tell how many subjects they’d expect to have returned to depression for each treatment (if the treatments are equally effective). They should see that those 60 subjects would be expected to divide 20-20-20 under .

Diagnosis Depression returned No sign of depression Total

Placebo 24 (20) 6 (10) 30

Treatment St. J W Posrex 22 14 (20) (20) 8 16 (10) (10) 30 30

Total 60 30 90

16-6 Part II Understanding Probability Distributions and Statistical Inference the null hypothesis. We can look at this three ways. Since 30/90 = 1/3 of all subjects got the placebo, 1/3 of the 60 with recurrent depression should have been in the placebo group.

Or, since 60/90 = 2/3 of the subjects suffered returning depression, we’d expect that to be true of 2/3 of the 30 who got the placebo. Or, there’s always the mindless explanation: row total column total/grand total = (60  30)/90 = 20.

• Display all the expected counts in the table (or make a separate table) to confirm that none is less than 5. It’s okay to proceed with a  2 test of homogeneity.

Mechanics. We now face another issue: degrees of freedom.

• Note that if you start with these row and column totals, you could freely assign a count to only two cells in any row. The remaining cell count in that row is now fixed (to make the row total come out right), as are the cell counts in the next row (to make all the column totals work). Draw a 3  5 grid on the board and make up some row and column totals.

Then ask students how many degrees of freedom there are. They should see that you can freely play with all but one row and all but one column. That’s (3 – 1)(5 – 1) = 8 degrees of freedom.

Here we have (2 – 1)(3 – 1) = 2 degrees of freedom. Write out the calculation of  2 .

 =

( 24 − 20) 2

( 22 − 20) 2

(14 − 20) 2

+ + 20 20 20 = 0.8 + 0.2 + 1.8 + 1.6 + 0.4 + 3.6 = 8.4 2

+

( 6 − 10) 2 10

+

(8 − 10) 2 10

+

(16 − 10) 2 10

P-value = P(  2 > 8.4) = 0.015 2

Conclusion. We’ll state a general conclusion and then note that one cell contributed greatly (3.6) to the high  2 value. Looking back at that cell in the table reveals more individuals there than we expected. Here goes: Because P < 0.05, we reject the null hypothesis. There is strong evidence that the tested treatments are not all equally effective in preventing the recurrence of depression. It appears that people who take the prescription drug Posrex are more likely to remain free of the disease than those who took a placebo or the natural remedy St. John’s Wort.

3. After having students do the calculations by hand at first, show them the calculator’s matrix approach as you do the test for Handedness Eye Left Right independence. As an example you can collect data about eye color color and handedness from your classes. Here are some sample data. Note Brown 6 36 that expected cell counts for lefties with green eyes and “other” are Blue 7 26 too small (3.83 and 2.67 are less than 5) to allow use of the  2 test.

Green 2 21 Other 4 12 We surmount this obstacle by combining “green” and “other,” as seen in the second table. We have one group of respondents who answered questions about two categorical variables, so this will be a test of independence.

Hypotheses. H0 : Handedness and eye color are independent.

H A : There is an association between handedness and eye color. .

Chapter 16 Inference for Counts: Chi-Square Tests

16-7

Model. We have counts of categorical variables. Our sample, though not random, may be representative of all students. After combining cells for students Handedness Eye color Left Right with green and other color eyes, all expected counts are greater 6 36 than 5.

Brown Mechanics. df= 2,  2 = 0.71, P-value = 0.70 Conclusion. With such a high P-value, we fail to reject the null hypothesis. There is no evidence of any association between handedness and eye color in high-school students.

Blue Other

(7) 7 (5.5) 6 (6.5)

(35) 26 (27.5) 33 (32.5)

Resources ActivStats*

• Chapter 15—“Chi-Square.” This lesson includes both a simulation illustrating independence and a return to the Titanic story.

Web Links • The Mars Candy Company has changed their website and no longer provides the color distribution for M&M’s. Here is part of an email written to a statistics teacher that may be helpful: Our color blends were selected by conducting consumer preference tests, which indicate the assortment of colors that pleased the greatest number of people and created the most attractive overall effect. On average, our mix of colors for M&M'S CHOCOLATE CANDIES is: M&M'S MILK CHOCOLATE: 24% cyan blue, 20% orange, 16% green, 14% bright yellow, 13% red, 13% brown.

M&M'S PEANUT: 23% cyan blue, 23% orange, 15% green, 15% bright yellow, 12% red, 12% brown.

M&M'S KIDS MINIS: 25% cyan blue, 25% orange, 12% green, 13% bright yellow, 12% red, 13% brown.

M&M'S DARK: 17% cyan blue, 16% orange, 16% green, 17% bright yellow, 17% red, 17% brown.

M&M'S PEANUT BUTTER and ALMOND: 20% cyan blue, 20% orange, 20% green, 20% bright yellow, 10% red, 10% brown.

Each large production batch is blended to those ratios and mixed thoroughly. However, since the individual packages are filled by weight on high-speed equipment, and not by count, it is possible to have an unusual color distribution.

•

GfK Roper Consulting gathers information on consumer preferences from around the world. Their website can provide information that can be used for this chapter: http://www.gfk.com/

* ActivStats is available at www.pearsonmylabandmastering.com/northamerica/mystatlab/

.

16-8 Part II Understanding Probability Distributions and Statistical Inference Basic Exercises 1.

For each of the following find the χ2 value. a. b. c. d.

2.

Suppose we have three categories A, B, and C. Assume that the historical distribution of observations among these three categories is 20%, 40%, and 40% respectively. A sample of size 250 is taken, and we find 75 observations in category A, 125 in category B, and 50 in category C. a. b. c. d.

3.

Calculate the standardized residual for category A. Calculate the standardized residual for category B. Calculate the standardized residual for category C. What do the standardized residuals tell us?

Suppose we have four categories A, B, C and D. Assume that the percentage of observations in each category is the same. A sample of size 500 is taken, and we find 100 in category A, 100 in category B, 140 in category C and 160 in category D. a. b. c. d.

5.

State the null hypothesis for the Chi-square goodness-of-fit test.

What are the expected frequencies for each category?

Calculate the χ2 statistic.

State your conclusion at α = .05.

For the Chi-square goodness-of-fit test performed in exercise 2, a. b. c. d.

4.

Right tail probability of .05 with 10 degrees of freedom.

Right tail probability of .01 with 10 degrees of freedom.

Right tail probability of .05 with 25 degrees of freedom.

Right tail probability of .01 with 25 degrees of freedom.

Which Chi-square test is appropriate in this situation? Explain.

What are the expected frequencies for each category?

Calculate the χ2 statistic.

State your conclusion at α = .01.

A survey about Internet usage included the following question “Do you bank online?” The data are shown below for adults over the age of 40 broken down by gender. Is the proportion of who banks online the same for men and women?

Bank online?

Yes No Total

Women 120 130 250

.

Men 157 93 250

Total 277 223 500

Chapter 16 Inference for Counts: Chi-Square Tests a. b. c. d.

6.

16-9

What is the correct null hypothesis?

What are the expected frequencies for each cell?

Calculate the χ2 statistic.

State your conclusion at α = .05.

The survey about online banking also included younger adults. Suppose the following results were obtained for adults under the age of 25.

Bank online?

Yes No Total

Women 175 125 300

Men 188 112 300

Total 363 237 600

Calculate the χ2 statistic to determine whether the proportions are equal.

State your conclusion at α = .05.

7.

A sample of size 1200 is taken and the following 3 by 4 contingency table shows the observed frequencies for the row and column variables. Use α = .05 to perform the Chisquare test for independence.

Row Variable X Y Z a. b. c. d.

A

Column Variable B C

D

100 60 40

120 100 80

200 100 100

80 100 120

What is the degrees of freedom for the χ2 statistic?

What are the expected frequencies for each cell?

Calculate the χ2 statistic.

State your conclusion at α = .01.

ANSWERS 1.

a. b. c. d.

18.307 23.209 37.652 44.314

2.

a. b. c. d.

The distribution of sample observations is the same as the historical distribution.

50 for A, 100 for B, and 100 for C 43.75 At α = .05, we reject the null hypothesis. The distribution of the sample observations is different from the historical distribution. .

16-10 Part II Understanding Probability Distributions and Statistical Inference

3.

a. b. c. d.

3.536 2.5 -5 There are large differences between observed and expected frequencies; we observe more than expected in A and B and less than expected in C.

4.

a.

Chi-square test of homogeneity. The null hypothesis is that the percentage of observations in each category is the same.

125 for each category 21.6 As P-value is small (alternatively,  2 = 21.6  cr2 = 11.345 ), we reject the null hypothesis; responses are not homogeneous (not equal across categories).

b. c. d.

5.

a. b.

The proportions of women and men who bank online are equal.

Bank online?

Yes No

c. d.

Women Men 138.50 138.50 111.50 111.50

11.081 As P-value =.0009 < α = .05 (alternatively,  2 = 11.081  cr2 = 3.841 ), we reject the null hypothesis; responses are not homogeneous (not equal across the gender categories.)

6.

As P-value =.278 > α = .05 (alternatively,  2 = 1.179  cr2 = 3.841 ), we fail to reject the null hypothesis; responses are homogeneous (equal across the gender categories.)

7.

a. b.

6

Row Variable X Y Z c. d.

Column Variable B C 125.00 125.00 90.00 90.00 85.00 85.00

A 83.33 60.00 56.67

D 166.67 120.00 113.33

53.133 As P-value = 1.10311E-09 < α = .05 (alternatively,  2 = 53.133  cr2 = 16.812 ), we reject the null hypothesis. There is strong evidence that the row and column variables are not independent.

.

17-1

Chapter 17 Nonparametric Methods What’s It About?

All of the inference methods we’ve seen thus far require that we make assumptions about the underlying distribution of the errors. There are times when those assumptions seem unwarranted. Nonparametric methods offer alternatives that don’t require distributional assumptions for many of the inference methods we have discussed. We pay for that freedom from assumptions with reduced statistical power and with methods that may not be as convenient or that may not come equipped with diagnostics.

Comments Unlike other chapters in the book, this one is not about any particular kind of data or kind of question about data. Instead, it is a survey of the situations already discussed, reconsidering each one from the point of view of distribution-free methods.

We first discuss data types, specifically numeric and ordinal data, and the usage of nonparametric statistics for these types of data.

The Wilcoxon Signed-Rank test is a nonparametric alternative for paired data. The examples here work with quantitative variables, but our focus is only on which one of each pair is bigger rather than on “by how much?” The Wilcoxon Rank-Sum or MannWhitney test compares two groups. We present an example in which only rank data are available, but the method could also be used by ranking quantitative values, for example, as a way to deal with skewness or outliers in the measurements. You might remind students of Tukey’s quick test, as another alternative for dealing with such data. Tukey’s test is a “slippage” test rather than an overall comparison of distributions, but the two methods are often used in the same situations.

The Kruskal-Wallis test extends these ideas to comparing several groups. Again, our example starts with rank data, but the method could apply to ranks found from measured responses.

If you have covered Chapter 15 on experimental design and ANOVA, then you might want to discuss the Friedman Test for a randomized block design. This test generalizes the Wilcoxon Signed-Rank test to the two-way ANOVA situation. If your students have not seen ANOVA, you may want to skip this section. It is written to address students who have that level of background.

Kendall’s Tau is a test for monotonicity in a two-variable relationship. It is worth discussing with your class that often when we consider the relationship between two quantitative variables, what we are really interested in is monotonicity rather than linearity. Linear regression doesn’t address that question.

Spearman’s Rank Correlation is a more direct replacement for correlation that simply substitutes ranks for quantitative values. It is especially useful when the data have skewed distributions or are non-linearly related for which the correlation coefficient is inappropriate.

.

17-2 Part II Understanding Probability Distributions and Statistical Inference The chapter concludes with a discussion of when nonparametric methods are most appropriate. That section can generate some good classroom discussion and is worth the attention.

Class Do’s Do relate nonparametric methods to their parametric counterparts. This is both a good way to present the nonparametric methods and a good review of the assumptions and conditions for parametric methods. It may be reassuring to students to have alternative methods to turn to when some of the assumptions and conditions fail.

Do point out that nonparametric methods don’t always test exactly the same hypotheses as their parametric alternatives. That’s neither good nor bad; just different.

Note that nonparametric methods often offer hypothesis tests without corresponding confidence intervals. There are methods for creating such intervals for some of the nonparametric approaches, but they are not as common in statistical software as the tests themselves. We have emphasized the value of using confidence intervals to assess effect sizes; that feature may be missing from some nonparametric methods.

The Importance of What You Don’t Say Don’t revert to talking about data “types.” Nonparametric methods were once presented as necessary in many social science and business applications because the available data were rankings and could not support the manipulations that quantitative data could support. But, in fact, nonparametric methods are also useful for quantitative data when the underlying assumptions of our parametric methods (such as being Normally distributed) are not met. Parametric methods are more powerful. For quantitative data, we should test whether the assumptions warrant the use of parametric methods first before, perhaps, choosing nonparametric methods instead.

