Business Statistics, First Canadian Edition with My StatLabNorean R. Sharpe Solution Manuals by TBtutors

Type:

Solution Manual

Resource:

Business Statistics

Edition:

1st Edition

Author(s):

Norean R. Sharpe

Instructor's Solutions Manual Linda Dawson David Wright University of Ottawa

for

Business Statistics Canadian Edition Norean R. Sharpe Babson College

Richard D. De Veaux Williams College

Paul F. Velleman Cornell University

David Wright University of Ottawa

Contents Part I Exploring and Collecting Data Chapter 2

Data 1

Chapter 3

Surveys and Sampling 6

Chapter 4

Displaying and Describing Categorical Data 15

Chapter 5

Displaying and Describing Quantitative Data 32

Chapter 6

Randomness and Probability 56

Chapter 7

Scatterplots, Association, and Correlation 69

Chapter 8

Introduction to Linear Regression 83

Part II Understanding Data and Distributions Chapter 9

Random Variables and Probability Distributions 113

Chapter 10

Sampling Distributions 132

Chapter 11

Confidence Intervals for Proportions 1434

Chapter 12

Testing Hypotheses About Proportions 165

Chapter 13

Confidence Intervals and Hypothesis Tests for Means 186

Chapter 14

Comparing Two Means 206

Chapter 15

Paired Samples and Blocks 226

Chapter 16

Interference for Counts: Chi-Square Tests 246

Chapter 17

Nonparametric Methods 269

Part III Exploring Relationships Among Variables Chapter 18

Inference for Regression 279

Chapter 19

Understanding Residuals 306

Chapter 20

Multiple Regression 319

Chapter 21

Building Multiple Regression Models

328

Part IV Building Models for Decision Making Chapter 22

Time Series Analysis 337

Chapter 23

Decision Making and Risk 353

Chapter 24

Design of Experiments and Analysis of Variance, ANOVA 362

Chapter 25

Quality Control 370

Chapter 2 – Data 1.

Canadian Labour Force. a. Someone on vacation from a full time job is employed (they are on the payroll of their employer) b. Someone who is not working, has a job offer, but is trying to find a better offer is unemployed because they are not working but are available and searching for work c. Someone who looked for work until 6 months ago, but then gave up looking is not in the labour force because they are not searching for work.

2. Domestic Credit in Canada. Imagine collecting the data and putting it in a Table with one row for each year and two columns corresponding to the two variables: domestic credit and GDP. The rows identify the WHO of the data (i.e. the years) and the column headings identify the WHAT (domestic credit and GDP). WHEN is recent years, WHERE is Canada, WHY is to investigate possible future trends. The two variables, domestic credit and GDP, are both quantitative and measured in $bn. 3.

Oil spills. The description of the study has to be broken down into its components in order to understand the study. Who–50 tankers having recent oil spills; What–what is being measured–date, spillage amount (no specified unit) and cause of puncture; When–recent years; Where–United States; Why–not specified but probably to determine whether or not spillage amount per oil spill has decreased since Congress passed the 1990 Oil Pollution Act and use that information in the design of new tankers; How–how was the study conducted–not specified, although it is mentioned that the data is online; Variables–what is the variable being measured–there are 3 variables–the date, the spillage amount which is quantitative, and the cause of the puncture which is categorical; Concerns–more detail needed on the specifics of the study.

Sales. The description of the study has to be broken down into its components in order to understand the study. Who– who or what was actually sampled–months at a major Canadian company; What–what is being measured–money spent on advertising ($ thousand) and sales ($ million); When–monthly for the last 3 years; Where–United States (assumed); Why–to compare money spent on advertising to sales; How–how was the study conducted–not specified; Variables–what is the variable being measured–there are 3 variables–the date, the amount of money spent on advertising which is quantitative, and sales which is quantitative; Concerns–none.

Food store. Who– who or what was actually sampled–existing stores; What– what is being measured– weekly sales ($), town population (thousands), median age of town (years), median income of town($), and whether or not the stores sell beer/wine; When–not specified; Where–United States; Why–the food retailer is interested in understanding if there is an association amongst these variables to help determine where to open the next store; How–how was the study conducted–data collected from their stores; Variables–what is the variable being measured– sales ($), town population (thousands), median age of town (years), median income of town($), which are all quantitative. Whether or not the stores sell beer/wine is categorical; Concerns–none.

Sales II. Who– who or what was actually sampled–quarterly data from a major U.S. company; What–what is being measured–quarterly sales ($ million), unemployment rate (%), inflation rate (%); When–quarterly from 2004–2006; Where–United States; Why–to determine how sales are affected by the unemployment rate and inflation rate; How–how was the study conducted–not specified; Variables–what is the variable being measured–quarterly sales ($ million), unemployment rate (%), and inflation rate (%) which are quantitative; Concerns–none.

Subway’s menu. Who–Subway’s sandwiches; What–type of meat, number of calories (in calories), and serving size (in ounces); When–not specified; Where–Arby’s restaurants; Why–assess the nutritional value of the different sandwiches; How–information gathered on each of the sandwiches offered on the menu; Variables–the number of calories and serving size (ounces) are quantitative, and the type of meat which is categorical; Concerns–none.

. 1

Part I Exploring and Collecting Data 8.

MBA admissions. Who–MBA applicants; What–sex, age, whether or not accepted, whether or not they attended, and the reasons for not attending (if they did not accept); When–not specified; Where–a school in the northeastern United States; Why–the researchers wanted to investigate any patterns in female student acceptance and attendance in the MBA program; How–data obtained from the admissions office; Variables––sex, whether or not the students accepted, whether or not they attended, and the reasons for not attending if they did not accept (all categorical) and age (years) which is quantitative; Concerns–none.

Climate. Who–385 species of flowers; What–date of first flowering (in days); When–data gathered over the course of 47 years; Where–southern England; Why–the researchers wanted to investigate if the first flowering is indicating a warming of the overall climate; How–not specified; Variables––date of first flowering is a quantitative variable; Concerns–date of first flowering should be measured in days from January 1 to address leap year issues.

10. MBA Admissions II. Who–MBA students; What–each student’s standardized test scores and GPA in the MBA program; When–2000–2005; Where–London; Why–to investigate the association between standardized test scores and performance in the MBA program over the last five years; How–not specified; Variables––standardized test scores and GPA, both quantitative variables; Concerns–none. 11. Schools. Who–students; What–age (years or years and months), race or ethnicity, number of days absent, grade level, reading score, math score, and any disabilities/special needs; When–ongoing and current; Where–a Canadian Province; Why–keeping this information is a state requirement; How–data collected and stored as part of school records; Variables––there are 7 variables. Race or ethnicity, grade level, and disabilities/special needs are categorical variables. Number of absences, age (years or years and months), reading scores, and math scores are quantitative variables; Concerns–what tests are used to measure reading and math ability and what are the units of measurement? 12. Pharmaceutical firm. Who–experimental participants; What–herbal cold remedy or sugar solution, and cold severity; When–not specified; Where–major pharmaceutical firm; Why–scientists were testing the effectiveness of an herbal compound on the severity of the common cold; How–scientists conducted a controlled experiment; Variables––there are 2 variables. Type of treatment (herbal or sugar solution) is categorical, and severity rating is quantitative; Concerns–the severity of a cold might be difficult to quantify (beneficial to add actual observations and measurements, such as body temperature). Also, scientists at a pharmaceutical firm could have a predisposed opinion about the herbal solution or may feel pressure to report negative findings about the herbal product. 13. Start-up company. Who–customers of a start-up company; What–customer name, ID number, region of the country, date of last purchased, amount of purchase ($), and item purchased; When–present day; Where–Canada (assumed); Why–the company is building a database of customers and sales information; How–assumed that the company records the needed information from each new customer; Variables–– there are 6 variables: name, ID number, region of the country, and item purchased which are categorical and date and amount of purchase ($) are quantitative; Concerns–although region is coded as a number, it is still a categorical variable. 14. Cars. Who–cars parked in executive and staff lots at a large company; What–make, country of origin, type of vehicle (car, van, SUV, etc.), and age of vehicle (probably in years); When–not specified; Where–a large company; Why–not specified; How–data recorded in executive and staff lots of a large company; Variables––make, country of origin, and type of vehicle are 3 categorical variables. Age is the single quantitative variable. Whether or not the vehicle is in an executive or staff lot is also a categorical variable; Concerns–none. 15. Vineyards. Who–vineyards; What–size (acres), number of years in existence, state, varieties of grapes grown, average case price ($), gross sales ($), and percent profit; When–not specified; Where–assume United States as state is recorded; Why–business analysts hope to provide information that would be helpful to grape growers in the United States; How–not specified; Variables––size of vineyard (acres), number of years in existence, average case price ($), gross sales ($), and percent profit are 5 quantitative variables. State and variety of grapes grown are categorical variables; Concerns–none. .

