Type:
Solution Manual
Resource:
Business Statistics
Edition:
4th Edition
Author(s):
Norean Sharpe Richard De Veaux Paul Velleman David Wright
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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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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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
.
# of Siblings
# of Provinces Visited
Shoe Size
Height
2-1
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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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
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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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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
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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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