Class Examples 1. It is a good idea to return to the examples of previous chapters and re-analyze them with the corresponding nonparametric methods. The differences and similarities in the analyses can improve understanding of both the nonparametric and the parametric methods.

2. Canadians are paying more for their communications services in a trend driven by increased spending on Wireless and Internet packages. In October 2015, the Canadian Radio-television and Telecommunications Commission said households spent an average of $203 per month on their communication services in 2014.

Suppose ten families in the city A are matched demographically with ten families in the city B, and their monthly communication expenses (rounded to $) are obtained. The data follow.

Family Pair 1 2

City A 215 204

City B 196 207 .

Chapter 17 Nonparametric Methods 3 4 5 6 7 8 9 10

232 178 199 223 203 208 175 231

17-3

196 186 154 187 205 207 150 189

An analyst uses α = .10 to determine whether there is a significant difference in monthly communication expenses between these two cities.

Suppose that difference is d = city A expenses − city B expenses.

Hypotheses. H0 : Md = 0

H A : Md  0 (two-tailed test) Model. The data consist of matched pairs, but we cannot use a paired t-test as the histogram below clearly shows that differences in the scores are not even approximately Normally distributed. Since the t-test is invalid, we have to use a nonparametric test instead. The Wilcoxon signed-rank test is designed for comparing paired data of this type. We test the null hypothesis that the median of differences (Md) is zero. We assume that the data are appropriately paired. We also assume that the distributions of expenses for both cities are symmetric.

Mechanics. We rank the pairwise differences according to their absolute values. As the differences for family pairs 3 and 6 are equal, we calculate their ranks as the average of their orders. See the table below.

Order 1 2 3 4 5 6

Difference 1 -2 -3 -8 19 25

Rank 1 2 3 4 5 6

.

17-4 Part II Understanding Probability Distributions and Statistical Inference 7 8 9 10

36 36 42 45

7.5 7.5 9 10

We sum up the ranks for all positive differences (T + ) and for all negative differences (T − ) .

T + = 1 + 5 + 6 + 7.5 + 7.5 + 9 + 10 = 46 T− = 2+3+ 4 = 9 We focus on the smaller of these two totals, and call it T = 9. This is our test statistic.

Conclusion. The Table W2 in Appendix B indicates the critical value of 11. As the test statistic is smaller than the critical value, at α = .10 we reject the null hypothesis, concluding that Md  0 and there is a statistically significant difference in monthly communication expenses between city A and city B. Resources ActivStats1 Web Links Many polling firms offer data that can be used for this chapter including Ipsos (http://ipsos.com/), Nanos Research (http://nanosresearch.com/), and Harris/Decima (http://www.harrisdecima.ca/).

Basic Exercises 1.

Suppose a company implemented a quality control program and has been operating under it for a year. The management wants to determine whether worker productivity has significantly increased since installation of the program. Company data contain figures for items produced per worker during a sample production runs one year ago. Productivity figures on the same workers are gathered now and compared to the previous records. The following data represent items produced per hour.

Worker 1 2 3 4 5 6

Before 5 4 9 6 9 8

After 11 9 9 8 4 7

1 ActivStats is available at www.pearsonmylabandmastering.com/northamerica/mystatlab/

.

Chapter 17 Nonparametric Methods 7 8 9 10 11 12 a. b. c. d.

2.

10 10 3 7 2 5

17-5

7 9 7 9 6 10

Why does the company’s statistical analyst choose the Wilcoxon signedrank test for analysis?

Write the null and alternative hypotheses.

Is it one-tailed or two-tailed test?

Given α = .01, what is the critical value for the test statistic?

For the Wilcoxon signed-rank test performed in exercise 1, a.

Calculate the sum of the ranks of all positive differences. b.

Calculate the sum of the ranks of all negative differences. c.

Find the test statistic. d.

If α = .01, what is the analyst’s conclusion?

ANSWERS 1.

a.

The data consist of matched pairs, but we cannot use a paired t-test as the histogram below clearly shows that differences in the production figures are not Normally distributed, but dramatically skewed to the right. We assume that the data are appropriately paired. We also assume that the distributions of productivity in both cases are symmetric.

b.

Suppose that difference is d = Before − After.

H0 : Md = 0 vs. H A : Md < 0 c.

One-tailed test .

17-6 Part II Understanding Probability Distributions and Statistical Inference

2.

d.

We set aside one tie (zero difference). The Table W2 in Appendix B indicates the critical value of 7 (one-tailed, α = .01, n = 11).

a.

We rank the pairwise differences according to their absolute values.

Order Difference 1 1 2 1 3 -2 4 -2 5 3 6 -4 7 -4 8 -5 9 -5 10 5 11 -6 + T = 1.5 + 1.5 + 5 + 9 = 17

Rank 1.5 1.5 3.5 3.5 5 6.5 6.5 9 9 9 11

b. c.

T − = 3.5 + 3.5 + 6.5 + 6.5 + 9 + 9 + 11 = 49 T = min(17 , 49) = 17

d.

Since the test statistic T = 17 is larger than the critical value of 7, at α = .01, the analyst fails to reject the null hypothesis. There is not sufficient evidence that the worker productivity has significantly increased since installation of the program.

.

18-1

Chapter 18 Inference for Regression What’s It About?

We return to the question of associations between quantitative variables, examining the regression model, with its underlying assumptions and conditions. We see that t-models describe the sampling distribution of slopes of regression lines. We present step-by-step examples of hypothesis tests and confidence intervals for slope with careful attention to proper interpretation of the results. Here we emphasize computer output, because that is the way that most people calculate and encounter regression.

Comments The last chapter closed with the chi-square test for independence, allowing us to check for evidence of an association between categorical variables. Now we learn to make inferences about associations between quantitative variables as well.

We last looked at linear regression back in Chapter 7. If you delayed introducing regression until now, students will need to see it and have a command of fitting regression by least square before attacking this chapter. Regression is a very important topic, involving a number of key concepts and skills. Students need to know how to create and describe a scatterplot and how to find and interpret the correlation. They need to understand the concept of residuals and be able to use a residuals plot to confirm the appropriateness of a linear model. They need to be able to find the equation of the regression line and interpret its slope and y-intercept.

Looking Ahead This is the first chapter of Part III, Exploring Relationships Among Variables. Chapters 18-21 are all about analyses in which a single quantitative response variable is modeled by one or more quantitative predictors. Chapter 18 introduces inference for regression and connects to the inference chapters that have preceded it. (Indeed, the close relationship is not surprising. With appropriately constructed indicator variables, inference for means and pooled variance inference for differences between means are just special cases of regression.)

Chapter 19 discusses some more sophisticated issues of fitting and evaluating regression models, concentrating especially on what we can learn by examining the residuals.

Chapters 20 and 21 discuss multiple regression models. In Chapter 20 we’ll see that the same principles as those introduced in Chapter 18 apply to multiple regression. In Chapter 21, we consider regression diagnostics and the challenges of building multiple regression models.

Class Do’s Start with a review of the key regression concepts from Chapters 6 and 7. If possible, have students work in groups, helping each other remember the concepts, the mechanics, and the interpretations.

• How do we make a scatterplot?

• What do we look for in a scatterplot?

• How do we find the correlation?

• What does the correlation mean? .

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Part III Exploring Relationships Among Variables

• What does R 2 mean?

• How do we create a linear model?

• How do we know if the linear model is appropriate?

• How do we create a residuals plot?

• What do we look for in the residuals plot?

• What does the slope mean?

• What does the y-intercept mean?

We also get to review the idea of a sampling distribution. Remind students about the sampling distribution models from Chapters 10 and 13. We know that sample means vary from sample to sample and follow a Normal model. Not knowing the population’s standard deviation, though, forces us to use the almost-Normal t-models when we do inference. Now we are interested in examining how slopes of regression lines vary from sample to sample. Recall the example of price and home size from Chapter 8. Have students imagine the scatterplot for the entire population, a dense cloud of points. We must believe that for this population there is an ideal line that captures the association between price and home size. Our sample produces a line that approximates the true line. Other samples would also produce such approximating lines, each with a slightly different slope. Slopes, then, vary from sample to sample. Each slope, b1, is an estimate of the true slope, 1 . The distribution of all such slopes has a t-model, with n – 2 degrees of freedom.

Knowing the model enables us to do inference about slope. Almost.

The t-model we want to use is appropriate only if the population behaves according to certain assumptions. When students understand the assumptions that create the perfect model of linear association (illustrated in Figure 18.4), they’re better able to remember the conditions they need to check.

• We believe that there is an underlying linear relationship between x and y that we can model with the least squares line. We check the Linearity Condition with a scatterplot. If the relationship is straight enough, we can find the least squares line.

• We believe that the errors – the (signed) distances of points from the ideal line, akin to residuals in the sample – are independent, that one person’s heavier-than-expected weight does not tell whether another person might be heavier or lighter than their height might predict.

(Check the residuals plot for random scatter.)

• We believe that at each value of x the y’s follow a Normal model. For example, a Normal model describes the prices of all the homes that are, say, 1200 square feet in size – or any other size. (Moreover, we believe that the means of all those Normal models lie on the line relating y and x, so this can be seen as a statement about the distribution of the errors around the ideal line.)

• We believe that all those Normal models have the same standard deviation. In other words, we believe the variability of weights is the same at each height. This assumption is called “homoscedasticity,” a term students don’t actually need to know. We can examine the residuals plot for uniform spread – checking whether the variance around the line is constant. If .

Chapter 18 Inference for Regression

18-3

the spread seems constant, then we can pool the residuals together and look at a histogram or Normal probability plot to check whether their distribution really does seem nearly Normal.

But what are we doing inference about? Help students see that even if there is no association present in the population, a sample might draw points that appear to show an association. The question we ask is whether the association we see in our sample is so strong that it could not just have popped up by chance during the sampling process. We want to know if the apparent association in our sample provides evidence that there must be an association in the population.

Our null hypothesis, then, is that in the population there is no association – in other words, that the ideal line is really horizontal: 1 = 0.

When a hypothesis test indicates that an association exists, have students create a confidence interval for the slope of the true line. Earlier in the course they learned to interpret slope and to interpret confidence intervals. That does not mean they’ll easily get the phrasing right when dealing with both.

• Slope: The model suggests that the price of houses in this market increase at about 113 dollars per square foot of living area.

• Confidence interval: We are 95% confident that the true mean home price in this market is between $207,322 and $216,611.

• Confidence interval for slope: We are 95% confident that the average price of houses in this market increase at between $108 and $118 per square foot of living area.

Invest time in looking at regression output. Most regression analyses are produced by statistics software packages. Be sure students can find, use, and properly interpret the key results.

It is a good idea to frame the model of this chapter as a model for the behaviour of one quantitative response variable based on another quantitative predictor variable. In later chapters we’ll see a quantitative variable modeled by categorical variables (ANOVA) and a quantitative response variable modeled by several quantitative predictors (multiple regression).

The Importance of What You Don’t Say Don’t say “dependent variable” and “independent variable.” We warned about this back in Chapter 8, and it is still true. We have by now made so much of checking for independence that to introduce the term in an entirely different context and meaning can only be harmful. Speak of response variables and predictors (or, when we get to ANOVA, possibly factors.)

Don’t use the phrase “Coefficient of Determination.” “R-squared” is what the statistics packages call it and what we’ve been calling it. This is another term the students don’t need to know at this point.

Don’t spend time on the formulas for calculating s, SE(b1), and so on. Students need to understand the concepts and be able to interpret regression output from software packages. Don’t distract them from these central issues with formulas. If they can conduct a hypothesis test and create a confidence interval using software or calculator to do the mechanics, that will be fine.

Class Examples 1. Review the concepts of regression by examining the association between anxiety level and math test scores. You can give students the worksheet and let students work on it in groups. By putting their heads together they should collectively remember enough to do everything, and .

18-4

Part III Exploring Relationships Among Variables

the discussion will help everyone get back up to speed on these concepts and skills.

Solution: We want to examine the relationship between anxiety level and math test score. There is a weak, negative association between anxiety level and math test score. The correlation between the scores is only – 0.525. Generally, lower anxiety levels are associated with higher math test scores. There is one student who scored much lower on the math test than we would have suspected, based on the student’s anxiety level. This point is close to a typical anxiety level, so it shouldn’t prove to be particularly influential. Both variables are quantitative, and the scatterplot shows an association that is roughly linear. We will create a linear model to predict the math test score from the anxiety level.

Source

Sum of Squares

df

Regression Residual

2925.03 7687.47

1 22

T e s t

80 60 40

2

4

6

8

Anxiety Level

r e s 20 i 0 d u a -20 l -40 s ( M / A )

Residuals Plot

Residuals

Dependent variable is: Math Test No Selector R squared = 27.6% R squared (adjusted) = 24.3% s = 18.69 with 24 - 2 = 22 degrees of freedom

100 M a t h

Mean Square

2925.03 349.431

F-ratio

8.37

Var iable

Coefficient

s.e. of Coeff

t-ratio

prob

Constant Anxiety Level

91.6583 -4.48624

11.02 1.551

8.32 -2.89

² 0.0001 0.0084

50

60

70

80

predicted(M/A)

The linear model is Math = 91.658 − 4.486( Anxiety) . This model only accounts for 27.6% of the variability in math test score. For each additional unit of anxiety level, the model predicts a decrease of about 4.5 points in math test score. The residuals plot has random scatter, indicating an appropriate model, but the low R 2 makes the model a poor predictor of math test score.