Chapter 2 Data

16. Environment. Who–streams; What–name of stream, substrate of the stream (limestone, shale, or mixed), acidity of the water (measured in PH), temperature (degrees Celsius), and BCI (a measure of biological diversity – unknown units); When–not specified; Where–Alberta; Why–research conducted for an ecology class; How–not specified; Variables–– there are 5 variables. Name of stream and substrate of the stream (limestone, shale, or mixed) are categorical variables. Acidity of the water (PH), temperature (degrees Celsius), and BCI (a measure of biological diversity – unknown units) are quantitative variables; Concerns–none. 17. Gallup Poll. Who–1,180 American voters; What–region (Northeast, South, etc.), age (in years), party affiliation, whether or not the person owned any shares of stock, and their attitude (scale 1 to 5) toward unions; When–not specified; Where–United States; Why–the information was gathered as part of a Gallup public opinion poll; How–telephone survey; Variables–– there are 5 variables. Region (Northeast, South, etc.), party affiliation, and whether or not the person owned any shares of stock are categorical variables. Age (in years), and their attitude (scale 1 to 5) toward unions are quantitative variables; Concerns–none. 18. FAA. Who–all airline flights in the United States; What–type of aircraft, number of passengers, whether departures and arrivals were on schedule, and mechanical problems; When–the information is currently recorded; Where–United States (assumed); Why–the information is required by the CTA; How–data is collected from airline flight information; Variables–– there are 4 variables. Type of aircraft, whether departures and arrivals were on schedule, and mechanical problems are categorical variables. Number of passengers is a quantitative variable; Concerns–none. 19. Mobile Phones. Who–mobile phone manufacturers; What–sales; When–last 3 years; Where–world wide; Why–to project the future of the mobile phone business. Variable–sales which is a quantitative variable measured in $. 20. Canadian Families. Who–all Canadians (since it is a census); What–family type; When–every 5 years; Where–Canada; Why–to investigate social trends. Variable–family type which is a categorical variable. 21. Canadian Oil and Gas Production. Who–crude oil, natural gas, natural gas by-products; What–Value and volume; When–every year; Where–Canada; Why–not specified. Variables–value and volume, both quantitative measured in $ and m3 respectively. 22. Overnight Visitors to Canada. Who–overnight visitors to Canada; What–number of nights spent in Canada and money spent in Canada; When–each year; Where–Canada; Why– to provide information for the tourism industry. Variables–number of nights spent in Canada and money spent in Canada both of which are quantitative variables with no units and units of $ respectively. 23. Stock market. Who–students in an MBA statistics class; What–total personal investment in stock market ($), number of different stocks held, total invested in mutual funds ($), and the name of each mutual fund; When–not specified; Where–a business school in Toronto; Why–the information was collected for use in classroom illustrations; How–an online survey was conducted, participation was probably required for all members of the class; Variables–– there are 4 variables. Total personal investment in stock market ($), number of different stocks held, total invested in mutual funds ($) are quantitative variables. The name of each mutual fund is a categorical variable; Concerns–none. 24. Theme park sites. Who–potential theme park locations; What–country of site, estimated cost ($), potential population size (counts), size of site (hectares), whether or not mass transportation within 5 minutes of site; When–2010; Where–Europe; Why–to to present to potential developers on the feasibility of various sites; How–not specified; Variables–– there are 5 variables. Country of site and whether or not mass transportation within 5 minutes of site are both categorical variables. Estimated cost (€), potential population size (counts) and size of site (hectares) are quantitative; Concerns–none. 25. Indy. Who–Indy 500 races; What–year, winner, car model, time (hrs), speed (mph), and car number; When–1911-2007; Where–Indianapolis, Indiana; Why–examine trends in Indy 500 race winners; How– official statistics kept for each race every year; Variables–– there are 6 variables. Winner, car model, and .

Part I Exploring and Collecting Data car number are categorical variables. Year, time (hrs) and speed (mph) are quantitative variables; Concerns–none. 26. Kentucky Derby. Who–Kentucky Derby races; What–date, winner, winning margin (in lengths), jockey, winner’s payoff ($), duration of the race (minutes and seconds), and track conditions; When–1875-2004; Where–Churchill Downs, Louisville, Kentucky; Why–examine trends in Kentucky Derby winners; How– official statistics kept for each race every year; Variables–– there are 7 variables. Winner, winning jockey, and track conditions are categorical variables. Date, winning margin (in lengths), winner’s payoff ($), and duration of the race (minutes and seconds) are quantitative variables; Concerns–none. 27. Mortgages. Each row represents each individual mortgage loan. Headings of the columns would be: borrower name, mortgage amount. 28. Employee performance. Each row represents each individual employee. Headings of the columns would be: Employee ID Number (to identify the row instead of name), contract average ($), supervisor’s rating (110), and years with the company. 29. Company performance. Each row represents a week. Headings of the columns would be: week number of the year (to identify each row), sales prediction ($), sales ($), and difference between predicted sales and realized sales ($). 30. Command performance. Each row represents a Broadway show. Headings of the columns would be: the show name (identifies the row), profit or loss ($), number of investors, and investment total ($). 31. Car sales. Cross-sectional are data taken from situations that vary over time but measured at a single time instant is said to be a cross-section of the time series. This problem focuses on data for September only which is a single time period. Therefore, the data are cross-sectional. 32. Motorcycle sales. Time series data are measured over time. Usually the time intervals are equally-spaced (e.g. every week, every quarter, or every year). This problem focuses on the number of motorcycles sold by the dealership in each month of last year; therefore, the data are measure over a period of time and are time series data. 33. Cross sections. Time series data are measured over time. Usually the time intervals are equally-spaced (e.g. every week, every quarter, or every year). This problem focuses on the average diameter of trees brought to a sawmill in each week of a year; therefore, the data are measure over a period of time and are time series data. 34. Series. Cross-sectional are data taken from situations that vary over time but measured at a single time instant is said to be a cross-section of the time series. This problem focuses on data for attendance of the third World Series game. Therefore, the data are cross-sectional.

Chapter 2 Data

MiniCase: Ottawa Senators PLAN DO

Setup Clarify the objective

Identify the types of data in the data file

Mechanics

WHAT: The answer comes from the column headings. The shooter’s first and last names, the team, the shooter’s position, total shots, total goals, shooting percentage and the number of game deciding goals.

Describe the Ws for the data

WHO: is the shooter, since this is what each row of the table is about. WHERE, The data is from the NHL for games played across North America. WHEN: Data entries are recorded after each game has been played for the 2007-08 season WHY: The data has been collected to record the results of NHL Shootouts. Variable Type

Identify the types of variables that the data consists of

The shooters names, team and position are primary data, since it has not been processed or summarized at all. The rest of the data is secondary since it is a summary of the results of shots taken during multiple games and has therefore been processed. The whole table is cross sectional data since it all applies to the 2007-08 season. The data on shooters names, team and position are categorical and the rest of the data are quantitative.

REPORT

Summary & Conclusions

We have identified the Ws and the variable types of our data.

Summarize the results

The data is partly primary and partly secondary The data is cross-sectional. The data is partly categorical and partly quantitative.

Chapter 3 – Surveys and Sampling 1.

Roper. a. No, they were not using a random sample. As stated, the survey was designed to get 500 males and 500 females. b. They used a stratified random sample, with the strata being the gender of the respondents.

Coffee shop survey. a. No, they were not using a random sample. It would be difficult and unlikely that they could get exactly 50 from each class via a random sample. b. They probably used a stratified random sample, with the strata being the class of the respondents.

Software licenses. a. This sample was a voluntary response, not a random sample. b. There is no confidence in the estimate sampled. Voluntary response samples are almost always biased, and so conclusions drawn from them are almost always wrong.

Drugs in baseball. a. This sample was a cluster sample, with the teams selected being the cluster. A cluster is recognized by a random sample selection of teams within all of baseball. A sample is then taken from the selected teams. b. It is a reasonable solution to the problem of randomly sampling players. You can sample an entire team at once relatively easily. You could select your random sample prior to showing up unannounced but it would be less efficient to search for the players randomly selected.

Gallup. a. The population of interest is all adults in the United States aged 18 and older. b. The sampling frame, a list of individuals from which the sample will be drawn, consists of adults with landline telephones, which are the only numbers available for a study like this. c.

An increasing number within the population (e.g., many college students and others with mobile only service) don’t have landline phones, which could create a bias.

Defining the survey. a. They are using a multistage design in which the countries selected are clusters, and then a random sample is drawn within each cluster. They don’t specify how the random samples are taken. b. The difference in population size has no effect on the precision of estimates from these surveys. Only the sample size matters and the sample chosen should be representative of the entire population.

HR directors. a. Population–Human resources directors of Fortune 500 companies. b. Parameter–Proportion of those surveyed who don’t feel that surveys intruded on their work day. c.

Sampling Frame–List of all Human Resource directors at Fortune 500 companies.

d. Sample–Those who responded (23% of all HR directors). e.

Sampling method–Questionnaire mailed to all HR directors (not random sample). . 6

Chapter 3 Surveys and Sampling

Bias–Voluntary response sample. Those who responded to the questionnaire that was mailed could be predisposed to a particular answer. Because they are responding to the survey, they may be more inclined to believe that surveys like this do not intrude on their workday.

Technology Forecasting. a. Population–Researchers b. Parameter–impact of new technologies on fuel efficiency for motor vehicles c.

Sampling Frame–established researchers in the automobile industry, the clean tech industry, government research labs and universities.

d. Sample–those who respond

Sampling Method–not stated, probably stratified random sample, so as to get researchers from each of the stated industries with enough famous (high profile) researchers

Bias–the most knowledgeable researchers may not have time to respond unless they are paid to participate.

Alternative medicine. a. Population–All Consumer Union subscribers. b. Parameter–Proportion of Consumer Union subscribers who have used and benefited from alternative medicine. c.

Sampling Frame–All Consumer Union subscribers.

d. Sample–Subscribers who responded. e.

Sampling method–Questionnaire to all subscribers.

Bias–Nonresponse. Those who respond could have strong feelings about the topic and affect the results.

10. Global warming. a. Population–adults. b. Parameter–Proportion of sample who believe that global warming has already begun and the proportion of sample who think global warming will never happen. c.

Sampling Frame–adults aged 18 and over.

d. Sample–1007 randomly selected adults. e.

Sampling method–Random selection method not specified.

Bias–Probably not biased. A large sample that was randomly selected was interviewed so it follows that the conclusions could be generalized.

11. At the bar. a. Population–Adult bar patrons. b. Parameter–Proportion of sample who thought drinking and driving was a serious problem. .

Part I Exploring and Collecting Data c.

Sampling Frame–All chosen bar patrons.

d. Sample–Every 10th person leaving the bar. e.

Sampling method–Systematic sampling (every 10th person).

Bias–Probably biased toward thinking drinking and driving is not a serious problem. The sample consisted of bar patrons leaving the bar. A large percentage of them had something to drink, most likely leading to a biased viewpoint. In addition, bar patrons don’t reflect what all adults think about drinking and driving.

12. Election poll. a. Population–City voters. b. Parameter–Not clearly specified; likely, the proportion of voters who think certain issues are important or favor certain issues. c.

Sampling Frame–All city resident voters.

d. Sample–Every city resident voter in one block from each district. e.

Sampling method–Multistage design for a combination approach. A single block chosen from each district (not clear what method used in selection) represents a cluster. The sample represents all of the residents that could be found and willing to participate. This is a convenience sample.

Bias–Parameter(s) of interest not clearly specified. Convenience sampling within block clusters is not random and could produce biased results.

13. Toxic waste. a. Population–Soil located near former waste dumps. b. Parameter–Concentrations of toxic chemicals. c.

Sampling Frame–Any accessible soil surrounding a former waste dump.

d. Sample–Soil samples taken from 16 locations near a former waste dump. e.

Sampling method–Not specified how the sample locations were chosen.

Bias–Not specified how soil sample locations were chosen and therefore cannot assume they were chosen randomly, perhaps accessibility or some other factors. Unless this is known, it is possible that bias can affect the results if soil taken is more or less polluted than a random selection would produce.

14. Housing discrimination. a. Population–Landlords in a particular area. b. Parameter–Proportion of landlords illegally denying fair access to rental apartments. c.

Sampling Frame–All advertised apartments.

d. Sample–Apartments actually visited and inquired about. e.

Sampling method–Not specified how the apartments visited were chosen.

Chapter 3 Surveys and Sampling f.

Bias–Likely to be a fair study as long as the apartments visited were randomly chosen and not all in one section of town.

15. Quality control. a. Population–Snack food packages. b. Parameter–Proportion of snack food packages passing inspection, weight of bags. c.

Sampling Frame–All snack food packages produced in a day.

d. Sample–Packages in 10 randomly selected cases, 1 bag from each case for inspection. e.

Sampling method–Multistage sampling due to a combination of methods. The selection of the 10 cases is a cluster and the sampling selection of an individual bag from each case is probably a random sample, although this is not specified.

Bias–Should be unbiased as long as the individual bag chosen is random. There could be differences in the first bag of a case versus the last bag.

16. Contaminated milk. a. Population–Dairy farms. b. Parameter–Not clearly specified although perhaps the proportion of dairy farms passing inspection. c.

Sampling Frame–All dairy farms although not specifically stated this way.

d. Sample–Not specified. Probably a random selection of farms and then a random selection of milk samples. e.

Sampling method–Multistage sampling due to a combination of methods. The selection of dairy farms is a cluster and the sampling selection of an individual sample(s) from each dairy is probably a random sample, although this is not specified.

Bias–Should be unbiased as long as the farms and the milk samples are randomly selected.

17. Pulse poll. The station’s faulty prediction is most likely the result of bias. Only people watching the local TV station news have the opportunity to respond. The responders who volunteered to participate may have different viewpoints than those of other voters, who either chose not to respond or didn’t have the opportunity to participate (didn’t see the news program). 18. Paper poll. The newspaper’s faulty prediction is most likely the result of sampling error. The description of the stratified sampling method does suggest that the sample is representative of the voting population. However, it is unclear whether the percentages by party, age, etc. were accurate when compared to the entire voting population. Random selection of individuals within each strata means that the sample statistics will vary from the population parameter. In addition, no measure of a sampling error percentage was given for the result. 19. Cable company market research. a. Sampling strategy is volunteer response. Bias is introduced because only those individuals who see the ad and feel strongly about the issue will respond. The opinions may not be representative of the rest of the public. b. Sampling strategy is a cluster of one town selected to be sampled. Bias is introduced because one town may not be representative of all towns. .

10 Part I Exploring and Collecting Data c.

Sampling strategy is an attempted census, accessing all customers. Bias is introduced because of nonresponse to the mailing survey.

d. Sampling strategy is stratified by town, selecting 20 customers at random from each town to be surveyed, including follow up. This strategy should be unbiased and representative of the public opinion about the cable issue. 20. Cable company market research 2. a. Sampling strategy is volunteer response. Bias is introduced because only those individuals who see the ad and feel strongly about the issue will respond. The opinions may not be representative of the rest of the public. b. Sampling strategy is still volunteer response. Bias is introduced because only those who are strongly motivated to express their opinions will attend the meetings. c.

Multistage sampling, with a cluster sample within each town, consisting of those who live on a randomly selected street. Bias is introduced if there is a large percentage of residents on the selected street who do not participate or if the selected street is unrepresentative of the town as a whole.

d. Sampling strategy is systematic sampling. This strategy should be unbiased and fairly representative of the public opinion about the cable issue. 21. Churches. a. This is a multistage design because of the combination of methods. A cluster sample consists of the 3 churches chosen at random. One hundred church members are randomly selected from each church to be surveyed. b. Since only 3 churches are chosen at random, if any one of the churches chosen is not representative of the entire 17 churches, bias is introduced in the form of sampling error. 22. Canadian Research in the Great Lakes. a. This is a systematic design since they sample at regular intervals, i.e. every 3rd day. b. The sampling is done only during July and August, so that other months are not included. Some fish species may be present in more or less numbers in different months. 23. Amusement park riders. a. This is a systematic sample (every 10th person in line). b. It is likely to be representative of all of those waiting in line to go on the roller coaster. It would be useful to compare those who have waited and are now at the front with those who are in the back of the line. Otherwise, survey every 10th person about to board the roller coaster for a more consistent response. c.

The sampling frame consists of persons willing to wait in line for the roller coaster on a particular day within a given time frame.

24. Playground. The managers will get responses only from those who bring children to the park. It is very possible that parents and others who are dissatisfied with the playground’s size and condition do not come to the playground. 25. Survey wording. a. Answers will vary. Question 1 is a straightforward question about the issue and certainly appropriate for the survey. Question 2 is biased in its wording and could in some way offend those surveyed because it connects the cost to a daily cost of a cappuccino. Many people don’t have .

Chapter 3 Surveys and Sampling 11 coffee drinks and so if they don’t spend that money, why would they want to spend it on cable? Those people who drink coffee most likely would not give up a coffee drink in order to have cable. The statement would be more neutral by just stating how much the cable would cost per day. b. Question 1 is neutrally worded because it is a simple, straightforward statement asking for the required response. 26. More words. a. Answers will vary. Both questions have biased wording because they both have statements that lead the responder to a particular answer. Question 3 asks if slow speed reduces enjoyment of the web services. We know that slow speed certainly doesn’t enhance a user’s enjoyment and, therefore, the responder will most likely respond favorably to the question about high speed Internet. Question 4 equates high speed Internet with children’s education. Most people support anything that enhances or is vital for a child’s education and, therefore, the responder will likely respond favorably to the question about high speed Internet. b. A more appropriate wording for this question would be “Do you think faster Internet access would be helpful to you for the things that you use the Internet for?” 27. Another ride. Biases exist because it could be that only those who think it is worth waiting for the roller coaster ride are likely to still be in line. Those who don’t like roller coasters or don’t want to stay in lines are not part of the sampling frame. Therefore, the poll won’t get a fair picture of whether park patrons overall would favor more roller coasters. 28. Playground bias. The first sentence points out problems and issues that the respondent may not have noticed, and might lead them to feel they should disagree. The last phrase mentions higher fees which could make people reject the proposed improvements to the playground. 29. (Possibly) Biased questions. a. This statement is biased because it leads the responder toward yes because of the word “pollute”. The word “pollute” conjures up a negative image leading the responder to agree that companies should pay for this behavior. Another way to phrase it would be “Should companies be responsible for costs of environmental cleanup?” b. This statement is biased because it leads the responder to no because of the words “enforce” and “strict” that conjure up images that could lead a responder to having negative reaction. Another way to phrase it would be “Should companies have dress codes?” 30. More possibly biased questions. a. This statement seems unbiased, stating the question without extra leading phrases. b. This statement is biased because it leads the responder to agreeing with space exploration because it has been a “great tradition” of the past. The responder is unlikely to disagree with an activity that is in line with a “great tradition” of the past. A better way to phrase the question would be “Do you favor continued funding for the space program?” 31. Phone surveys. a. It would be difficult to achieve a random sample in this case because not everyone in the sampling frame has an equal chance of being chosen. People with unlisted phone numbers, without phones, and those at work or away from the home at the designated calling time cannot be contacted. b. Another strategy would be to generate random numbers and call at random times or select random numbers from the phonebook and call at random times (this doesn’t solve the unlisted phone number issue). .

12 Part I Exploring and Collecting Data

In the original plan, families that have one person at home are more likely to be included in the study. Using the second plan, more people are potentially included although people without phones or those not home when called are still not included.

d. This change does improve the chance of selected households being included in the study. e.

The random digit dialing does address all existing phone numbers, including unlisted numbers. However, there is still the issue of residents not being home at the time of the call. In addition, people without phones are still left out of the study.

32. Cell phone survey. Cell phones are not used equally in all demographic groups. Retired persons may not have cell phones. Individuals who cannot afford a cell phone or those who choose not to have one except for emergency purposes would also be left out of the study. As cell phones replace land lines and as they become more affordable and commonplace, this strategy would be a viable option. 33. Change. a. Answers will vary b. The parameter being estimated is the true mean amount of change that you carry daily just before lunch. c.

Population is now the amount of change carried by your friends. The average parameter estimates the mean of these amounts.

d. The 10 measurements in c) are more likely to be representative of your class (peer group with similar needs) but unlikely for larger groups outside of your circle of friends. 34. Fuel economy. a. The mean fuel efficiency for your last 6 fill-ups (sample statistic). b. The mean fuel efficiency for your vehicle (population statistic). c.

The results may not represent my typical driving habits Consumer Reports: speed, highway or city driving, aggressive driving, etc.

d. Consumer Reports would be trying to estimate the mean gas mileage for all cars of this make and model. 35. Accounting. a. Assign numbers 001 to 120 (3 digits required because the maximum number is 120) representing each order in a day. Use random numbers to select 10 transactions to check for accuracy. b. Separate the transactions and sample each type (wholesale and retail) proportionately. This would be a stratified random sample. 36. Happy workers? a. If all types of employees are sampled every month in relative proportions to their job types, the study should be without bias. It is not specified how the sample will be chosen. Because the company is doing the study on itself, there could be some inherent bias introduced due to selfinterest. b. The random sample approach assigns numbers to all employees, 001 to 439 (3 digits are required because the maximum number is 439). Then a random number table or random number generator software would be used to select the sample. .