2. Students need to understand that a low correlation may be evidence of a weak association between two variables. But it may also simply arise from sampling error even when there is no association at all between the two variables. To demonstrate, have each student create a list of 20 random anxiety scores between 1 and 10 and a list of 20 random math test scores between 30 and 100. Look at the scatterplot and find the correlation. Clearly, there should be no association between random numbers, yet some will see patterns and many will have correlations of 0.20 or more. We need a way to see if the correlation is high enough to be meaningful – a hypothesis test, in other words. It turns out that we can test correlations by looking at the slope of the regression line. After all, a true correlation of 0 would make the slope of the ideal line also 0. We suspect that the slopes of the lines we fit to our random scatterplots, though not exactly 0, aren’t statistically significant.

3. Perform a hypothesis test and create a 95% confidence interval for slope (including careful attention to the assumptions and conditions) for the anxiety and math test data.

Solution: .

Chapter 18 Inference for Regression

18-5

Plan. We wonder whether the weak relationship we see between anxiety level and math test score is evidence of an actual relationship or simply the appearance of a relationship that is due merely to sampling error.

Hypotheses.

H0 : There is no association between math test score and anxiety level. 1 = 0

HA : There is a linear association between math test score and anxiety level. 1  0 Model.

Linearity Condition: There is no obvious bend in the scatterplot (from #1).

Independence Assumption: There is no reason to believe that one student’s score affected any other student’s score. The residuals plot (from #1) shows random scatter.

Equal Variance Assumption: The residuals plot shows relatively consistent spread.

8 Normal Population Assumption: The histogram of the residuals is unimodal and symmetric, with the exception of one large 6 negative residual. This residual isn’t that large, relative to the 4 others, so it shouldn’t be a problem.

2 Under these conditions, the sampling distribution of the regression slope can be modeled with a Student’s t-model, with (n – 2) = 22 -50.0 -12.5 25.0 degrees of freedom. We’ll do a regression slope t-test. residuals(M/A)

Mechanics.

Dependent variable is: Math Test No Selector R squared = 27.6% R squared (adjusted) = 24.3% s = 18.69 with 24 - 2 = 22 degrees of freedom Source

Sum of Squares

df

Regression Residual

2925.03 7687.47

1 22

Mean Square

2925.03 349.431

F-ratio

8.37

Var iable

Coefficient

s.e. of Coeff

t-ratio

prob

Constant Anxiety Level

91.6583 -4.48624

11.02 1.551

8.32 -2.89

² 0.0001 0.0084

The linear model is Math = 91.658 − 4.486( Anxiety) .

Conclusion. Because the P-value is low, we reject the null hypothesis. There is strong evidence that, on average, lower anxiety scores are associated with higher math scores.

Follow-up . 95% Confidence Interval * b1  t22 (SE (b1 )) = −4.486  2.074(1.551) = (−7.70, −1.27)

We are 95% confident that the interval (−7.70, −1.27) contains the average math test score difference associated with difference in anxiety level (the slope). In other words, each additional unit of anxiety level is associated with an average decrease of between 1.27 and 7.70 points on the math test.

4. The large selection of homework exercises will provide you with plenty of additional examples for discussion. .

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Part III Exploring Relationships Among Variables

Resources ActivStats*

• Lesson 16 – “Inference for Regression.” Simulations in this chapter can help students understand what inference for the slope is about. Another activity helps them explore the regression output for the selected statistics package.

Web Links Statscan: http://www.statcan.gc.ca/ Bank of Canada: http://www.bankofcanada.ca/ OECD: www.oecd.org amongst others offer a multitude of data that can be used for this chapter.

Basic Exercises 1.

Consider the following data. x 35 40 50 65 40 30 a. b. c. d.

2.

Make a scatterplot for these data. What does it tell you about the relationship between these two variables? Is the Linearity Condition satisfied?

The line of best fit for these data has b0 = 18.915 and b1 = −0.2596 . Use the estimated regression equation to predict y for all six values of x.

Find the residuals ei.

Calculate the residual standard deviation, se.

Refer to exercise 1. Suppose that you are testing the hypotheses H0: β1 = 0 vs. HA: β1 ≠ 0. a. b. c. d.

3.

y 9 8 4 3 10 12

What is the standard error of the regression slope?

Compute the value of the t-statistic.

What is the number of degrees of freedom associated with the t-statistic?

At α = .05, is the slope of the regression line significantly different from zero?

Explain.

A data set of 5 observations for x and y results in the following estimated regression line yˆ = 4.3 + 0.265x . The standard error of the regression slope is 0.035.

* ActivStats is available at www.pearsonmylabandmastering.com/northamerica/mystatlab/

.

Chapter 18 Inference for Regression a. b. c. d.

4.

Compute the 90% confidence interval for the regression slope. Interpret.

Compute the 95% confidence interval for the regression slope. Interpret.

Compute the 99% confidence interval for the regression slope. Interpret.

Explain how the confidence interval results can be used to test H0: β1 = 0 vs.

HA: β1 ≠ 0.

A data set of 20 observations on x and y results in a correlation coefficient r = − 0.225.

Suppose we want to test if the correlation between x and y is zero. a. b. c. d.

6.

Compute the value of the t-statistic to test if there is a significant relationship between x and y.

What is the number of degrees of freedom associated with the t-statistic?

What is the P-value associated with the t-statistic?

At α = .01, is there a significant relationship between x and y? Explain.

Refer to exercise 3. a. b. c. d.

5.

State the null and alternative hypotheses.

Compute the value of the t-statistic.

What is the number of degrees of freedom associated with the t-statistic?

At α = .01, is the correlation between x and y zero? Explain.

Consider the following data for which the estimated regression line is yˆ = 3.525 + 0.525x with se = 0.500 and SE(b1) = .056. x 1 4 8 10 12 a. b. c. d.

7.

18-7

y 4 6 7 9 10

Find the standard error for the predicted mean value of y when x = 7.

Find the standard error for the predicted mean value of y when x = 14.

Develop 95% confidence intervals for the predicted mean value of y when x = 7 and when x = 14.

Which confidence interval is wider? Why?

Refer to exercise 6. a. b. c. d.

Find the predicted individual value of y when x = 9.

Develop a 95% prediction interval for the predicted individual value of y when x = 9.

Develop a 99% prediction interval for the predicted individual value of y when x = 9.

Which confidence interval is wider? Why? .

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Part III Exploring Relationships Among Variables

8.

Consider the following data for which the estimated regression line is yˆ = 19.283 − 0.795x with se = 1.300 and SE(b1) = .182. x 12 13 17 19 20 a. b. c. d.

y 10 8 7 5 2

What is the predicted value of y when x = 16?

Develop a 95% interval for the predicted mean value of y when x = 16.

Develop a 95% interval for the predicted individual value of y when x = 16.

Which interval is wider? Why?

ANSWERS 1.

a.

Scatterplot of y vs x 12

10

y

8

6

4

2 30

40

50 x

60

70

The direction of association is negative. The Linearly Condition appears to be satisfied. b.

9.8, 8.5, 5.9, 2.0, 8.5, 11.1 − 0.8, − 0.5, − 1.9, 1.0, 1.5, 0.9 c. d.

1.462 2.

a.

For the given sx = 12.517 and se = 1.463. Thus, .

Chapter 18 Inference for Regression

SE ( b1 ) =

18-9

1.462  0.052 12.517 6 − 1

b. c. d.

− 4.99 4 Yes, the P-value is .008 < α = .05 so we can reject the null hypothesis.

3.

a. b. c. d.

7.57 3 .005 Yes, the P-value = .005 is less than .01 so we reject the null hypothesis. At α = .01, there is strong evidence that y is linearly related to x.

4.

a.

For  = .10 and df = 3 : t3* = 2.353.

0.265  2.353 0.035 = (0.183 , 0.347) We are 90% confident that y increases, on average, between 0.183 and 0.347 for each additional unit of x.

For  = .05 and df = 3 : t3* = 3.182.

0.265  3.182  0.035 = (0.154 , 0.376) We are 95% confident that y increases, on average, between 0.154 and 0.376 for each additional unit of x.

For  = .01 and df = 3 : t3* = 5.841.

0.265  5.841 0.035 = (0.061 , 0.469) We are 99% confident that y increases, on average, between 0.061 and 0.469 for each additional unit of x.

If the confidence interval does not include zero, then the slope is significantly different from zero.

b.

c.

d.

5.

a. b. c. d.

H0: ρ = 0 vs. HA: ρ ≠ 0. − .9797 18 No. As P-value =.340 > .01, at α = .01, we fail to reject the null hypothesis. There is no evidence that the correlation is significantly different from zero.

6.

a. b. c.

0.224 0.451 For  = .05 and df = 3 : t3* = 3.182. As x = 7 , we find yˆ v = 3.525 + 0.525(7) = 7.2 7.2  3.182  0.224 = (6.487 , 7.913) When x = 14 , we find yˆ v = 3.525 + 0.525(14) = 10.875 10.875  3.182  0.451 = (9.440 , 12.310)

d.

The confidence interval is wider for x = 14 since this value is further away from the mean of the x’s.

7.

a.

yˆ v = 3.525 + 0.525(9) = 8.25 .

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Part III Exploring Relationships Among Variables

d.

For  = .05 and df = 3 : t3* = 3.182. When x = 9 , we get SE( yˆ v ) = 0.559.

8.25  3.182  0.559 = (6.471 , 10.029) For  = .01 and df = 3 : t3* = 5.841.

8.25  5.841 0.559 = (4.985 , 11.515) The 99% prediction interval is wider because the level of confidence is higher.

a.

yˆ v = 19.283 − 0.795(16) = 6.563

b. c.

8.

b. c. d.

For  = .05 and df = 3 : t3* = 3.182. When x = 16 , we get SE(ˆ v ) = 0.583.

6.563  3.182  0.583 = (4.708 , 8.418) For  = .05 and df = 3 : t3* = 3.182. When x = 16 , we get SE( yˆ v ) = 1.425.

6.563  3.182  1.425 = (2.029 , 11.097) Prediction interval for the individual value (see c.) is wider than the confidence interval for the predicted mean value (see b.), because of greater uncertainty when predicting an individual value versus a mean value.

.

19-1

Chapter 19 Understanding Regression Residuals What’s It About?

Chapters 7 and 18 have briefly touched on What Can Go Wrong? with regression. Now we examine those issues in greater detail. We look at patterns in the residuals; the dangers of extrapolation; the possible effects of outliers, high leverage, and influential points; the problem of regression of summary data; autocorrelation in time-dependent data; and re-expressions to improve linearity. The exercises offer students more practice with regression while helping them learn to notice and deal with potential problems.

Comments Earlier in this Guide we described Statistics as the art of distilling meaning from data. In choosing that term we mean to imply that what we do is not merely algorithmic or formulaic. We have to Plan. We have to consider the circumstances, check the conditions, and choose our approach wisely, and then interpret the results carefully. We cannot rush into regression full speed ahead; there are many pitfalls. Chapter 8 taught students the concepts, the mechanics, and the correct interpretations of simple regression mostly by looking at situations where everything works. In Chapter 18 we extended those results so we could perform inference from the data to the population at large. The real world, though, is rarely so cooperative. Students should be cautious when applying regression techniques and interpreting the results. This chapter offers insights into many of the common problems that can arise, and provides ways of dealing with them. In some sense, there’s nothing dramatically new here; after all, we are still just doing linear regression. But what we’re aiming at is much deeper—the wisdom to use our statistical tools properly.

Looking Ahead We’re about to expand our horizons from bivariate data to talk about multiple regression. Things get more complex there, so it is important that students be comfortable with bivariate regression.

The underlying lesson here is that regression models are meant to be understood, not just fit blindly with a computer. By looking at the residuals and thinking about the patterns we find, we can often understand our data more deeply than we could if we only looked at the regression model. We hope that attitude will carry over into the discussions of multiple regression.

Class Do’s Emphasize the fact that we always need to Plan before rushing into regression. Plot the data. Look for trouble. Think of a way to deal with any problems.

Changes in a pattern over time may be a signal that something has changed in the underlying situation. Think about what might have changed historically. Divide the data into subsets that show a consistent trend. Answer the questions you have about the situation by creating a model for the relevant subset of the data. Don’t extrapolate much – or at all if you can avoid it.

Always look at the residuals plot. Many times important features of the association show up more clearly there than in the original scatterplot, including outliers, bends, and noteworthy subsets of the data. Students may think first of plotting the residuals against the x-values, but most computer software plots them against the predicted y-values (the yˆ ’s). The pattern seen in the plot is exactly the same either way ( yˆ being a linear transformation of x). The reason for this other method will be clear in the next chapter. When we use several x’s to predict y. We can’t put them all on the .