Chapter 3 Surveys and Sampling 13 c.

The random number approach does not recognize the separate labor groups where there are few managers and mostly laborers. A random sample would favor the largest group, the laborers.

d. A better solution would be to stratify by job type (proportionately to the numbers within each type). e.

Answers will vary. Assign numbers 01 to 14 to each person (2 digits are required because the maximum number is 14). Use a random number table or random number generator software to select 2 managers.

37. Quality control. a. Randomly select 3 cases and then randomly select one jar from each case. b. Assign numbers 01 to 20 to cases 07N61 to 07N80 respectively. Then generate three random numbers between 01 and 20 and select the appropriate case. Then assign random numbers 01 to 12 to each of the 12 jars within each case. For each case selected, generate a random number between 01 and 12 and select the corresponding jar within each case. c.

The method described involves two separate sampling methods and, therefore, it is multistage sampling.

38. Fish quality. There are some sources of biases for the results. The scientists asked people fishing to bring any fish they caught to the field station for inspection. They did not require it and, therefore, the people that participated volunteered to do so and possibly do not represent the catch of all. People with discoloured fish might be more likely to bring them in for inspection. In addition, it would have to be assumed that the fish caught and brought in for inspection are representative of all fish downstream of the chemical plant. It could be misleading to suggest that 18% of fish in the river have discoloured scales. 39. Sampling methods. a. Yellow pages may not include all doctor listings. If regular line listings are used, the list may include all doctors. If ads are used, not all doctors would be included and the ones with ad would not be typical of all doctors. b. This sampling method is not appropriate. The cluster sample chosen (the randomly selected page) will only contain a handful of businesses and maybe only one or two business types. 40. More sampling methods. a. It does not specify the method for calling local businesses and whether all or a random sample were called. Some people will say that they are willing to sign a petition when asked over then phone but later may not be willing to sign it. The mention of signing a petition may bias business owners to respond positively. b. If the food court is the largest and perhaps only food court in the airport, then the results would be fairly representative. If travelers don’t like the food available, they probably aren’t eating there and made other choices.

14 Part I Exploring and Collecting Data

MiniCase: Canadian Labour Force Survey PLAN

Setup State the Objectives State the Population and Parameters

The objective is to obtain information about the Canadian Labour Force The population is the Canadian Labour Force. The parameters include: Unemployment Rate, shifts of employees from one industry to another, hours worked, and demographic information about employees including age, sex, marital status, education level and Province or Territory of residence.

State the Sampling Plan

A SRS is inappropriate because the population is very large compared to the sample size and with SRS we might therefore omit some important groups. A better plan is to use stratified sampling in which we first ensure we have enough people in the sample to represent each Province & Territory, then each age group etc.

Mechanics. Specify When and Where the sampling will be performed. Specify How the survey will be conducted

REPORT

When–second half of each month Where–across Canada How–We could contact people chosen in the stratified sample by phone and email, contacting the non-respondents repeatedly until we get a response. If this becomes too time consuming, we will request a budget to pay people to respond.

Summary & Conclusion Give an overall plan of how the survey will be conducted

We will use a random sample from across Canada, stratified by province/territory and demographic group. We will call and email people selected in the sample repeatedly until we get a response.

Chapter 4 – Displaying and Describing Categorical Data 1.

Graphs in the news. Answers will vary.

Graphs in the news, part 2. Answers will vary.

Tables in the news. Answers will vary.

Tables in the news, part 2. Answers will vary.

U.S. market share. a) Yes, this is an appropriate display for these data because all categories of one variable (sellers of carbonated drinks) are displayed. The categories divide the whole and the category Other combines the smaller shares. b) The company with the largest share is Coca-Cola.

World market share. a) Yes, this is an appropriate display for these data. All categories of one variable (distributors of confectionery products) are displayed. The categories divide the whole and the category “Other” combines the smaller distributors. b) The company with the largest share is Cadbury who just edges out Mars.

Market share again. a) The pie chart does a better job of comparing portions of the whole. b) The “Other” category is missing and without it, the results could be misleading.

World market share again. a) Either a bar chart or a pie chart would be acceptable, but probably the bar chart does a better job because the “Other” category is so large and takes up almost two thirds of the pie. In addition, the close categories are hard to compare directly because they are so close. b) Cadbury Schweppes has the bigger market share but you would probably need to look at the bar chart to verify.

Insurance company. a) Yes, it is reasonable to conclude that deaths due to heart OR respiratory diseases is equal to 30.3% plus 7.9%, which equals 38.2%. The percentages can be added because the categories do not overlap. There can only be one cause of death. b) The percentages listed in the table only add up to 73.7%. Therefore, other causes must account for 26.3% of U.S. deaths. c)

An appropriate display could either be a bar graph or a pie graph, using an “Other” category for the remaining 26.3% causes of death.

16 Part I Exploring and Collecting Data 10. Canadian Election Results. a)

b) A pie graph is appropriate since its shows clearly that the percentage seats for the Conservatives is less than 50%. This would not be clear in a bar graph. 11. Web conferencing. WebEx Communications, Inc. has the majority of the market share for web conferencing (58.4%), and Microsoft has approximately a quarter of the market share. Other companies comprise about 15% of market share so there appears to be opportunity for either company to grow. Either a bar chart or pie chart would be an appropriate graphic display.

12. Mattel. Mattel received nearly equal amounts of revenue from their Mattel and Fisher-Price brands (about 43-44%). They received the remaining 12% from their American Girl Brand. Either a pie chart of a bar chart would be appropriate.

Chapter 4 Displaying and Describing Categorical Data 17

13. Small business productivity. a) The percentages total more than 100%. The percentages do not represent parts of one variable. They simply represent the percentages of businesses who took certain actions. They could also take more than one action. b) Bar chart:

A pie chart would not be appropriate because the percentages do not represent parts of a whole and do not total 100%.

d) (Answers will vary) Over 50% of the business owners said that they have updated their computers or made noncomputer capital investments to increase productivity (or both). A smaller percentages of business owners (from 10-37%) made changes in their hiring or salary structures. 14. Canadian Election Results a)

b) The Conservatives and the Bloc are over-represented by having a higher percentage of seats compared to their percentage of votes. 15. Environmental hazard. The bar chart shows that grounding is the most frequent cause of oil spillage (118) for these 312 spills, while collisions (97) are ranked a close second. The other causes that with lower values (24-43) were due to hull failures, fires and explosions, and unknown causes. In order to differentiate .

18 Part I Exploring and Collecting Data between close counts, a bar chart is easier to read unless the pie chart gives the actual percentages. It is difficult to determine differences between similar areas in the pie chart. To showcase the causes of oil spills as a fraction of all 312 spills, the pie chart is the better choice. 16. Winter Olympics. a) There are too many categories to make a meaningful display of the data―too many bars and too many slices in the pie. It would be especially difficult to read the countries with low numbers. b) One way to simply the graph would be to showcase only those who earned 9 or more medals and include the smaller awards under an “Other” category.

17. Importance of wealth. a) India 76.1% - USA 45.3% = 30.8% b) The vertical axis on the display starts at 40% which makes the comparison between countries difficult and the areas disproportionate. For example, the India bar looks about 5-6 times as big as the USA bar but the actual values are not even twice as big. c)

The display would be improved by starting the vertical axis at 0%, not 40%.

The percentage of people who say that wealth is important to them is highest in China and India (over 70%), followed by France (close to 60%) and then the USA and U.K. where the percentages were close to 45%.

18. Importance of power. a) The percentages don’t add up to 100% so a pie chart is not appropriate. Showing the pie chart three dimensionally on a slant violates the Area Principle and makes it much more difficult to compare fractions of the whole. b) A bar chart is more appropriate. (See next page.)

Chapter 4 Displaying and Describing Categorical Data 19

The percentage of people who say that power is important to them is highest in India (almost 75%), followed by China (close to 50%) and then France (almost 45%). The lowest percentages occur in USA and the UK (28-36%).

19. Google financials. a) These are column percentages because the column sums add up to 100% and the row percentages add up to more than 100%. b) A stacked bar chart is appropriate.

The main source of revenue for Google is from their own websites, which in 2002 was 70%, dropping down to 50% in 2004, and back up to 60% in 2006. The second largest source of revenue is from other network websites. While the Google websites have remained the main source of revenue, the revenue from the Google network websites has been increasing. Licensing and other revenue was 6% in 2002 but since 2004 has only been 1%.

20. Real estate pricing. a) These are column percentages because the column sums add up to 100% and the row percentages add up to more than 100%. b) 2.4% c)

This cannot be determined. We are only given the percentages of size within each Price category.

d) Small 61.5% + Med Small 30.4% = 91.9%. e)

The higher the Price of a house, the higher the Size. A stacked bar chart is shown below illustrating the changing conditional distributions.

20 Part I Exploring and Collecting Data

21. Stock performance. a) 62.5% (25/40) b) 35% (14/40) c)

15% (6/40)

d) 50% (20/40) e)

61% (14/23)

65% (11/17)

g) There does not appear to be much, if any, relationship between the performance of a stock on a single day and its performance over the previous year. 22. New product. a) 4.0% (56/1415) b) 34% (481/1415) c)

3.7% (18/481)

d) 32.1% (18/56) e)

Marginal Distributions – total % of the categories: Students 64%; Faculty/Staff 23.9%; Alumni 4%; Town Residents 8.2%.

Conditional Distributions – percentages for Very Likely column: Students 66.5%; Faculty/Staff 20.4%; Alumni 3.7%; Town Residents 9.4%.

g) The likelihood to buy seems independent of campus group (compare percentages for Very Likely in each category). However, there are more students, so focusing advertising in that group may have a greater impact on revenue. 23. Causes of Death in Canada. a) The percent of deaths due to diabetes is 3.42% = (3981+3900)/(116006+114126) b) The percent of deaths in men due to stroke is 4.97% = 5767/116006 c)

The percent of cancer deaths that were females is 47.3% = 31862/(31862+35481)

d) It is important to convert the data to percentages in order to make the comparison.