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Part III Exploring Relationships Among Variables

horizontal axis. It makes sense to find a linear combination of the x’s to plot against, and what combination could be better than yˆ ? After all, that’s the linear combination of the x’s that is most highly correlated with y.

Beware of unusual points. They come in three flavours.

• Least harmful are the points that confirm the pattern, but are extreme examples—high-leverage points with small residuals. A person who is really tall and weighs a lot may fit the overall pattern perfectly. The slope and intercept may be largely unaffected by that point (it’s still the same line). Such a point, however, will increase the value of R 2 possibly misleading you into believing that the model is stronger than the rest of the data may warrant. Of course, it is best to check that the model for the other points is the same. If the other points don’t determine a clear model, one extreme point can completely dictate the model. Be wary of overstating your case on the strength of one point.

• Outliers that aren’t high-leverage points but still show up with large residuals may not affect the slope much but are clearly not consistent with the pattern seen in the other points. They often change the intercept. Never throw them away without just cause. Sometimes we know or strongly suspect that a datum might be inappropriate to use— mistakes in measurement, an individual known to be somehow different from the rest of the group, etc. However, it’s not defensible to pare away anomalous measurements just to strengthen a model.

Most troublesome are influential points. These are high-leverage points that aren’t consistent with the model that describes the other points. These distort the model and may mislead us about the underlying relationship between the variables. They are hardest to detect. Influential points may have very small residuals, precisely because they are influential. A point that is an outlier in the x direction may exert so much leverage that it pulls the regression line toward it. We need to look at the scatterplot, covering up a point or two at a time to see if our perception of the pattern changes.

When we suspect a point is influential we should omit that point and recalculate the regression. If the slope of the line changes dramatically we know that the omitted point was influential. We can be pretty sure that a model for all the data won’t be worth much, so we need to set any influential points aside and find a model for the remaining points. But what should we do with the influential points? We need to Think more about those. They may simply be a mistake, or they may represent a fluke occurrence. However, we cannot overlook the possibility that data inconsistent with the rest of the pattern reflect a real and important aspect of the situation that we do not clearly understand.

Often the points that don’t fit the model tell us the most about it. Think of the saying that “the exception proves the rule”—but interpret it correctly. The word “prove” in that maxim means “to test” (as in proof reading), not “to demonstrate the truth of.”

Don’t be misled:

• Any regression based on summaries is problematic. In theory, if there is an underlying linear relationship, then the means of all the y-values for each x are collinear. (The picture of this — just for your own teaching background for now — is Figure 18.4 on p. 608.) We should not be surprised, then, to see an apparently strong association when we use only the means for the values of the response variable (instead of all the data). By overlooking the variability in the y’s we make the pattern more obvious but also make the regression seem misleadingly strong.

• Even strong associations cannot be attributed to cause and effect. Often there are lurking variables present. Students should always be thinking about plausible explanations that avoid .

Chapter 19 Understanding Residuals

19-3

implying that X causes Y. Perhaps Y causes X, or (more insidious) some other factor Z causes both X and Y. The Importance of What You Don’t Say Don’t talk about populations, inference, and so on. There's no need to wonder whether our regression model is “true.” That's easy: All models are wrong. We hope that our regression models are useful, but they certainly – and obviously – simplify the relationship. Not only do they fail to fit every point exactly, most likely, there are no zero residuals at all, so no point is fit exactly.

Don’t say that we “throw out” unusual points. When talking about outliers, try to resist suggesting that we would ever “throw them out.” It's vivid and tempting language, but we should always keep them around to examine and think about. If we find they are genuinely errors, then, of course we are justified in correcting them or deleting them. But otherwise, we “set outliers aside,” comparing the analysis with and without the unusual points.

Class Examples 1. Lots of population data that changes over time will show ups and downs, especially where the baby boomers are involved. The idea that we should describe different eras with different models will not strike the students as strange.

2. The best way to help students understand the impact that an influential point can exert is with a demonstration. They can enter four fixed points into a computer package, then position a fifth point in several different places to see how the correlation and the regression line change.

• Play with the Correlation activity (Chapter 6) from ActivStats*. You are allowed to position several points and then drag one around to see how the correlation changes.

• Even better, play with ActivStat’s Linear Regression tool (Chapter 7). Given a scatterplot with regression line drawn, you can drag one point around to see how the line moves. Note that points farther to the left or right sides of the plot exert greater leverage than points in the middle.

3. Tell your class that a vocabulary achievement test given to elementary school children discovered a strong association between weight and vocabulary score. Look for explanations that make causality mistakes. Ignoring political correctness, students may suggest that kids who are overweight probably stay indoors and read instead of playing sports (weight indirectly causing a better vocabulary). Certainly that chain of causality could be reversed – kids who read a lot spend less time in fitness-enhancing activities. Neither of those explanations considers age as a lurking variable: Older kids are both bigger and have better vocabularies, so there may be a clear association between weight and scores on a vocabulary test given across many grade levels.

Resources ActivStats

• Chapter 6 has the Correlation tool.

• Chapter 7 offers the Linear Regression tool. Notice that the tools are available in the toolbar at the top of the lesson pages. * ActivStats is available at www.pearsonmylabandmastering.com/northamerica/mystatlab/

.

19-4

Part III Exploring Relationships Among Variables

Web Links

• The OECD: www.oecd.org ; Statscan: http://www.statcan.gc.ca/; The World Bank: http://www.worldbank.org/; and the UN: http://www.un.org/en/databases/index.html all offer useful data for research.

Basic Exercises 1.

Consider the following scatterplots and describe the unusual point in each. a.

Scatterplot of y vs x 20

y

15

10

5

0 0

2

4

6

8

10

x

b.

Scatterplot of y vs x 20

y

15

10

5

0 0

5

.

10 x

15

20

Chapter 19 Understanding Residuals c.

Scatterplot of y vs x 20

y

15

10

5

0 10

12

14

16

18

20

x

2.

Consider the following data set whose estimated regression line is ŷ = 12.5 – 0.576 x. y 10 12 8 16 9 11 7 8 6 8 4 5 5 7 2 4 a. b. c. d.

3.

x (time) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Find the predicted values of y for x = 1 through 16.

Find the residuals.

Plot the residuals versus time order.

What does an examination of the residual plot reveal?

Refer to exercise 2. a. b.

Calculate the Durbin-Watson statistic.

At α = .05, what are the values of dL and dU. .

19-5

19-6

Part III Exploring Relationships Among Variables c. d.

4.

A regression line fit to sales data using time as the independent variable results in a calculated Durbin-Watson statistic of 0.977. The series consists of 40 observations. a. b. c.

5.

Is there evidence of positive autocorrelation? Explain.

Is there evidence of negative autocorrelation? Explain.

At α = .01, what are the values of dL and dU. Is there evidence of positive autocorrelation? Explain.

Is there evidence of negative autocorrelation? Explain.

Consider the following plots. Indicate in each case whether a re-expressing of the data is necessary. If yes, indicate what data transformation might be appropriate. a.

b.

.

Chapter 19 Understanding Residuals

19-7

c.

ANSWERS 1.

a. b. c.

high-leverage, influential, large residual high-leverage, not influential, small residual not high-leverage, can be influential, large residual

2.

a.

11.924, 11.348, 10.772, 10.196, 9.620, 9.044, 8.468, 7.892, 7.316, 6.740, 6.164, 5.588, 5.012, 4.436, 3.860, 3.284 –1.924, 0.652, –2.772, 5.804, –0.620, 1.956, –1.468, 0.108, –1.316, 1.260, –2.164, –0.588, –0.012, 2.564, –1.860, 0.716

b. c.

Versus Order (response is y)

6 5 4

Residual

3 2 1 0 -1 -2 -3 1

2

3

4

5

6

7 8 9 10 11 Observation Order

12

13

14

Copyright © 2021 Pearson Canada Inc.

15

16

19-8

Part III Exploring Relationships Among Variables d.

There is a repeating pattern in the residuals. It would make sense to check the model for autocorrelation.

3.

a. b. c. d.

2.968 1.10, 1.37 No, D is greater than the upper critical value (D = 2.968 > dU = 1.37).

Yes, D is greater than 4 minus the lower critical value (D = 2.968 > 4 – dL = 2.90).

4.

a. b. c.

1.25, 1.34 Yes, D is less than the lower critical value (D = 0.977 < dL = 1.25).

No, D is less than 4 minus the upper critical value (D = 0.977 < 4 – dU = 2.66).

5.

a. b. c.

Yes, log to straighten.

Yes, square to deal with left skewness.

Yes, square root to straighten.

.

20-1

Chapter 20 Multiple Regression What’s It About?

We first looked at simple regression in Chapter 7. There we learned about interpreting regression coefficients and R 2 . Then we discussed inference for regression in Chapter 18 and explained most of the other numbers in the computer regression table. Chapter 19 has discussed patterns in residuals. But the simple linear model isn’t rich enough to deal with the complexities of the real world. Linear models (almost) never account for all of the variation in the data (as the R 2 shows us). Adding more predictors to the model can help account for more of the behaviour of the response variable. And the linear model accommodates more predictors naturally.

With a statistics program it’s easy and natural to add more predictors to a regression. Fortunately, most of what we learned in Chapter 18 still holds. The R 2 still gives the fraction of variability accounted for by the regression. The coefficients table gives standard errors, t-statistics, and corresponding P-values for each coefficient.

But the regression coefficients have a subtly different interpretation that needs to be understood.

The chapter provides a careful discussion of what multiple regression coefficients mean.

The assumptions and conditions also receive renewed attention. The linearity assumption is now an assumption about the multiple regression model, and is harder to check with corresponding conditions. And it matters what order we do things in. Students should pay attention to the order in which we check conditions and compute results.

Comments The theme of this chapter is that multiple regression is pretty much just like what we’ve been doing with inference and what we’ve already seen for simple regression inference. Students who have used a statistics package for simple regression should be able to figure out on their own how to throw in a few more predictors. All students, having seen package output for simple regression, should be able to read most of the regression table for a multiple regression.

The most important new concepts here relate to interpreting multiple regression coefficients. A multiple regression coefficient measures the linear association between its predictor and the response variable after allowing for the linear effects of the other predictors.

Don’t let your students interpret coefficients either as causally predicting what will happen if a variable is changed or as the relationship “when other variables are held constant.” Both interpretations are common; neither is really correct.

Looking Ahead Chapter 21 will deal with some of the more sophisticated issues raised by building multiple regression models. Analysis of Variance (which we saw in Chapter 15) and multiple regression are the foundations of the most widely used statistics methods. Together they are aspects of a statistics model known as the General Linear Model. Students who take more advanced statistics courses are bound to encounter General Linear Model applications and generalizations. The better their understanding of ANOVA and multiple regression at this point in their study of Statistics, the sounder will be their understanding of the more advanced methods. .

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Part III Exploring Relationships Among Variables

Class Do’s Do encourage students to interpret multiple regression models, but insist that they do so carefully and correctly. Do insist that they plot residuals and check conditions. And be sure that they understand that it is a good idea to maintain alternative regression models for the same data while trying to understand which model is the most useful.

It is a good idea to work through one of the step-by-step examples. None of it is really new, but once put together into a full multiple regression analysis, there’s quite a bit here.

Start with a review of the key regression concepts from Chapters 6, 7, 18, and 19. A set of data and a list of questions should do it. Any of the data sets for the Mini Case Study projects of those chapters would be suitable. If possible, have students work in groups helping each other remember the concepts, the mechanics, and the interpretations.

• What does R 2 mean?

• How do we find a linear model?

• How do we know whether the linear model is appropriate?

• How do we plot residuals? (Remind them about plotting residuals against the predicted values rather than against x; that’s what we’ll need to do for the multiple regression.)

• What do we look for in the residuals plot?

• What does the slope of a simple regression mean?

• What does the y-intercept mean?

• What does it mean for the coefficients to have standard errors?

• How do we interpret the t-tests that come with each coefficient in a simple regression?

• What additional conditions did we need to check for those t-tests to be appropriate? How do we check them?

All of these concepts continue into the multiple regression framework. The mechanics are increasingly turned over to the computer. The interpretations are the place where we need to think a bit more carefully.

Consider the example of trying to predict a person’s shoe size (y) based on the person’s height (x1) and weight (x2). We must believe that for this population there is an ideal regression model that captures the association between shoe size and height and weight. Our sample produces a model that approximates this true model. Other samples would also produce such approximating models, each with slightly different coefficients. Coefficients, then, vary from sample to sample. Each coefficient, bj, is an estimate of a population model coefficient,  j . The sampling model of each of these coefficients has a t-model, with n – k – 1 degrees of freedom. Knowing the model enables us to do inference about the coefficients. Almost.

The t-model we want to use is appropriate only if the population behaves according to certain assumptions. When students understand the assumptions we needed for simple regression inference in Chapter 19, they’re better able to remember the conditions for multiple regression. .

Chapter 20 Multiple Regression

20-3

• We believe that at every possible combination of values of the x variables, the y’s follow a Normal model. In other words, a Normal model describes the shoe sizes of all the people who are, say, 60 inches tall and weigh 150 pounds. Or any other height and weight. (Checking a histogram of the residuals for approximate normality addresses this question, provided the linear model itself is appropriate and the variation around the model is the same everywhere.)

• We believe that the means of all those Normal models can be modeled with a linear model, and that the ideal model they determine is approximated by the regression model we fit to our sample. (Check the scatterplot of residuals against the predicted value for structure that suggests we don’t have the right model.)

• We believe that all those Normal models have the same standard deviation. In other words, we believe the variability of shoe sizes is the same at each height and weight combination. This assumption is called “homoscedasticity,” a term students don’t need to know, but usually enjoy knowing. (Check the residuals plot for uniform spread.)

• We believe that the errors – the distances of points from the ideal model, akin to residuals in the sample – are independent, that one person’s bigger than expected feet do not tell whether another person might have bigger or smaller feet than their height and weight might predict.

(Check the residuals plot for random scatter.)

But what are we doing inference about? Help students see that even if there is no association present in the population, a sample might draw points that appear to show an association. The question we ask is whether the association we see in our sample is so strong that it could not just have popped up by chance during the sampling process. We want to know if the apparent association in our sample provides evidence that there must be an association in the population.



Now, however, life is more interesting than it was back in Chapters 18 and 19 when we had only one predictor. Now we have a null hypothesis associated with each predictor. We hypothesize that each  j = 0 in the model. Be sure that students understand that this is not the same as the simple hypothesis that there is no linear association between y and that predictor. These null hypotheses talk about the relationship between y and the predictor after allowing for the linear effects of the other predictors. The null hypothesis can be viewed as saying that after doing the best we can with the other predictors, there’s nothing more that this one brings to the model.

When a hypothesis test rejects the null hypothesis, have students interpret the coefficient carefully.

The coefficient talks about the slope of the linear relationship between the y variable and that predictor after allowing for the linear effects of the other predictors.

Invest time in looking at regression output. Students are most likely to be given a multiple regression analysis produced by a software package. Be sure they can find, use, and properly interpret the key results.

The Importance of What You Don’t Say Don’t imply that changing x will cause a change in y corresponding to the regression coefficient.

Regression does not establish a causal relationship, it only models a linear association. For example (as discussed below), although weight is a good predictor of shoe size in a multiple regression, nobody goes on a diet to get dainty feet. And while taller people generally have bigger feet, it makes little sense to talk about the consequences of making an adult shorter. .

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Part III Exploring Relationships Among Variables

Don’t be lured into prediction – especially into the future. It is tempting to interpret regression models that have time as a predictor as if they provided a computerized crystal ball. We warned against this back in Chapter 7. But it is so tempting to do that it is worth repeating the warning here. Regression models do not have “inertia.” Just because they are headed in a particular direction when we lose sight of them off the edge of our plots, we can’t assume that they’ll continue indefinitely on the same path—indeed, a little thought usually shows that they couldn’t possibly do that.

Don’t even think about the formulas for calculating SE(b). Students need to understand the concepts and be able to interpret regression output from software packages. Just point out that for multiple regression, the formulas are just not worth considering at all. It is technically possible to write expressions for the standard errors in a regression with two predictors, but it doesn’t generalize. The general formula needs matrix algebra; you don’t want to go there.

Don’t say “holding the other predictors constant…” Yes, mathematically, it is technically true that if all the other x’s are held constant, then a coefficient tells how our prediction for y changes were the corresponding x to change. But in most business applications it is often meaningless to hold things constant. Workers can’t gain experience without also getting older. Advertising campaigns occur while other aspects of the economy change. Unless you are discussing a situation in which the predictors are under control and can be held constant while one other predictor is varied (e.g. an industrial quality experiment in which the speed of the conveyor belt, the oven temperature, and the amounts of various chemicals in the mix can be controlled independently) it probably isn’t possible to hold the other predictors constant while one changes. Regression models describe the population as it is (or was when the data were collected), not as it might be under changes. Houses with fewer rooms usually cost less, but that’s partly because they also are generally smaller.

Removing the walls between two small bedrooms to make a master bedroom suite probably won’t reduce the value of your house. That’s just not what the regression says.

Class Examples Students in a large statistics class were surveyed. Here is a regression predicting shoe size from height (inches), and weight (lbs.): Dependent variable is: shoe size R squared = 71.2% R squared (adjusted) = 71.0% s = 1.033 with 216 - 3 = 213 degrees of freedom Source Regression Residual

Sum of Squares df 562.446 2 227.247 213

Mean Square F-ratio 281.223 264 1.06689

Variable Intercept height weight

Coefficient -13.2086 0.290390 0.020321

t-ratio -10.6 13.2 6.75

SE(Coeff.)

1.245 0.0220 0.0030

P-value ≤ 0.0001 ≤ 0.0001 ≤ 0.0001

Students should be able to find the coefficients in a typical regression output and write the regression model. They should be able to interpret R 2 as the fraction of variability in shoe size accounted for by the regression model. .

Chapter 20 Multiple Regression

20-5

1. Students need to understand that a high R 2 may be evidence of a good regression model, but it may also simply arise from sampling error even when there is no association at all between the two variables. To demonstrate, have each student create 11 sets of 50 random numbers. Clearly there should be no association among random numbers, yet a regression of one of these “variables” on the 9 others will have a reasonably high R 2 , possibly around 20%. Is the R 2 high enough to be meaningful? We can address this question with hypothesis tests on the coefficients. After all, a true correlation of 0 would make the coefficients of the ideal regression model also 0. We suspect that the coefficients of the model we fit to our random data, though not exactly 0, aren’t statistically significant.

2. Run through the key steps of a hypothesis test and confidence interval (including careful attention to the assumptions and conditions) using the shoe size vs. height and weight example.

3. Once students understand the linear regression t-test, have them check the hypotheses associated with the coefficients of the regression on their random numbers. Almost all will turn out to be too weak to reject the null hypothesis. And when the occasional significant correlation does appear you can remind them of the risk of Type I Error that’s always present – we’re willing to take a 5% risk of rejecting a true null.

4. The large selection of homework exercises will provide you with plenty of additional examples for discussion.

Resources ActivStats*

• Lesson 18: Multiple Regression. ActivStats offers a number of visualizations that can be useful for individual student use or as classroom demonstrations.

Statistics software

• The user manual of whatever statistics software your students are using will discuss the statistics and graphics associated with multiple regression that the package makes available. Most also provide examples. Some offer additional wisdom.

Web Links: •

The OECD: www.oecd.org ; Statscan: http://www.statcan.gc.ca/ ; The World Bank: http://www.worldbank.org/ ; and the UN: http://www.un.org/en/databases/index.html all offer useful data for research.

Basic Exercises 1.

Consider the following data. y

x1

x2

35 40

9 8

8 10

* ActivStats is available at www.pearsonmylabandmastering.com/northamerica/mystatlab/

.

20-6

Part III Exploring Relationships Among Variables 50 65 40 30 a. b. c. d.

2.

d.

c. d.

c. d.

Interpret the meaning of b1, and b2 in the estimated multiple regression equation.

The results indicate that SE(b1) = .731. Construct a 95% confidence interval for β1.

Interpret.

The results indicate that SE(b2) = .715. Construct a 95% confidence interval for β2.

Interpret.

Would you recommend dropping either independent variable from the model?

Explain.

Find SSE, MSR and MSE. Calculate the F-statistic to test the overall significance of the multiple regression model.

State the null and alternative hypotheses.

At α = .05, is the multiple regression model significant overall? Explain.

Refer to exercise 4. a. b. c. d.

6.

Calculate SSE. The results indicate that SST = 783.33. Find SSR, MSR and MSE. Calculate the F-statistic to test the overall significance of the multiple regression model.

At α = .05, is the multiple regression model significant overall? Explain.

A data set of 12 observations for y, x1, x2 and x3 results in the estimated multiple regression equation y = 88.9 + 0.34 x1 – 2.49 x2 – 0.137 x3 with SST = 2198.92 and SSR = 1552.55. a. b.

5.

The estimated multiple regression equation for these data has b0 = 63.4, b1 = − 3.33, and b2 = 0.417. What is the predicted value of y when x1 = 9 and x2 = 8?

Find the predicted values of y for the other pairs of x1 and x2 values in the table.

Find the residuals ei.

Calculate the residual standard deviation, s.

Refer to the example in exercises 1 and 2. a. b.

4.

15 14 13 18

Refer to exercise 1. a. b. c.

3.

4 3 10 12

Compute R2.

Compute the adjusted R2.

Interpret their meaning.

At α = .05, can we conclude that R2 is significantly different from zero. Explain.

For the example in exercises 4 and 5, the results also show that SE(b1) = 2.374, .

Chapter 20 Multiple Regression

20-7

SE(b2) = 0.628 and SE(b3) = 0.140. a. b. c. d.

Calculate the t-ratio to test if β1 = 0.

Calculate the t-ratio to test if β2 = 0.

Calculate the t-ratio to test if β3 = 0.

At α = .05, would you recommend dropping any of the independent variables from the model? Explain.

ANSWERS 1.

a. b. c. d.

36.766 40.930, 56.335, 59.248, 35.521, 30.946 − 1.766, − 0.930, − 6.335, 5.752, 4.479, − 0.946 5.720

2.

a. b. c. d.

98.16 685.17, 342.59, 32.72 10.47 Yes, we reject the null hypothesis at α = .05 with a P-value = .044.

3.

a.

b.

For a given level of x2, with each additional unit of x1 the value of y decreases by 3.33, on average.

For a given level of x1, with each additional unit of x2 the value of y increases by .417, on average. −3.33  3.182  .731 = ( −5.656 , − 1.004 )

c.

0.417  3.182  0.715 = ( −1.858 , 2.692 )

d.

We would consider dropping x2 because it is not significant as the interval for its slope coefficient contains zero.

4.

a. b. c. d.

646.37, 517.52, 80.80 6.40 H0 : β1 = β2 = β3 = 0 vs. HA : at least one β  0 Yes, the P-value for the F-statistic is .016 so we can reject the null hypothesis.

5.

a. b. c.

70.6 % 59.6 % R2 = 70.6 % of variability in y explained by the regression model.

R2adj = 59.6 % reduces the overall proportion of variability in y explained by the regression model by 70.6 − 59.6 = 11.0% . R2adj is smaller than R2 because it shrinks when a predictor added, and the model contains three predictors.

Yes, as the F-test is equivalent to testing whether R2 is different from zero. Using the result of exercise 4 d, we conclude that R2 is significantly different from zero.

d.

6.

a.

0.14 .

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Part III Exploring Relationships Among Variables b. c. d.

− 3.96 − 0.98 Only x2 has a slope coefficient that is significantly different from zero, so the other variables can be dropped.

.

21-1

Chapter 21 Building Multiple Regression Models What’s It About?

We’ve now looked at least squares regression in Chapters 6, 7, 18, 19 and 20. The methods of multiple regression are central to much of statistical analysis of business data, so the effort is wellspent. This is the last of the chapters in this book to deal directly with regression, but the concepts will continue to appear in later chapters in different forms. In this chapter we consider some of the more sophisticated ideas, methods, and challenges related to multiple regression.

Comments This chapter covers a collection of useful methods and diagnostics, all related to multiple regression. You may want to view it as a survey of these topics. In a course that focused on multiple regression, each of these topics would deserve its own expanded section. Of course, that means there are opportunities to expand on each of the topics if you choose.

Here are some suggestions of things not said in the chapter that you might choose to add to classroom discussions of each of the major topics in the chapter: •

Indicator (Dummy) variables. We show indicators working to represent categories in a binary, and later in the chapter, in a multiple-level categorical variable. Students might like to see the mechanics of indicator variables at work. Perform a regression of a single quantitative variable on the indicator that is all 1’s, omitting the intercept term. If possible, work a toy example by hand so students can see why the resulting “slope” is just the mean.

Mention that the mean also has the least squares property—and this is one demonstration of why it does.

Then consider an indicator for just one level of a categorical variable—say, for just the women—and do the simple regression again. Now we find the mean only for the identified group. Suggest that this might be a way to find the means of the different groups needed to compute the Analysis of Variance. (Indeed, that’s exactly what statistics packages do.)

You can also show that including this indicator of all values in a multiple regression with other predictors, but with the estimation of the intercept turned off (each statistics package does this differently) results in the coefficient of the indicator being the intercept. That’s why many statistics packages label the intercept “constant” instead of “intercept”; they actually estimate it by including an indicator that is all ones with the data.

•

Leverage. One frustration about teaching leverage is that we can’t show students the full calculation without using matrix algebra. However, for a simple regression on a single predictor, the leverage of the ith point is

(x i − x ) + 1 hi = (x i − x )2 n 2

where the 1/n term accounts for the intercept. Students can see that this is the contribution of the ith point to the sum of squared deviations from the mean in the x direction.