Chapter 4 Displaying and Describing Categorical Data 21

The proportions of males and females dying from these causes is pretty similar. There are slightly less females than males dying of accidents and respiratory causes. 24. Google financials, part 2. a) 62.5% ($2,577,088/$4,121,282); 59.9% ($4,225,027/$7,054,921) b) 14.5% ($599,510/$4,121,282); 17.4% ($1,228,589/$7,054,921) c)

As a percentage over time: 12.3%, 8.4%, 7.4%, 9.4% and 10.7%. The costs dropped from 2002 to 2004 and have risen since, but have not returned to the 2002 level.

25. Movie ratings. a) Conditional distribution (in percentages) of movie ratings for action/adventure films: G 11.4%; PG 14.3%; PG-13 48.6%; R 25.7%. b) Conditional distribution (in percentages) of movie ratings for thriller/horror films: G 0%; PG 0%; PG-13 57.9%; R 42.1%. c)

Stacked bar chart:

d) Genre and Rating are not independent, in other words, Genre and Rating are related to each other. Thriller/Horror movies are all PG-13 or R and Drama is similar. For example, there is a 5% chance (6/120) that a randomly selected movie is rated G. However, if you were told that the movie was a Thriller/Horror film, there would be a 0% change that it was rated G. Thus, knowing the genre, does affect the rating – they are not independent.

22 Part I Exploring and Collecting Data 26. Canadian Transport Accidents. a) Marginal distributions for the rows and columns. Road Rail Air

Total

British Columbia

Manitoba

Saskatchewan

Alberta

Total

106

b) Saskatchewan

Road

Rail

Air

82%

12%

97.2%

1.4%

c) Alberta d)

Chapter 4 Displaying and Describing Categorical Data 23

27. MBAs. a) 62.7% (168/268) b) 62.8% (103/164) c)

62.5% (65/104)

d) The marginal distribution of origin: 23.9% from Asia; 1.9% from Europe; 7.8% from Latin America; 3.7% from the Middle East; 62.7% from North America. e)

The column percentages: Asia/Pacific Rim Europe Latin America Middle East/Africa North America Total

Full Time 18.90 3.05 12.20 3.05 62.80 100.00

Part Time 31.73 0.00 0.96 4.81 62.50 100.00

Total 23.88 1.87 7.84 3.73 62.69 100.00

They are not independent. For example, there is less than a 19% chance (31/164) that a randomly selected Full Time MBA student is an Asian/Pacific Rim student . However, there is more than a 31% chance (33/104) that a randomly selected Part Time MBA student is an Asian/Pacific Rim .

24 Part I Exploring and Collecting Data student. This is over a 50% increase in the likelihood that a student is an Asian/Pacific Rim student. Thus knowing the kind of MBA program does affect the likelihood of the origin of the MBA student. 28. MBAs, part 2. a) 32.1% (86/268) b) 29.3% (48/164) c)

36.5% (38/104)

d) There seems to be a slightly higher percentage of Evening MBAs who are women. This may be because women have other commitments during the day (such as work, family, etc.) that limit their choices. 29. Top producing movies. a) 7.0% (14/200) b) 5.0% (1/20) c)

3.5% (7/200)

d) 57.5% (69/120) e)

36.3% (29/80)

PG-13 films increased from 36.3% in 1996-1999 to 57.5% in 2000-2005 while R films decreased from 36.3% in 1996-1999 to 15.8% in 2000-2005. 2000-2005 1996-1999

G 5.0% 10.0%

PG 21.7% 17.5%

PG-13 57.5% 36.3%

R 15.8% 36.3%

Total 100.0% 100.0%

30. Movie admissions. a) 38.4% (2659/6921) b) 39.8% (567/1423) c)

36.4% (485/1334)

d) 9.3% (124/1334) e)

20.8% (130/626)

The age distribution of movie admissions stayed fairly constant over the 5 year period from 2002 to 2006. The distribution is shown in the stacked bar chart:

31. Tattoos. The study by the University of Texas Southwestern Medical Centre provides evidence of an association between having a tattoo and contracting hepatitis C. Approximately 33% of the subjects who were tattooed in a commercial parlor had .

Chapter 4 Displaying and Describing Categorical Data 25 hepatitis C, compared with 13% of those tattooed elsewhere, and only 3.5% of those with no tattoo. If having a tattoo and having hepatitis C were independent, we would have expected these percentages to be roughly the same.

32. Working parents. The Gallup poll doesn’t provide strong evidence of a change in people’s opinions regarding the ideal family in today’s society between the years of 1991 and 2001. The conditional distributions of opinion by year appear roughly the same. For example, the percentage of respondents in 1991 who thought the ideal family had two parents that worked fulltime was 14%, and in 2001, the percentage was 13%. There does not seem to be strong evidence to conclude that opinions have changed in the two years of study.

33. Moviegoers and ethnicity. a) Hispanic 14.5%; African-American 12.5%; Caucasian 73.0%. b) For 2006, Hispanic 15.7%; African American 12.0%; Caucasian 72.3%. c)

26 Part I Exploring and Collecting Data d) The conditional distribution of Ethnicity is almost the same across the five Years; however, there seems to be a slight increase in the percentage of Hispanics who go to the movies from 13.1% in 2002 to 15.7% in 2006. 34. Department store. a) Low 20.0%; Moderate 48.9%; High 31.0%. b) Under 30: Low 27.6%; Moderate 49.0%; High 23.5% 30-49: Low 20.7%; Moderate 50.8%; High 28.5% Over 50: Low 15.7%; Moderate 47.2%; High 37.1% c)

d) As age increases, the percentage of customers reporting a high frequency of shopping increases, and the percentage who report a low frequency of shopping decreases. e)

No. An association between two variables does not imply a cause-and-effect relationship.

35. Women’s business centres. a) Row percentages (add up to 100%). b) The stacked bar chart compares the locations of women’s business centres. The result is that the percentages are similar for both Established and Less Established for both locations.

Chapter 4 Displaying and Describing Categorical Data 27 36. Natural Gas Drilling in Canada. a) Number of wells drilled.

b) Percentage of wells drilled in these 3 gas fields, 2006, 2007 and 2008.

Encana changed the focus of its drilling activity in 2007 which was very different from 2006, with much less emphasis on Cutbank Ridge and a corresponding increase in Bighorn. The pattern then remained about the same for 2008, with only a slight reduction in Cutbank Ridge and a slight increase in Bighorn.

37. Worldwide toy sales. a) Row percentages (add up to 100%). b) No. We are given only the conditional distributions. We have no idea how much are sold in either Europe or America. .

28 Part I Exploring and Collecting Data c)

d) In America, more than 50% of all toys are sold by large mass merchant discounters and food hypermarkets and only 25% are sold in toy specialty stores. In Europe, 36% of all toys are sold in toy specialty stores while a relatively small 24% are sold in the large discount and hypermarket chains. 38. Internet Users. a) The percentage of South Americans that use the Internet is 136/393 = 34.6% b) The percentage of Internet users in the Americas that are in The Caribbean is 9/432 = 2.1% c) Distribution of population compared with the distribution of Internet users.

39. Health Care. a) The marginal totals were added. 160 of 1300 or 12.3% had a delayed discharge. Large Hospital Small Hospital Total 120 of 800 10 of 50 Major surgery 130 of 850 10 of 200 20 of 250 Minor surgery 30 of 450 Total 130 of 1000 30 of 300 160 of 1300 b) Major surgery patients were delayed 15.3% of the time. Minor surgery patients were delayed 6.7% of the time. .

Chapter 4 Displaying and Describing Categorical Data 29 c)

Large Hospital had a delay rate of 13%. Small Hospital had a delay rate of 10%. The small hospital has the lower overall rate of delayed discharge.

d) Large Hospital: Major Surgery 15% and Minor Surgery 5%. Small Hospital: Major Surgery 20% and Minor Surgery 8%. 40. Delivery service. a) Pack Rats has delivered a total of 28 late packages (12 Regular + 16 Overnight), out of a total of 500 deliveries (400 Regular + 100 Overnight). 28/500 = 5.6% of the packages are late. Boxes R Us has delivered a total of 30 late packages (2 Regular + 28 Overnight) out of a total of 500 deliveries (100 Regular + 400 Overnight). 30/500 = 6% of the packages are late. b) The company should have hired Boxes R Us instead of Pack Rats. Boxes R Us only delivers 2% (2 out of 100) of its Regular packages late, compared to Pack Rats, who deliver 3% (12 out of 400) of its Regular packages late. Additionally, Boxes R Us only delivers 7% (28 out of 400) of its Overnight packages late, compared to Pack Rats, who delivers 16% of its Overnight packages late. Boxes R Us is better at delivering Regular and Overnight packages. c)

This is an instance of Simpson’s Paradox, because the overall late delivery rates are unfair averages. Boxes R Us delivers a greater percentage of its packages Overnight, where it is comparatively harder to deliver on time. Pack Rats delivers many Regular packages, where it is easier to make an on-time delivery.

41. Graduate admissions. a) 1284 applicants were admitted out of a total of 3014 applicants. 1284/3014 = 42.6% b) 1022 of 2165 (47.2%) of males were admitted. 262 of 849 (30.9%) of females were admitted. c)

Since there are four comparisons to make, the table below organizes the percentages of males and females accepted in each program. Females are accepted at a higher rate in every program.

d) The comparison of acceptance rate within each program is most valid. The overall percentage is an unfair average. It fails to take the different numbers of applicants and different acceptance rates of each program. Women tended to apply to the programs in which gaining acceptance was difficult for everyone. This is an example of Simpson’s Paradox. 42. Simpson’s Paradox. Answers will vary. The three-way table below shows one possibility. The number of local hires out of new hires is shown in each cell.

30 Part I Exploring and Collecting Data 43. Revenue growth, last one. a) 8% b) No, because we aren’t given counts or totals. The percentages are within revenue categories. c)

92% (48% + 44%)

d) Although the percent with only a high school education is “relatively” constant over the three revenue categories, the study shows that firms having higher revenues had a significantly higher percentage of women CEO’s with graduate educations.