Unfortunately, this formula  doesn’t generalize to several x’s without the use of matrix algebra, so we must rely on technology for the calculations. .

21-2 Part III Exploring Relationships Among Variables •

Studentized residuals. There are two kinds of Studentized residuals. Internally studentized residuals use the estimate of the standard deviation of the errors that ordinarily appears in the regression table. But they don’t follow a Student’s t distribution. (Technically, Studentizing requires only that the numerator and denominator be statistically independent, not that the resulting ratio follow Student’s t.) Externally studentized residuals estimate the standard deviation of the errors by omitting the case whose residual we are Studentizing.

That’s not as hard as it seems (there’s a closed form calculation) and it yields a statistic that does follow a t distribution. We’ve elected to introduce only the externally Studentized residuals. Except for helping students to find the externally studentized residuals in their statistics software, it probably isn’t worth discussing any of this with them.

•

Influential cases. There are several statistics that attempt to assess how influential a case is.

The two most widely available are Cook’s distance and DFFITS. Both combine leverage and studentized residuals. Cook’s distance works with internally studentized residuals.

DFFITS works with externally studentized residuals. Most statistics packages offer both statistics.

•

Omitting influential cases. We come down on the side of identifying influential and extraordinary cases and isolating them from the estimation of a linear model for the rest of the data. The technique we illustrate—that of including an indicator for each case to be omitted—has the advantage of displaying the omitted points with the model rather than hiding them from view. It also provides a t-test of the hypothesis that the case belongs with the others; that’s just the t-test on its coefficient. We do not say that you must understand why a case is extraordinary before deciding to isolate it from the regression model.

Violation of the tacit assumption that the “Who” are a homogeneous population about whom it makes sense to make statements and build models is sufficient reason for holding aside cases that just don’t belong with the others. Often raising the question “why doesn’t this case seem to fit with the others?” leads to important insights about the relationships underlying the data. We think the raising of new interesting questions is often more important than the answering of the old ones that constituted the original “Why” of the study. But schools of thought differ on this subject. A discussion in class can be lively and instructional.

•

Stepwise Regression. The book only introduces this topic. Stepwise regression was a standard method when computers were big, expensive, and hard to use. Researchers wanted to get the most out of each attempt to fit a multiple regression model. With interactive computing, the fact that a stepwise solution is rarely as good as one found with human help made stepwise methods less attractive. Now, however, they have returned as a tool in the arsenal of data miners, who may have hundreds of variables to choose among.

We’ll mention them again in that context in Chapter 25.

•

Collinearity. Economic data are particularly prone to collinearity, so this topic is an important one. We suggest the Variance Inflation Factor, offered by most statistics software, as a good measure of collinearity. It is not, however, a statistical test, so the data analyst must use judgment in deciding whether the degree of collinearity is worrisome.

.

Chapter 21 Building Multiple Regression Models

21-3

Looking Ahead We’ll use multiple regression methods for fitting some time series models in Chapter 22.

Class Do’s Do encourage students to interpret multiple regression models, but insist that they do so carefully and correctly. Do insist that they plot residuals and check conditions. It is a good idea to work through one of the step-by-step or homework examples, if only to illustrate how to use the statistics package of choice to do the job.

Spend time looking at regression and diagnostic results. Help students find the right words to discuss collinearity, influential cases, and the competition of alternative models for the same data.

The Importance of What You Don’t Say Multicollinearity This isn’t a vital point, but there is no need for the “multi” prefix; there’s no such thing as “unicollinearity.” Just say collinearity. Encourage students to recognize collinearity when they find it. (You might also remind them that collinearity has two “l’s”, a feature it shares with words like colleague, collect, and collapse, in which the prefix “col-”, meaning “together,” is followed by a suffix that begins with an “l”.)

Best regression model. One of the lessons of this chapter—and of Statistics in general—is that there may not be a single “best” model. That’s especially true when we build multiple regression models. We may find several models for the same data, each appropriate for a different purpose or representing a particular point of view. Or a new predictor may become available, and it may change everything. Absolute truth is a rare commodity in general. It may be especially rare in Statistics.

Matrix algebra. Yes, that’s the way we handle the math when we handle the math. But there’s a big divide between approaches to multiple regression that rely on technology and avoid matrices, as we do, and those that slog through the math. Seeing the regression model in matrix terms does offer additional insights. But it requires a major investment in teaching practical linear algebra.

Don’t go there for this class.

Don’t even think about the formulas for calculating leverage or the other diagnostic statistics.

Students need to understand the concepts and be able to interpret regression output from software packages. The calculation details are only manageable with matrix algebra.

Don’t suggest that collinearity can be judged from scatterplots or correlations of pairs of predictors. Collinearity is a multivariate property. Although it is true that two highly correlated predictors are probably collinear, an absence of large pairwise correlations does not guarantee that there is no collinearity.

Class Examples Several large examples are worked through in the exercises. Any of them would make a good class example. Of course, students must be comfortable making regressions and displays with their statistics package of choice. Class examples to illustrate how to find a multiple regression, generate diagnostic plots and statistics and then try an alternative model are very valuable.

Some examples can be thought experiments: An article appearing in Education Next (“Exploring the Costs of Accountability,” Peyser and Costrell; Spring 2004, see p. 27) argued, based on a regression, that there was no relationship between school spending and mean (school level) test .

21-4 Part III Exploring Relationships Among Variables performance in math and language arts after allowing for district median income and percentage of students eligible for reduce-price lunch.

•

Do you think median income and poverty (as measured by eligibility for reduced-price lunch) are likely to be collinear with money spent on schools across districts?

•

If so, how might this collinearity affect the coefficient of school spending in predicting test scores?

Resources ActivStats* • Lesson 19: Building Multiple Regression Models • The user manual of whatever statistics software your students are using will discuss the diagnostic statistics available for multiple regression. Most also provide examples. Some offer additional wisdom.

Assignments There are several new ideas in this chapter: indicator variables, interactions with indicator variables, leverage and influential cases in multiple regression, model building, and collinearity.

Some of these new ideas (interactions, leverage, influential cases) aren’t really new, they’re just extensions of what we learned in earlier chapters. We’ve covered each of these ideas only briefly (we could easily write a chapter on each). To cover each in only the detail that we’ve provided could take 4-5 days. If you don’t have that kind of time, consider presenting a few examples demonstrating each of these ideas over two days of class.

Web Links

• The OECD: www.oecd.org ; Statscan: http://www.statcan.gc.ca/ ; The World Bank: http://www.worldbank.org/ ; and the UN: http://www.un.org/en/databases/index.html all offer useful data for research.

Basic Exercises 1.

Two independent variables, x1 and x2, are considered possible predictors of y. Suppose that x2 is an indicator variable that is coded 0, 1 (0 = absent, 1 = present). a.

Below is the output that shows the regression model using x1 only. At α =.05, is x1 significant in predicting y? Explain.

The regression equation is yˆ = 30.0 + 3.55 x1 Predictor Coef. SE Coef. T P Constant 30.025 8.171 3.67 0.003 x1 3.5490 0.3757 9.45 0.000 S = 9.63136 R-Sq = 88.1% R-Sq(adj) = 87.2% * ActivStats is available at www.pearsonmylabandmastering.com/northamerica/mystatlab/

.

Chapter 21 Building Multiple Regression Models b.

21-5

Below is the output that shows the regression model using both x1 and x2. Is the model better when it includes the indicator variable? Explain.

The regression equation is yˆ = 40.4 + 3.39 x1 − 14.2 x2 Predictor Constant x1 x2

Coef.

40.448 3.3873 -14.170

SE Coef.

5.843 0.2462 3.374

T 6.92 13.76 -4.20

P 0.000 0.000 0.001

S = 6.23454 R-Sq = 95.4% R-Sq(adj) = 94.6%

c.

2.

For each of the following examples below, show how you would code dummy variables to include in a regression model. a. b. c. d.

3.

How does the presence of x2 affect the predicted value of y?

Political party affiliation (Conservative, Liberal, NDP, Bloc Québécois, Green) Type of residence (Apartment, Condominium, Townhouse, Single family home) Gender (Female, Male) Class Standing (Freshman, Sophomore, Junior, Senior)

Using the data set for exercise 1, an interaction term is included in the model. Below is the output that shows the regression analysis results.

The regression equation is yˆ = 27.1 + 4.00 x1 + 10.9 x2 −1.22 x1  x2 Predictor Coef.

SE Coef.

T P Constant 27.123 5.574 4.87 0.001 x1 4.0009 0.2452 16.32 0.000 x2 10.931 7.508 1.46 0.176 x1  x2 -1.2154 0.3451 -3.52 0.006 S = 4.36833 R-Sq = 98.0% R-Sq(adj) = 97.4%

a. b. c.

4.

At α = .05, is the indicator variable significant? Explain.

At α = .05, is the interaction term significant? Explain.

Interpret the coefficient for the interaction term.

A regression model is being built for y using 5 potential predictors (x1, x2, x3, x4, and x5).

One of these predictor variables (x5) is an indicator variable. a.

Below are the results from stepwise regression. Write out the regression model.

Response is y on 5 predictors, with N = 79 N(cases with missing observations) = 2

.

N(all cases) = 81

21-6 Part III Exploring Relationships Among Variables Step 1 Constant 40730

2 26875

3 23154

x1 T-Value P-Value

14567 22.76 0.000

14157 21.64 0.000

9.6 7.16 0.000

8.5 6.07 0.000

15945 20.32 0.000

x2 T-Value P-Value x4 T-Value P-Value

1547 2.14 0.036

S 9171 R-Sq 84.29 R-Sq(adj) 84.08 Mallows Cp 56.1

b. c.

7135 90.61 90.36 5.3

6973 91.15 90.80 2.8

How much of the variability in y is being explained by the stepwise regression model?

Below are the results from best subsets regression. Do these results agree with the stepwise model? Explain.

Best Subsets Regression: y versus x1, x2, x3, x4, x5 Response is y 79 cases used, 2 cases contain missing values Vars 1 1 2 2 3 3 4 4 5

5.

Mallows xxxxx R-Sq R-Sq(adj) Cp 84.3 84.1 56.1 26.8 25.9 535.3 90.6 90.4 5.3 87.9 87.5 28.3 91.2 90.8 2.8 91.1 90.8 3.1 91.2 90.7 4.2 91.2 90.7 4.6 91.2 90.6 6.0

S 9171.5 19791 7135.3 8114.6 6972.9 6988.3 6992.7 7013.1 7029.8

1 2 3 4 5 X X XX X X XX X XXX XXXX XX XX XXXXX

Consider the following regression results for y and x.

The regression equation is yˆ = 80.9 − 0.241 x1 Predictor Coef SE Coef T P Constant 80.86 25.87 3.13 0.012 x1 -0.2409 0.8437 -0.29 0.782 S = 17.6981 R-Sq = 0.9% R-Sq(adj) = 0.0%

.

Chapter 21 Building Multiple Regression Models a.

b.

21-7

At α = .05, is x significant in predicting y? Explain.

A residual plot shows a curvilinear pattern. Consequently a quadratic term is added. Below are the regression results. Is the model better when it includes the quadratic term? Explain.

Regression Analysis: y versus x1, x12 The regression equation is yˆ = −293 + 25.8 x1 − 0.434 x12 Predictor Constant x1 x 12

Coef.

-292.76 25.826 -0.43444

SE Coef.

39.95 2.746 0.04556

T -7.33 9.41 -9.53

P 0.000 0.000 0.000

S = 5.33860 R-Sq = 92.0% R-Sq(adj) = 90.0%

c.

The correlation between x1 and x12 is .966. Is this of concern? Explain.

ANSWERS 1.

a. b. c.

2.

3.

Yes, the P-value associated with the t-statistic for the slope is less than .001.

Yes, both variables are significant (t-statistics for each slope coefficient have small P-values). In addition, the R2 adjusted has increased from 87.2% to 94.6%.

The presence of x2 decreases the predicted value of y by 14.2 if x2 = 1, and does not affect y if x2 = 0.

a.

Let x1 = 1 if Conservative, 0 otherwise Let x2 = 1 if Liberal, 0 otherwise Let x3 = 1 if NDP, 0 otherwise Let x4 = 1 if Bloc Québécois, 0 otherwise

b.

Let x1 = 1 if Apartment, 0 otherwise Let x2 = 1 if Condominium, 0 otherwise Let x3 = 1 if Townhouse, 0 otherwise

c.

Let x1 = 1 if Female, 0 otherwise

d.

Let x1 = 1 if Freshman, 0 otherwise Let x2 = 1 if Sophomore, 0 otherwise Let x3 = 1 if Junior, 0 otherwise

a. b. c.

No, the P-value associated with the t-statistic for its slope is 0.176 > 0.05.

Yes, the P-value associated with the t-statistic for its slope is 0.006 < 0.05.

When x2 is present, the slope relating y to x1 is less steep by 1.22. .

21-8 Part III Exploring Relationships Among Variables 4.