MiniCase―Loblaw PLAN

Setup State the objectives

Mechanics

To compare the number and percentage of Loblaw stores of different types across Canada (a)

(a) Compare the number of stores

(b) Compare the percentage of stores.

(b)

Chapter 4 Displaying and Describing Categorical Data 31

(d) Highest number of stores (e) Second highest percentage of corporate stores (f) Compare Ontario and Quebec

(c) A bar graph is better than a pie graph when we are comparing a large number of provinces and territories, since everything is on a single graph. We would have needed 12 pie charts. A stacked bar is particularly good for the percentages in (b) so that its clear that everything adds to 100%. Any kind of bar chart can be used for (a). (d) Quebec (e) Northwest Territories (f)

32 Part I Exploring and Collecting Data

When we are only comparing 2 provinces, we can use a pie chart as an alternative to a bar chart, since it is easy to see the comparison between only 2 charts. REPORT

Summary & Conclusion

The results are summarised in the above graphs.

Summarise graphical results.

Bar charts are used to compare between all provinces and territories. Alternatively pie charts can be use when only comparing 2 provinces. Comparisons among the numbers and percentages of the different types of stores in each province and territory can easily be seen from the graphs.

Chapter 5–Displaying and Describing Quantitative Data 1.

Statistics in business. Answers will vary.

Statistics in business, part 2. Answers will vary.

Ottawa Gas prices. Shape–the distribution is slightly skewed to the right with a single peak, making it unimodal. Centre–the centre of any distribution not perfectly symmetric is best represented by the median, the exact middle data point when the dataset is ordered either in ascending or descending order. There are 17 data points representing the 17 gas stations. The centre of the data would be the 9th data point (giving 8 data points on either side of the median). It can be estimated which bin contains the centre value by adding up the values in each bin. For example, the first bin $0.80-$0.85 contains 4 data points. The next bin ($0.85 to $0.90) contains 7 values therefore the median must be in this bin. Spread–the spread is determined from the range of data, low to high, or $0.80-1.00 = approximately $0.20. The exact range cannot be determined from the histogram because the intervals or bins do not represent the exact data points. There are no outliers or other unusual features.

Vineyards. a) Shape–the distribution is skewed to the right. Centre–the centre of a skewed distribution is best represented by the median, the exact middle data point when the dataset is ordered either in ascending or descending order. There are 36 data points representing the 36 wineries. The centre of the data would be close to the 18th value (between the 18th and 19th values). It can be estimated which bin contains the median value by adding up the values in each bin. The 0-30 acre bin contains 15 data points and the next bin (30-60 acres) contains about 13 data points. The total so far is 15+30 values = 45. The median value has to be contained in the 30 to 60 bin because that interval contains the 16th through 30 data points. Spread–the spread is determined from the range of data, high value minus low value, or 240-0 = approximately 240. Unusual features–there is a possible outlier in the 240-270 acre interval. b) The mean is expected to be larger in a right skewed distribution because the larger values affect the mean, pulling it towards the right tail of the distribution. c)

Because the distribution is skewed, the median is a better representation of the centre of the distribution.

Mutual funds. Shape–the distribution is approximately symmetric with a single peak, making it unimodal. Centre–the centre of the distribution is close to the single peak but to be sure, the exact number of values and the median should be determined. The bin values can be added up as follows: 1+3+0+7+18+14+8+4+3+2+2+1+1 = 64. The median occurs to the right of the 32nd data point. Twenty nine data points are contained in the first five bins. The 32nd data point is therefore contained in the 6th bin representing approximately 10%. Spread–the spread is determined from the range of data, high minus low, or 90%-(-15%) = approximately 105%. The two highest bins at 80% and 85% contain outliers.

Car discounts. Shape–the distribution has two peaks, making it bimodal. One mode occurs near $800 and the other mode occurs near $1600. The two modes could be due to that fact that men and women receive different discounts that are mixed together in this data set. To examine the differences, separate histograms should be created for female discounts and male discounts. Centre–the centre of the distribution is the median value which would be just past the 50th data point. The bin values can be added up as follows: 2+3+7+11+14+8 = 45. Forty five data points are contained in the first six bins. The median occurs in the next bin ($1200) which contains 11 data points. The 50th data point is therefore contained in the bin .

34 Part I Exploring and Collecting Data representing $1200-$1400. For a bimodal distribution, the centre is usually between the two peaks. Spread– the spread is determined from the range of data, high minus low, or $2400-0 = approximately $2400. There are no outliers or other unusual features. 7.

Mutual funds, part 2. a) Five-Number Summary (Quartile calculations may differ slightly using different software) Min –10.820

1st Qtr 6.965

Median 3rd Qtr Max 11.275 17.440 94.940

b) Centre can be represented by the median which is 11.275%. The spread can be represented by the maximum–minimum values which is 94.940–(–10.820) = 105.76%. The IQR (3rd quartile–1st quartile values) summarizes the spread by focusing on the middle 50% of the data. For this data set, the IQR = 17.440–6.965 = 10.475%. Another measure of spread is the standard deviation but this measure is reserved for symmetric distributions. The shape of the distribution can be determined from the histogram in Problem 7. The main distribution is fairly symmetric but there are high outliers, meaning that it is not appropriate to use standard deviation as a measure of spread. In addition, due to the outliers, it is also not accurate to represent the data by a simple calculation of maximum–minimum values. For this distribution, it is better to represent the spread using the IQR value. c)

Boxplot

d) The histogram clearly shows the outlier values as well as the skewness. Depending on the software package used to create a boxplot, the outliers may be identified using special symbols as shown above. 8.

Car discounts, part 2. a) Five-Number Summary (Quartile calculations may differ slightly using different software) Min $131.00

1st Qtr $831.50

Median $1327.50

3rd Qtr $1668.00

Max $2520.00

Chapter 5 Displaying and Describing Quantitative Data 35

b) Boxplot

c) 9.

The histogram shows that the distribution is bimodal (2 peaks).

Vineyards, part 2. a) The distribution can be described as skewed to the right. Symmetry can be determined by comparing the mean and the median. The mean is 46.50 and the median is 33.50. The mean is much larger than the median indicating a right skew (the higher values are pulling the mean value higher than the median). In addition, symmetry can be determined by comparing the difference between the first quartile and the median and the third quartile and the median. If the distribution is symmetric, these values should be fairly equal. In this summary, the median–Q1 = 33.50–18.50 = 15 compared to Q3–median = 55–33.50 = 21.5. The right side of the distribution is wider than the left which indicates a right skew. Finally, the maximum value of 250 is very high compared to the median of 33.50 while the minimum value of 6 compared to the median of 33.50 is a much smaller number also indicating a skew to the right. b) Yes, there is one high outlier at 250. This is clearly shown in the histogram from Exercise 6 and is also clear from the fact that 250>Q3+1.5*IQR. c)

The boxplot shows the outlier at 250 but without the data set, the length of the upper whisker going to the upper fence (limit for the outliers) cannot be determined.

36 Part I Exploring and Collecting Data

10. Graduation. a) The distribution can be described at least as roughly symmetric. Symmetry can be determined by comparing the mean and the median. The mean is 68.35 and the median is 69.90. These values are fairly close indicating at least a roughly symmetric distribution. In addition, symmetry can be determined by comparing the difference between the first quartile and the median and the third quartile and the median. If the distribution is symmetric, these values should be fairly equal. In this summary, the median–Q1 = 69.90–59.15 = 10.75 compared to Q3–median = 74.75–69.90 = 4.85. The left side of the distribution is slightly wider than the left but not by a large margin. Finally, the maximum value of 87.40 compared to the median of 69.90 is similar to the minimum value of 43.20 compared to the median of 69.90 although the left side again is wider. b) Outliers can be determined mathematically by adding the term 1.5*IQR to Q3 to see if any data values fall above or by subtracting 1.5*IQR from Q1 to see if any data values fall below. In this case, IQR = Q3–Q1 = 74.75–59.15 = 15.6. 1.5*IQR = 1.5*15.6 = 23.4. To check for high outliers, add 1.5*IQR to Q3 = 23.4 + 74.75 = 98.15. The maximum data point falls below that value so there are no high outliers. To check for low outliers, subtract 1.5*IQR from Q1 = 59.15– 23.4 = 35.75. The minimum data point is above this value so there are no low outliers. c)

Boxplot

11. Hockey. a) Stemplot – The stem and leaf plot has split stems representing 0-4 and 5-9.

Chapter 5 Displaying and Describing Quantitative Data 37

b) Boxplot

The distribution of the number of games played per season by Wayne Gretzky is skewed to the left toward lower values and has low outliers. The median value is 79 games and the range is 37 games.

d) The outliers identified at 45 and 48 games could have been caused by injuries. The season with 64 games has a gap to both lower and higher values. Most of his seasons were played with games totaling in the 70s and 80s. 12. Baseball. a) Stemplot–The stem and leaf plot has split stems representing 0-4 and 5-9.

b) Boxplot

38 Part I Exploring and Collecting Data

The distribution of Mark McGwire home runs is at least somewhat skewed to the right with typical season home runs in the 30s. McGwire had three seasons when he hit less than 10 home runs. With the exception of those values, the total number of home runs per season was between 22 and 70. The season with 70 home runs is identified as an outlier however it is only slightly higher than the season with 65 home runs.

d) The three low values are not officially identified as outliers but they can be considered to be unusual compared to the productivity of the rest of the seasons. Injury may have caused the low number of home runs. 13. Gretzky returns. a) The distribution is skewed therefore the median is used to describe the centre. b) The mean should be pulled toward the tail of the distribution, in this case, toward the lower values. c)

The chart displayed is not a histogram. It is a time series plot using bars to represent each data point. A histogram would arrange the data into bin intervals rather than displaying the number of games over time.

14. McGwire, again. a) The distribution is somewhat skewed so the median can be used to represent the centre. b) The median value is 33. c)

The mean value should be close to the median because the distribution is fairly symmetric. The mean should be slightly higher due to the high outlier.

d) The chart displayed is not a histogram. It is a time series plot using bars to represent each data point. A histogram would arrange the data into bin intervals rather than displaying the number of games over time. 15. Pizza prices. a) Vancouver Five-Number Summary (Quartile calculations may differ slightly using different software) Min $2.21

1st Qtr $2.51

Median $2.61

3rd Qtr $2.72

Max $3.05

b) Range = max–min = $3.05 - $2.21 = $0.84; IQR = Q3–Q1 = $2.72 - $2.51 = $0.21 c)

Boxplot

Chapter 5 Displaying and Describing Quantitative Data 39

d) This distribution is fairly symmetric with a high outlier at $3.05. The mean is $2.62 and the standard deviation is $0.156. e)

There is one observation classified as an outlier at $3.05 per frozen pizza.

16. Pizza prices, part 2. a) Montreal Five-Number Summary (Quartile calculations may differ slightly using different software) Min $1.55

1st Qtr $2.525

Median $2.65

3rd Qtr $2.74

Max $3.03

b) Range = max–min = $3.03 - $1.55 = $1.48; IQR = Q3–Q1 = $2.74 - $2.525 = $0.215 c)

Boxplot

d) This main part of the distribution is approximately symmetric with four low outliers identified. The median price is $2.65 and the IQR is $0.22. Because of the outliers, the best representation of the centre of the distribution is the median price. The middle half of the prices fell in the range of $2.53 to $2.74. e)

There are four low outliers. All but five of the prices were above $2.20.

17. Earnings of Canadians. The fact that 2.2% earned over $150K implies that the distribution is skewed to the right. For skewed distributions, the median is the appropriate measure of the centre. The mean is affected by the few people with very large earnings and is therefore less representative of the distribution as a whole. The appropriate measure for the spread is the IQR or Inter Quartile Range, which is used whenever the median is used for the centre of the data distribution. 18. Canadian Consumer Price Index. Mean(Xbar) = ∑x/n = 2007.7/18 = 111.5389 Variance = ∑(X-Xbar)^2/n-1 = [(110.7-111.5389)^2 + (113.2-111.5389)^2 + ... + (110.8-111.5389)^2] /17 = 108.5428/17 = 6.3849 SD = SQRT(Variance) = 2.53 Coefficient of Variation = 2.53/111.5389 = 0.0227 .

40 Part I Exploring and Collecting Data 19. Canadian Weekly Earnings. A survey of 13 randomly selected Canadians obtained the following figures for their average weekly earnings. a) Calculate the mean weekly earnings. b) Calculate the standard deviation of weekly earnings. c) Calculate the coefficient of variation. d) Calculate the z-scores for Subjects # 6 & 13 and interpret their meaning. Subject # 1 2 3 4 5 6 7 8 9 10 11 12 13

Weekly Earnings 714.65 628.9 673.38 707.93 725.29 803.46 701.93 724.03 835.52 761.01 882.47 1,004.63 948.68

The mean weekly earnings = x = ∑x/N = (714.65+628.9+ … +948.68) / 13 = 10111.88 / 13 = 777.84 $ per week b) Standard deviation is calculated from the formula for a sample: s^2 = [Σ(x- x )^2 ] / (n-1) = [(714.65-777.84)^2 + (628.90-777.84)^2 + … + (948.68-777.84)^2)]/12 = 12435.73 s= 111.52 $ per week c) Coefficient of variation = 111.52/777.84 = 0.1434 d)Z-Score Z = (x – u) / σ For Subject #6 z = (803.46 - 777.84) / 111.52 = 0.2298 For Subject #13 z = (948.68 - 777.84) / 111.52 = 1.532 Subject #6 is 0.2298 standard deviations above the mean and Subject #13 is 1.532 standard deviations above the mean.

Chapter 5 Displaying and Describing Quantitative Data 41 20. GDP Growth. A report for a given data set should include summaries and graphs with analysis. Five-Number Summary (Quartile calculations may differ slightly using different software) Min 0.0

1st Qtr 0.019

Median 0.029

3rd Qtr 0.04

Max 0.074

Range = max – min = 0.074 – 0.0 = 0.074 or 7.4%; IQR = Q3 – Q1 = 0.04 – 0.019 = 0.021 or 2.1%

The shape of the distribution of GDP growth rates can be seen in the histogram as being unimodal (single peak) and slightly skewed to the right. In addition, there is a possible high outlier showing on the boxplot. The value is not identified as a “far outlier” (more than 3 IQRs from Q3). The median growth rate is 2.9% with and IQR of 2.1%. The lowest growth rate is Italy with a 0% growth rate and the maximum growth is in Turkey with a growth rate of 7.4%. This value is not far from the upper fence in the boxplot. The middle 50% of growth rates among these countries is between 1.9% and 4%. 21. Start-up. a) Range: max – min = 6796 – 5185 = 1611 yards b) The middle 50% of the distribution lies between Quartile 1 (5585.75 yards) and Quartile 3 (6131 yards) (Quartile calculations may differ slightly using different software). c)

Summary statistics should include the representation of the centre, in this case, the mean (5893 yards) because the distribution is approximately symmetric. In addition, a measure of the spread for this dataset is the standard deviation (386.6 yards), chosen also because the distribution is approximately symmetric.

d) Shape–the distribution of the lengths of all of the Vermont golf courses is roughly symmetric and unimodal. Centre–the mean is 5893 yards, approximately 5900 yards. Spread–represented by the standard deviation of 386.6 yards. 22. Real estate. a) Range: max – min = 5228 – 672 = 4556 sq. ft. b) The middle 50% of the distribution lies between Quartile 1 (1342 sq. ft.) and Quartile 3 (2223 sq. ft.) (Quartile calculations may differ slightly using different software). c) Summary statistics should include the representation of the centre, in this case, the median (1675 sq. ft.) because the distribution is skewed to the right. In addition, a measure of the spread for this dataset is the IQR (Q3 – Q1 = 2223 – 1342 = 881 sq. ft.), chosen also because the distribution is right skewed. d) Shape–the distribution of the sizes of all of the houses in a specific area is skewed to the right and unimodal. Centre–the median is 1675 sq. ft. Spread–represented by the IQR (881 sq. ft.) and a range of 4556 from 672 sq. ft. (minimum) to 5228 sq. ft. (maximum). .

42 Part I Exploring and Collecting Data 23. Food sales. a) Suitable displays of a single quantitative variable are either a histogram or a boxplot. Both are shown here.

b) The median for the data set is $95,974.5 and the mean is $107,844.94. The mean is pulled toward the higher values because the distribution is right skewed and the higher values have the effect of increasing the mean value.

The median does a better job of depicting typical store sales because the distribution is skewed with high outliers.

d) Standard Deviation = $46,962.19 and the IQR = $43,300 (Quartile calculations may differ slightly using different software) e)

The IQR is a better measure of the spread for a skewed distribution with outliers. The standard deviation is affected by outliers.

If the outliers were removed, the mean would decrease in value because the higher numbers would not be included in the calculation. The standard deviation would also decrease, indicating a smaller spread when the high values are excluded. The median and IQR would remain relatively unaffected because the calculations are not affected by outliers unless there are a large number of them.

Chapter 5 Displaying and Describing Quantitative Data 43 24. Insurance profits. a) Suitable displays of a single quantitative variable are either a histogram or a boxplot. Both are shown here.

b) The median for the data set is $1645 and the mean is $2562. The mean is pulled toward the higher values because the distribution is right skewed and the higher values have the effect of increasing the mean value. c)

The median is a better indicator of the centre of the distribution because the distribution is skewed.

d) The standard deviation is $2683.10; the IQR is $3059.5 (Quartile calculations may differ slightly using different software). e)

The IQR is resistant to the effects of outliers.

The mean and standard deviation would decrease. The median and IQR would be relatively unaffected.

25. Ipod failures. The failure rate is calculated by dividing the number failed by the total number of iPods (both failed and OK) for each model. The result is then multiplied by 100% to get a rate.

The median value for the failure rate for all 17 models is 16.2%. An appropriate graphical display of the distribution of a single quantitative variable is either a histogram or a boxplot.

44 Part I Exploring and Collecting Data

The distribution is left skewed. The centre is best represented by the median at 16.2% and the spread is best represented by the IQR which is 10.7% (Quartile calculations may differ slightly using different software). The middle 50% of failure rates are between 10.6% and 21.3%. The lowest value or best failure rate is 3.2% for the 60GB Video model, and the worst failure rate is 29.9% for the 40GB Click Wheel. 26. Unemployment. An appropriate graphical display of the distribution of a single quantitative variable is either a histogram or a boxplot.

The histogram shows that the distribution is unimodal and skewed to the right. Outliers sometimes can only be determined from a boxplot.

The boxplot shows a high outlier representing Poland at 13.1%. The lowest rate is Switzerland with 2.2%. The range (maximum–minimum) is 10.9%. The middle 50% of rates represented by the IQR is between 4.4% and 7.1%. The IQR spread is 2.7% (Quartile calculations may differ slightly using different software). .