5.

a.

yˆ = 23154 + 14157 x1 + 8.5x2 + 1547 x4

b. c.

91.15 % Yes, the same model has the highest R2 adjusted and the lowest S.

a. b. c.

No, the P-value associated with the t-statistic for the slope is 0.782 > 0.05.

Yes. Now both variables are significant and the R2 adjusted increased to 90.0 %.

Yes, may be indicative of collinearity.

.

22-1

Chapter 22 Time Series Analysis What’s It About?

Time series are a special kind of data. And they are especially common in business situations.

Some analyses focus on understanding the components of a time series. Others attempt to forecast into the future. Some methods build on the multiple regression methods discussed in Chapters 20 and 21, so this chapter offers a nice application of those methods.

Comments This chapter provides an overview of useful methods. An entire course could be devoted to time series methods. We first discuss what time series analysis is about and how useful it is to transform raw data into Index Numbers. We then discuss the Trend, Seasonal, Cyclic, and Irregular components of time series. These are useful concepts for understanding the methods to follow.

Smoothing methods are purely descriptive. We discuss moving averages, weighted moving averages, and then the special kind of weighted moving averages known as exponential smoothers.

We illustrate how well each method follows features of a time series according to the parameters selected for the smoother and how the ability to follow such features trades off with the ability to smooth out irregular fluctuations.

We then introduce autoregressive models. This is a topic that could be expanded much more. We present only an introduction to the concepts and methods. Autoregressive models introduce the idea of modeling a time series and, thus, of being able to forecast beyond the next time point in the series.

With autoregressive models as our bridge, we address models in which a trend and seasonal component are estimated by multiple regression. Unlike smoothing methods, these models require an understanding of the period of the seasonal effects. They also carry assumptions and conditions—in particular linearity and constant variance (called stationarity for time series data.)

Consideration of these assumptions leads us to compare additive and multiplicative models, which differ primarily by whether we fit a model for the data or for a logarithmic transformation of the data.

Teaching Notes The material of this chapter is presented in other orders in other texts. Our topic order reflects the approach of the entire book. We first discuss time series in general, identifying components and important attributes. Then we discuss models for time series, starting with the simplest descriptive models (smoothing methods), then more sophisticated descriptive methods (exponential smoothing, autoregressive models), and then on to models that support inference (additive and multiplicative models fit by multiple regression). If you have skipped Chapters 20 and 21 on your way to Chapter 22, then students will be prepared for the earlier sections, and you’ll probably stop with Section 22.8.

The Whole Foods Market data with which we open the chapter, and which we follow through, has a punch line.—an explanation for the seasonal pattern that is unexpected. We reveal the truth in Section 22.11, and admit that the authors missed it at first. The lesson is that we shouldn’t jump to conclusions to provide “explanations” for patterns in time series; there may be more there than meets the eye. .

22-2 Part IV Using Statistics for Decision Making We have been explicit about a few notational quirks that other texts leave ambiguous. (For example, the time subscript of the last observed point used by a smoothing model and the notation for the first value forecast by that model.) Don’t let that be a sticking point.

Forecasting A principal concern with time series—especially in business applications—is the attempt to forecast into the future. In previous chapters, we’ve cautioned students against extrapolations beyond the range of the available data. But, of course, that’s just what we’d like to do. Now a more sophisticated approach is appropriate. Although it is surely the case that much data have no “inertia,” and thus cannot be expected to continue changing in whatever way they have changed in the past, it is also true that some patterns are understandable and reliable. The seasonal component of a time series is often one such reliable pattern. We can expect sales of skis to continue to be strong in the Fall and weak in the Spring, as they have been in the past, because we understand the mechanism and that it is driven by seasonal changes we can safely assume will continue.

On the other hand, it is especially risky to assume that exponential growth in a trend component will continue. Exponential growth cannot continue for long before consuming everything in sight.

The intelligent use of time series models is a valuable business planning skill.

Looking Ahead We may not be able to forecast into the indefinite future, but we can state that this chapter is the end of the sequence of chapters that began back in Chapters 6 and 7 dealing with regression-based models. These have formed a consistent theme throughout the course. There is (always) more that can be said, but we won’t be saying it in this book.

Class Do’s Do encourage students to compare descriptions and models for the same time series data by different methods and with different choices of model parameters. Make lots of graphs. Then make some more. Graph the data with the model traced over it. Then graph the residuals away from the smooth trace or model. There are few rules here, so it is important for students to get an intuitive feeling for how these methods compare. Of course, a computer program is essential.

The discussion of Choosing a Time Series Forecasting Method (section 22.10) can provide a good basis for a class discussion.

The Importance of What You Don’t Say Predict We generally have preferred “forecast” although we use both terms. The primary caution is not to give students the impression that we have a magic crystal ball. Students (and, indeed, many professionals) are far too willing to believe that the computer is more powerful than it could possibly be and can answer questions that cannot be answered by the data we have available.

Human beings have wanted to see into the future; every culture has fairy tales and myths that include omens and forecasts. Be sure that students know that this chapter has not suspended the laws of the universe. Time still proceeds in only one direction; we cannot see into the future even when we can make shrewd guesses.

Class Examples •

We work through several large applications in the exercises. Any of them would make a good class example. Try out several methods. Ask the class to guess what would happen .

Chapter 22 Time Series Analysis 22-3 with various changes in parameters. How would changing the length of a moving average or the value of  in an exponential smoother be likely to change the result?

•

Emphasize that time series analysis requires much judgment, so students should expect to understand their data. What is the seasonal component likely to represent? Is that a phenomenon you’d expect to continue into the period in which you’d like to forecast? How about the trend? Can that be expected to continue?

•

Look back at the oil price example of Chapter 19, Section 2. Would the methods of this chapter have done any better? For a real scare, have students look up oil prices since that example was printed.

Resources ActivStats* • Lesson 20: Time Series • The user manual of whatever statistics software your students are using will discuss the time series methods and displays available. Most also provide examples. Some offer only some of these methods. Others offer greater sophistication, for example, in autoregressive methods.

Assignments Experience counts here. Have students do several analyses comparing different methods and explaining how and why they differ.

Web Links •

The Toronto Stock exchange offers time series based data that students can reference under-price history: http://www.tsx.com/. The Bank of Canada also has much time series based economic data: http://www.bankofcanada.ca/rates/

Basic Exercises 1. Consider the following showing the average yearly price of homes ($ thousands) in a city.

Year 2009 2010 2011 2012 2013 2014 2015

Average Price 420 435 470 510 505 530 550

Construct a price index using 2011 as the base year. * ActivStats is available at www.pearsonmylabandmastering.com/northamerica/mystatlab/

.

22-4 Part IV Using Statistics for Decision Making 2. Consider the following monthly time series.

Month 1 2 3 4 5 6 7 8 9 10 11 12 a. b.

Value 120 121 125 123 126 127 132 125 130 134 140 136

d.

Prepare a time series plot. Are the data stationary in the mean? Explain.

Develop a 2-month moving average for this time series. What is the forecast for month 13?

Develop a 4-month moving average for these data. What is the forecast for month 13?

What is the naïve forecast for month 13?

3.

Refer to exercise 2.

c.

a. b. c. d.

4.

Refer again to exercise 2. a. b. c. d.

5.

Compute the MSE for the 2-month moving average.

Compute the MSE for the 4-month moving average.

Compute the MSE for the naïve method.

Which method is best? Explain.

Compute the MAPE for the 2-month moving average.

Compute the MAPE for the 4-month moving average.

Compute the MAPE for the naïve method.

Which method is best with respect to MAPE? Do MSE and MAPE give the same result? Explain.

Consider the following daily time series. Assume the initial smoothed value is the initial observed value for forecasting using SES. Day 1 2 3 4 .

Value 6 10 8 5

Chapter 22 Time Series Analysis 5 6 a. b. c. d.

6.

22-5

7 11

Use a single exponential smoothing (SES) model with α = 0.1 for this time series. What is the forecast for day 7?

Use a single exponential smoothing (SES) model with α = 0.5 for this time series. What is the forecast for day 7?

Use a single exponential smoothing (SES) model with α = 0.9 for this time series. What is the forecast for day 7?

Which model produces the smoothest and most stable forecast?

Refer to exercise 5. a. b. c. d.

Compute the MSE for the SES model with α = 0.1.

Compute the MSE for the SES model with α = 0.5.

Compute the MSE for the SES model with α = 0.9.

Which method is best? Explain.

7.

Refer again to exercise 5.

a. b. c. d.

Compute the MAPE for the SES model with α = 0.1.

Compute the MAPE for the SES model with α = 0.5.

Compute the MAPE for the SES model with α = 0.9.

Which method is best? Explain. Do MSE and MAPE give the same results?

8.

Autoregressive models were fit to a time series consisting of monthly sales data.

The most recent values of this series are 58, 63, 64, 64, 63, 55, 54 and 44.

a.

The resulting AR(1) model is yˆ = 7.46 + 0.857 ylag1 . Use this model to forecast the

b. c.

most recent values of this series.

Compute MSE and MAD. The resulting AR(4) model is yˆ = 9.13 + 1.05 ylag1 – 0.092 ylag2 – 0.266 ylag3 + 0.131ylag4

d.

9.

a. b. c. d.

Use this model to forecast the most recent values of this series.

Compute MSE and MAD. Which AR model is better? Explain.

A regression model was fit to a time series that exhibits both trend and seasonality.

The resulting model, based on 60 observations, is yˆt = 55.5 + 12.8 Time – 32.5 Q1 − 42.5 Q2 – 20 Q3 Is there a positive or negative trend in the series? Explain.

Which quarter has the highest values, on average? Explain.

Use this model to forecast for the next period (it is the first quarter of the next year).

Use this model to forecast for the fourth quarter of the next year.

.

22-6 Part IV Using Statistics for Decision Making ANSWERS 1.

Year 2009 2010 2011 2012 2013 2014 2015

Average Price 89.4 92.6 100 108.5 107.4 112.8 117.0

2. a. The series has an increasing trend with some fluctuations around the trend. There are also some outlying points that may be worth investigating.

b.

Month 1 2 3 4 5 6 7 8 9 10 11 12 .

Value 120 121 125 123 126 127 132 125 130 134 140 136

MA (2) * 120.5 123.0 124.0 124.5 126.5 129.5 128.5 127.5 132.0 137.0 138.0

Chapter 22 Time Series Analysis

22-7

The forecast for period 13 is 138.0. c.

Month 1 2 3 4 5 6 7 8 9 10 11 12

Value 120 121 125 123 126 127 132 125 130 134 140 136

MA (4) * * * 122.25 123.75 125.25 127.00 127.50 128.50 130.25 132.25 135.00

The forecast for period 13 is 135.00. d.

The naïve forecast for period 13 is 136.

3.

a. b. c. d.

19.05 27.4766 18 The naïve method is the best with the lowest MSE.

4.

a. b. c. d.

2.7377 3.4998 2.9333 MA (2) has the lowest MAPE. So the two accuracy measures do not give the same results as MSE favours the naïve approach and MAPE the MA(2).

5.

a. b. c. d.

6.91724 8.875 10.5832 SES with α = 0.1 produces the smoothest and most stable forecast.

6.

a. b. c. d.

6.988 7.2188 8.1262 SES with α = 0.1 has the lowest MSE.

7.

a. b. c.

23.4924 24.2965 30.8586 .

22-8 Part IV Using Statistics for Decision Making d.

SES with α = 0.1 has the lowest MAPE. Yes, both accuracy measures favor the same model.

8.

a. b. c. d.

57.166, 61.451, 62.308, 62.308, 61.451, 54.595, 53.738 MSE = 25.8104, MAD = 3.93586 61.282, 60.621, 52.444, 52.396 MSE = 26.8653, MAD = 4.32275 The AR(1) model is preferred with lower MSE and MAD values.

9.

a. b.

Positive trend since the slope associated with Time is positive.

The fourth quarter has the highest values, on average, because all of the regression coefficients associated with quarters 1 through 3 are negative.

803.8 874.7

c. d.

.

23-1

Chapter 23 Decision Making and Risk What’s It About?

We discuss Actions and States of Nature displayed in Payoff Tables or Decision Trees.

When we attach probabilities to the states of nature, we can find expected values associated with each action. Introducing conditional probabilities allows more complex situations to be analyzed.

Comments This is a business-centered application of the ideas of expected value and variance that might have seemed abstract in Chapter 9. Here the outcomes have units of dollars, making the expectations more concrete.

The specific numbers in the examples in this chapter are, almost by necessity, invented for pedagogical purposes, but they are realistic and based on real decisions made by business managers. One of the authors is a co-founder of the company used for the main example (Data Description) and the insurance problem in the Guided Example is from a recent case study published in the Journal of the Academy of Business Education.

Class Do’s Be sure to open the discussion with the example, which is based on a real potential decision. In the context of these data, emphasize the following definitions: • Actions vs. states of nature • Payoffs (including units) Then set up the example both in the form of a payoff table and a decision tree, so the students understand the relationship of the two forms of display. Using the payoff table and the Data Description example, the students should learn how to compare the results for the following strategies: • Minimax • Maximax • Minimin The concept of Expected Value can be explained using both the payoff table format (usually using Excel) and the decision tree format by multiplying across the tree as shown in Figure 23.2.

The concept of Expected Value of Perfect Information can be challenging for students because they understand that we can’t know for sure outcomes before the states of nature occur. However, they can usually understand that if we had access to “expert” or “inside” information that could theoretically provide advance knowledge or warning of what will happen, we can then ask what would be the expected value. It is also important to explain the difference between EVPI and EVwPI using the equation in Section 23.5.

In the absence of perfect information, it is common for companies to improve their knowledge base by hiring a marketing firm to conduct surveys or market research.

Therefore, the section of sample information is applicable and usually interesting to .

23-2 Part IV Using Statistics for Decision Making students. At this point in the course, it’s a good idea to review the concept of conditional probability because the students haven’t seen this since early in the course.

The concepts of variation build on the computations in Chapter 9, and it’s an easy extension to the concepts of: • Coefficient of variation (CV) • Return to risk ration (RRR) In addition to covering these concepts in the context of the Data Description example, it’s a good idea to go through the Guided Example (GE) to provide another application.

The Importance of What You Don’t Say Don’t give the impression that we are certain of our decision – or that there is only one correct decision. The probabilities used in decision analysis are estimates – either based on expert knowledge, historical patterns, or benchmarking. It’s a good idea to emphasize the notion of sensitivity analysis, which involves computing the expected outcomes using a range of different probabilities. This is easy to do in Excel and students are often interested to see the impact of even small changes in the probabilities.

Class Examples 1. If you cover the Guided Example on the Insurance Services, then a good followup is to use this same example to conduct a sensitivity analysis and observe the impact on revenue. This is the topic of a Mini Case Study Project at the end of the chapter. While the probabilities of weak and strong tourism are estimates, the rest of the monetary values are based on the published case and a real business decision. Depending on the current state of the economy when this chapter is being taught, both the probabilities of the strength of tourism and the potential growth rates can be varied over a range.

•

The classic example for this chapter is a comparison of the outcome for a company on how best to provide technical support for its customers.

Namely, through an online FAQ, a free help desk, or a pay help desk.

This example complements the information the students are getting in their business math, finance and economics courses. It is a fairly easy example for them to understand and that they should be able to relate to.

A sensitivity analysis can be conducted using a range of probabilities.

Supplementary Software • PrecisionTree 7.5 by Palisade uses decision analysis techniques and decision trees in Excel to identify outcomes for alternative actions. This software is easy to learn for students and is currently in use in a wide range of industries, including the defense, finance, insurance, and medical industry.

• Crystal Ball by Oracle is spreadsheet based software that can be used for simulations and sensitivity analysis in decision making. It is fairly easy to learn and is in use in the finance, manufacturing, and pharmaceutical industries to inform financial decisions.

• @Risk by Palisade is also spreadsheet based software that uses Excel for simulation and optimization techniques to perform sensitivity analysis in decision .

Chapter 23 Decision Making and Risk 23-3 making. It provides a wide range of probability distributions that facilitate the level and accuracy of the sensitivity calculations and is a good complement to Precision Tree 7.5.

•

MyLab Statistics allows students to practice many of this chapter’s exercises. For many of the exercises, it features step-by-step guided solutions.

Basic Exercises 1.

The payoff table below shows costs for a decision that involves two actions and three states of nature.

States of Nature Actions a1 a2 a. b. c. d.

2.

s1 220 150

s2 350 240

s3 200 390

What is the minimax choice?

What is the minimin choice?

Suppose P(s1) = .2, P(s2) = .2, and P(s3) = .6. What is the expected value of each action?

What is the best choice using the expected value approach?

The payoff table below shows profits for a decision that involves three actions and three states of nature.

States of Nature Actions a1 a2 a3 a. b. c. d.

3.

s1 550 410 750

s2 600 590 710

s3 710 800 590

What is the maximin choice?

What is the maximax choice?

Suppose P(s1) = .5, P(s2) = .3, and P(s3) = .2. What is the expected value of each action?

What is the best choice using the expected value approach?

The payoff table below shows profits for a decision that involves two actions and three states of nature. Assume that P(s1) = .3, P(s2) = .3, and P(s3) = .4.

States of Nature Actions a1 a2

s1 110 50

s2 150 200 .

s3 220 350

23-4 Part IV Using Statistics for Decision Making a. b. c. d.

4.

A decision involves two actions and two states of nature. Moreover, the decision maker has the option to obtain additional information. Without any additional information, a1 is the best choice with an expected profit of $10,000. If the additional information reveals that s1 is more likely, then the best choice is a2 with an expected profit of $12,000. If the additional information reveals that s2 is more likely, then the best choice is a1 with an expected profit of $9000. a.

b. c.

d.

5.

What is the expected value of each action?

What is the best choice using the expected value approach?

What is the expected value with perfect information?

What is the expected value of perfect information (EVPI)?

Suppose that the probability of additional information revealing s1 as more likely is .75 and s2 as more likely is .25, what is the expected value with sample information (EVwSI)?

What is the expected value of sample information (EVSI)?

Suppose that the probability of additional information revealing s1 as more likely changed from .75 to .20. Compute the expected value with sample information (EVwSI).

Under which scenario should the decision maker obtain additional information? Explain.

Consider the following payoff table showing costs for a decision that involves three actions and three states of nature. Assume that P(s1) = .7, P(s2) = .2, and P(s3) = .1.

States of Nature Actions a1 a2 a3 a. b. c. d.

6.

s1 22 30 10

s2 40 30 20

s3 35 25 55

What is the expected value of each action?

Based on the expected value approach, what is the best choice?

What is the standard deviation in outcomes for each action?

Which action has the least variable outcomes?

Consider the following payoff table showing profits for a decision that involves three actions and two states of nature. Assume that P(s1) = .8 and P(s2) = .2.

States of Nature Actions a1 a2 a3

s1 450 800 400

s2 425 150 650

.

Chapter 23 Decision Making and Risk 23-5 a. b. c. d.

7.

Refer to Exercise 6. a. b. c. d.

8.

What is the expected value of each action?

Based on the expected value approach, what is the best choice?

Compute the CV (coefficient of variation) for each action.

If the decision maker is risk averse, would the CV approach lead to the same choice as the expected value approach? Explain.

What is the return to risk ratio for a1?

What is the return to risk ratio for a2?

What is the return to risk ratio for a3?

Based on the RRR, which action is favoured? Explain why the RRR and CV lead to the same choice.

Consider the following joint probability table.

Event x y a. b. c.

A .10 .10

B .40 .10

C .15 .15

Find P(A), P(B) and P(C).

Find the following conditional probabilities: P(x/A), P(x/B), P(x/C), P(y/A), P(y/B) and P(y/C).

Use Bayes Theorem to find the reverse conditional probabilities: P(A/x), P(A/y), P(B/x), P(B/y), P(C/x) and P(C/y).

ANSWERS 1.

a. b. c. d.

a1 a2 EV(a1) = 234, EV(d2) = 312 a1

2.

a. b. c. d.

a3 a2 EV(a1) = 597, EV(a2) = 542, EV(a3) = 706 a3

3.

a. b. c. d.

EV(a1) = 166, EV(a2) = 215 a2 233 18

4.

a. b. c.

$11,250 $1250 $9600 .

23-6 Part IV Using Statistics for Decision Making d.

The decision maker should not obtain additional information when the probability revealing s1 changes from .75 to .20 because the expected value with sample information is less than the expected value without sample information.

a. b. c.

EV(a1) = 26.9, EV(a2) = 29.5, EV(a3) = 16.5 a3 SD ( a1 ) = 7.595 , SD ( a2 ) = 1.5 , SD ( a3 ) = 13.426

d.

a2

6.

a. b. c. d.

EV(a1) = 445, EV(a2) = 670, EV(a3) = 450 a2 CV(a1) = .022, CV(a2) = .388, CV(a3) = .222 No. The lowest CV leads to the least risky choice which is a1.

7.

a. b. c. d.

44.5 2.58 4.5 The RRR is the reciprocal of the CV. Again a1 is favoured as it is the least risky for its expected return.

8.

a.

P ( A) = 0.20 , P ( B ) = 0.50 , P ( C ) = 0.30

b.

P ( x A) = 0.50 , P ( x B ) = 0.80 , P ( x C ) = 0.50

5.

P ( y A) = 0.50 , P ( y B ) = 0.20 , P ( y C ) = 0.50 c.

P ( A x ) = 0.154 , P ( A y ) = 0.286 P ( B x ) = 0.615 , P ( B y ) = 0.286 P ( C x ) = 0.231 , P ( C y ) = 0.429

.

24-1

Chapter 24 Quality Control What’s It About?

Quality control methods have a long history in the field of statistics. Based on basic properties of the Normal distribution and on application of the Central Limit Theorem, quality control methods guide us in monitoring measures of quality of products and services. They form part of a larger effort of total quality management to help ensure that the products and services produced meet the standards of those who are responsible for them.

Comments This chapter serves as an introduction to the field of quality control. There are more advanced and efficient charts, but the methods shown here should provide a basis for a more thorough reading and understanding. Here are some thoughts for classroom discussion of the major points.

•

Basic Charting Ideas. All charts share certain characteristics. The chart is a time series plot of some characteristic of the process to be monitored. The process needs to be stationary (in the sense of time series) and in-control when one sets up the control chart. One of the primary purposes of a control chart is to detect and correct the causes (called special causes) that may make a process change and go out-of-control. The central line of a control chart is based on the past data of the process when it is known to be in-control. The lines that tell the user when to intervene are called control or specification limits depending on whether they are based on the process data (control) or on external design considerations (specifications).

•

Errors. As with hypothesis tests, charts are always subject to making Type I or Type II errors. A type I error is a false alarm in this context because it would be taking an action when unnecessary. A type II error would be a failure to detect a change in the process. The expected amount of time the chart will monitor an in-control process without giving a false alarm is called the average run length (or ARL). The ability of a chart to detect shifts can be described by its operating characteristic (or OC) curve which displays the probability of failing to detect shifts of various sizes (measured in standard deviation units).

•

Process Capability Studies. There are various measures that judge how well a process behaves compared to its specification limits. These process capability measures are crucial for assessing the stability of the process and its capacity for staying in control.

•

Runs Charts. The simplest quality control chart is simply a plot of an individual quality measure of a process over time. The upper and lower control limits are typically set either at a distance of 3 from the center line, or are based on external specifications. There are many rules for deciding when a process is out of control. The most common are the Western Electric Handbook (WECO) rules. These are commonly used in practice and various software packages implement different combinations of them

•

X-bar, R and S Charts. Rather than plot the individual observations, the X-bar chart plots the averages of the quality measure from a sample of size n. As with runs charts, various rules for determining when the process is out of control are used. The R chart plots the range of the sample and is used in conjunction with an X-bar chart to monitor stationarity in the standard deviation of the process. An alternative to the R chart is the S chart, based on the standard deviation of the sample instead of the range. More advanced charts for Copyright © 2021 Pearson Canada, Inc.

25-2 Part IV Using Statistics for Decision Making detecting a shift include the cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) chart.

•

Actions for Out-of-Control Processes. Once an out-of-control signal is seen, there are various methods for diagnosing and correcting the problem. We present some of the basics in this section and a Guided Example to illustrate.

•

Control Charts for Attributes. Control charts can also be used for non-quantitative measures. We present basic charts for proportions (p-charts) and counts (c-charts) of defects, counts.

•

Quality Control in Industry. As mentioned earlier, quality control charts are a small, but important, part of improving quality. In this section we introduce the ideas of W. Edwards Deming and Joseph M. Juran and the important philosophy of the Six Sigma movement.

The chapter includes a Guided Example that leads the students through a real case study of control charting, detecting an out-of-control process and the steps that led to correcting the problem. Also, there are two problems in Mini Case Studies section.

Looking Ahead This chapter doesn’t lead to any of the others in the book. It is a special topic that builds on the material in earlier chapters, in particular the Normal model.

Class Do’s Do encourage students to think about the real world challenges of monitoring quality.

Emphasize that the mechanics of a quantitative variable (runs, X-bar) control chart are really simple applications of the properties of a Normal model. The rules are different ways of determining rare events.

Emphasize that the probabilities of making a Type I or Type II error are the main characteristics of a control chart that must be balanced, and that the operating characteristic curve is a way of seeing the probability of a Type II error easily.

Encourage students to read about modern quality control philosophy in the works of Deming, Juran and other quality “gurus”.

The Importance of What You Don’t Say Other charts. There are more complicated charts that students may encounter in practice. Don’t dwell on the technical details, but emphasize that the principles are the same for all control charts.

Class Examples Quality control examples take up a lot of space (for the graphs) and time (for the analysis). We therefore prefer to spend time doing one example in detail instead of several smaller examples which may only cover a small part of this topic. The Guided Example KomTeK Technologies is excellent for this purpose.

Copyright © 2021 Pearson Canada, Inc.