Chapter 5 Displaying and Describing Quantitative Data 45

27. Sales. In describing side-by-side boxplots, there are two important things to mention–the description of the each data distribution and the comparison of the distributions to each other. Location #1 is roughly symmetric with some high outliers, one exceeding $320,000. The median sales value is approximately $240,000 and the minimum sales value is approximately $160,000. Location #2 is also roughly symmetric with high outliers close to $150,000 to $180,000. The median sales value is approximately $110,000 and the minimum sales value below $100,000. Location #1 clearly has higher sales than Location #2 in every week except for the high outlier in Location #2. 28. Sales, part 2. In describing side-by-side boxplots, there are two important things to mention–the description of the each data distribution and the comparison of the distributions to each other. The distribution of stores in Ontario (ON) is right skewed with several high outliers. The median sales value is approximately $115,000 and the minimum sales value is approximately $85,000. High outliers extend up to about $220,000. The distribution of stores in British Columbia (BC) is also right skewed with several outliers. The median sales value is approximately $100,000 and the minimum sales value is approximately $65,000. High outliers extend up to about $170,000. Overall, the stores in ON generally have higher sales than the stores in BC. The median value in ON is higher than the third quartile in BC. The lowest performing store in ON was higher than nearly 25% of the stores in BC. 29. Gas prices, part 2. a) It is obvious that gas prices have steadily increased over the three-year period from 2002–2004. The data spread has also increased over these three years. The 2002 distribution of prices is skewed to the left with several low outliers. Starting in 2003, the distributions have become increasingly skewed to the right. There is a high outlier in 2004 which is close to the upper fence. b) The evidence for prices showing more volatility would be prices showing a greater spread and a larger IQR. The year with the greatest spread and IQR is 2004. 30. Fuel economy. The IQR is the range in which 50% of the car’s fuel economy would lie. An IQR of 0.01 L/100km would imply that the figure of 3.8 L/100km hardly changes at all with different drivers and cities, and is therefore unlikely to be correct. Similarly an IQR of 5.0 L/100km implies such a large range that the figure of 3.8 L/100Km is pretty meaningless. We are therefore left with the option of 1.0 L/100km as the best choice. The IQR is usually between 1 and 2 times the standard deviation, for unimodal data. For city driving our data is almost certainly unimodal (although if we combined city and highway driving, we might well get a bimodal distribution.) Assuming the IQR is 1.0 L/100km, the standard deviation is therefore in the range 0.5 to 1.0 L/100km, giving a variance in the range 0.25 to 1.0 L2/100km2. Note that the units are squared for the variance. 31. Wine prices. a) Identify the highest value for the three regions–occurs in Seneca Lake. b) Identify the lowest value for the three regions–occurs in Seneca Lake. c)

To answer the question about which wines are generally more expensive, look at the IQR box and determine which region has the middle 50% of its prices higher than the others. That region is clearly Keuka Lake.

d) Cayuga Lake and Seneca Lake vineyards have approximately the same median price at about $200 per case. The middle 50% of prices for these two regions are also similar, from approximately $150 to $220 per case. A typical Keuka Lake vineyard case has a much higher price of about $260 and the middle 50% values are between $240 and $280 per case with one outlier at approximately $170 per case. Seneca Lake vineyard prices are the most varied and include both the lowest and the highest prices for all three regions (from $100 to $300).

46 Part I Exploring and Collecting Data 32. Ozone. a) Identify the highest value for all of the months―occurs in both March and April at approximately 430. b) The largest IQR representing the middle 50% of values is shown by the length of the box which is the largest for February at approximately 50. c)

The smallest range is represented by the smallest distance from low to high values which looks like August with a range of less than 50.

d) January had median levels slightly lower than June, with values at 340 and 350 respectively. June’s ozone levels were more consistent, represented by the compact distribution. January’s ozone range was 300 to 400 while June’s range was 310 to 380. June has both low and high outlier values. e)

Generally, ozone levels rose during the winter and were highest in the spring, then fell throughout the summer months and were lowest in the fall. Ozone levels were consistent in the summer (represented by the compact distribution) and became more variable in the fall and most variable in winter (represented by the expanded distribution). Ozone levels then became more consistent in the spring with levels starting to drop.

33. Test scores. a) The highest mean score can be determined from looking at the shape of the test histograms. Class 1 is symmetric with a mean in the 60 interval. Class 2 is somewhat symmetric with the centre also being in the 60 interval. Class 3 is left skewed with most of the data points to the right of a score of 60 so the mean for Class 3 would be the highest. b) The same logic can be applied to find the median score. For symmetric distributions, the mean is the same as the median and close to for approximately symmetric distributions. Most of the data points for Class 3 are to the right of a score of 60 therefore the median for Class 3 would be the highest. c)

For symmetric and approximately symmetric distributions, the mean and the median are almost the same. That applies to both Class 1 and Class 2. Class 3 will have a mean pulled toward its distribution tail which is to the left for a left skewed distribution. Therefore, the difference between the mean and the median will be greatest for Class 3.

34. Test scores, again. a) Class 3 has the middle 50% of the data (IQR) nearly above the median of the other two classes. Clearly Class 3 has the overall higher scores. b) Class 1 has a symmetric distribution and Class 2 has a roughly symmetric distribution. Both are unimodal. Class 3 has a distribution of scores that is skewed to the left and also unimodal. 35. Quality control. In order to analyze the data, it is appropriate to create summaries separately for Fast and Slow data sets. In addition, create side-by-side boxplots for comparison of distributions.

Chapter 5 Displaying and Describing Quantitative Data 47

The slow drilling data contains an extremely high outlier indicating that one hole was drilled almost an inch away from the centre of the target. If this data point is correct, the engineers should investigate the slow speed drilling process closely for any extreme, intermittent inaccuracy. The outlier is so extreme that no graphical display can show the distributions in a meaningful way if that data point is included. The outlier can be removed in order to look at the remaining data points. However, it should be noted that it is never good scientific practice to eliminate a data point because it doesn’t fit the distribution. All outliers should be investigated to determine if they are a result of human error, input error, or some problems with the measurements that need to be addressed. It seems apparent that the entry of .9756000 was in error and was meant to be .000009576. This should be investigated and if found to be true, the summary statistics should be recalculated. But with the outlier removed, the slow drilling process is shown to be more accurate. The entire distribution with the exception of the extreme outlier not pictured lies well below the fast distribution. The greatest distance from the target for the slow drilling process is 0.000098 inches which is more accurate than the smallest distance for the fast drilling process, 0.000100 inches. 36. Fire sale. In order to analyze the data, it is appropriate to create summaries separately for the No Fireplace and Fireplace data sets. In addition, create side-by-side boxplots for comparison of distributions.

The house prices with a fireplace contain a seemingly impossible high outlier at $235,105,000. This price is beyond the maximum selling price of a house in 2006. The outlier can be removed in order to look at the remaining data points. However, it should be noted that it is never good scientific practice to eliminate a data point because it doesn’t fit the distribution. All outliers should be investigated to determine if they are .

48 Part I Exploring and Collecting Data a result of human error, input error, or some problems with the measurements that need to be addressed. It seems apparent that the entry of 235,105,000 was in error and was meant to be 235,105. This should be investigated and if found to be true, the summary statistics should be recalculated. But with the outlier removed, the side-by-side boxplot shows that the prices for houses with fireplaces are generally higher than those without fireplaces. The median price for houses with fireplaces is close to $140,000 compared to a value close to $110,000 for houses without. The spread of sale prices for house without fireplaces is much greater than houses with fireplaces. There are only 3 houses with fireplaces that are more expensive than the most expensive house without a fireplace. 37. Customer database. a) The mean of 54.41 is meaningless. The data set is categorical, not quantitative. b) Typically, the mean and standard deviation are influenced by outliers and skewness, however, once again, these results are meaningless in this instance because the values are categorical. c)

No. Summary statistics are only appropriate for quantitative data.

d) The intern analysed a categorical data set as if it was quantitative. 38. CEOs. a) Industry codes are categories. Therefore, it is not appropriate to create a histogram of the data. The graph shown is a bar graph. b) Histograms require a single quantitative variable for analysis. The complaints bar graph simply identify which codes have the largest number of complaints. It is not appropriate to consider the distribution of this data set. 39. Houses for sale. a) Even though MLS ID numbers are categorical identifiers, they are assigned sequentially, so this graph can be analyzed as shown. Most if not almost all of the houses listed a long time ago have sold and are no longer listed. The older numbers are represented by the lower values giving the distribution the left skewed shape. b) Although some information could be gathered from this graph, a histogram is generally not an appropriate display for categorical data. The MLS ID numbers are categorical identifiers. 40. Car discounts, part 3. In general, women received larger discounts than men. The median discount for women (approximately $1750) was higher than the third quartile of the men’s discounts (approximately $1400). The smallest discount received by a woman was larger than the median of the male discounts.

Chapter 5 Displaying and Describing Quantitative Data 49

41. Hurricanes. a) Histogram

b) The distribution is fairly uniform and appears somewhat right skewed. There do not appear to be any outliers. c)

d) The timeplot does not support the claim that the number of hurricanes has increased in recent decades. There was a peak in the 1940s but numbers have decreased since that time. 42. Hurricanes, part 2. a) Histogram

b) The distribution is fairly uniform from 4 to 10 hurricanes with one outlying decade with only one major hurricane. .

50 Part I Exploring and Collecting Data

Timeplot

d) Most decades have 5 to 8 major hurricanes with no obvious trend over time. This does not support the scientists’ claim, at least not up to the year 2000. 43. Productivity study. Questions include: What is plotted on the x-axis? If it is time, what are the units? Months? Years? Decades? How is productivity measured? 44. Productivity study revisited. Questions include: What is plotted on the x-axis and y-axis? What are the units? Since productivity and wages have different units, how does the graph make sense? We are unable to compare the two variables. 45. Assets. a) The distribution is right skewed. That makes it difficult to estimate the mean which is pulled toward the right. The median can be determined from previous methods: there are 79 total data points and the median is located at the 40th data point. The first interval (0–$4000) contains over 40 data points and therefore contains the median value. The spread for a skewed distribution cannot be easily determined. b) It might be possible to transform the data using logs or square roots which can make the data more symmetric. 46. Assets, again. a) The transformation using logs makes the data more symmetric, so this is a better transformation in this case. b) The value of 50 corresponds to 502, or 2500 ($ million) = $2.5 billion. 47. Real estate, part 2. To answer this question, the values have to be standardized and the z values

(y  y) where y is the value being compared to y which is s the mean value and s is the standard deviation. The house that sells for $400,000 has a z -score of (400000 – 167900)/77158 = 3.01. The house with 4000 sq. ft. of living space has a z -score of (4000 – compared according to the formula: z 

1819)/663 = 3.29. The value of 3.29 is further away from the centre of a normal distribution than 3.01 and, therefore, is the more unusual value. 48. World bank. a) To answer this question, the values have to be standardized and the z values compared. Procedures Time Cost Total z -score Spain (10–7.9)/2.9= 0.724 (47–27.9)/19.6= 0.974 (15.1–14.2)/12.9= 0.07 1.768 Guatemala (11–7.9)/2.9= 1.069 (26–27.9)/19.6= –0.010 (47.3–14.2)/12.9= 2.566 3.625 .