Our topic in this chapter is relationships between two variables. We measure both variables on the same cases. Often, we take the view that one of the variables explains or influences the other.

Statistical summaries of relationships are used to inform decisions in business and economics in many different settings.

A response variable measures an outcome of a study. An explanatory variable explains or influences changes in a response variable.

independent variable

dependent variable

EXAMPLE 2.1

The Best Price?

Price is important to consumers and, therefore, to retailers. Sales of an item typically increase as its price falls, except for some luxury items, where high price suggests exclusivity. The seller’s profits for an item often increase as the price is reduced, due to increased sales, until the point at which lower profit per item cancels rising sales. Thus, a retail chain introduces a new TV that can respond to voice commands at several different price points and monitors sales. The chain wants to discover the price at which its profits are greatest. Price is the explanatory variable, and total profit from sales of the TV is the response variable.

EXAMPLE 2.2

Inventory and Sales

Emily is a district manager for a retail chain. She wants to know how the average monthly inventory and monthly sales for the stores in her district are related to each other. Emily doesn’t think that either inventory level or sales explains the other. She has two related variables, and neither is an explanatory variable.

Zachary manages another district for the same chain. He asks, “Can I predict a store’s monthly sales if I know its inventory level?” Zachary is treating the inventory level as the explanatory variable and the monthly sales as the response variable.

Question 2.1

1. What are the cases?
2. Would your data set have a label variable? If yes, describe it.
3. What are the variables? Are they quantitative or categorical?
4. Is there an explanatory variable and a response variable? Explain your answer.

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Question 2.2

1. What are the variables and cases?
2. Which variable is the explanatory variable? Which is the response variable? Explain your answers.
3. The Tall contains 12 ounces of beverage, the Grande contains 16 ounces, and the Venti contains 20 ounces. Answer parts (a) and (b) with ounces in place of the names for the sizes.

For Exercises 2.44 and 2.45, see page 82; for 2.46, see page 84; for 2.47, see page 86; for 2.48 and 2.49, see page 88; for 2.50, see page 90; for 2.51, see page 90; for 2.52, see page 91; and for 2.53, see page 94.

Question 2.54

EXAMPLE 2.22

Predicting the Number of Target Stores in 2008 and 2014

Here are data on the number of Target stores in operation at the end of each year in the early 1990s in 2008 and in 2014:¹²

A plot of these data is given in Figure 2.19. The data for 1990 through 1993 lie almost exactly on a straight line, which we calculated using only the data from 1990 to 1993. The equation of this line is y = −88,136 1 44.5x and r² = 0.9992. We know that 99.92% of the variation in stores is explained by year for these years. The equation predicts 1220 stores for 2008, but the actual number of stores is much higher, 1682. It predicts 1487 for 2014, also an underestimate by a large amount. The predictions are very poor because the very strong linear trend evident in the 1990 to 1993 data did not continue to the years 2008 and 2014.

Extrapolation is the use of a regression line for prediction far outside the range of values of the explanatory variable x that you used to obtain the line. Such predictions are often not accurate.

A lurking variable is a variable that is not among the explanatory or response variables in a study and yet may influence the interpretation of relationships among those variables.

EXAMPLE 2.23

Gas and Electricity Bills

A single-family household receives bills for gas and electricity each month. The 12 observations for a recent year are plotted with the least-squares regression line in Figure 2.20. We have arbitrarily chosen to put the electricity bill on the x axis and the gas bill on the y axis. There is a clear negative association. Does this mean that a high electricity bill causes the gas bill to be low, and vice versa?

To understand the association in this example, we need to know a little more about the two variables. In this household, heating is done by gas and cooling by electricity. Therefore, in the winter months, the gas bill will be relatively high and the electricity bill will be relatively low. The pattern is reversed in the summer months. The association that we see in this example is due to a lurking variable: time of year.

Question 2.77

EXAMPLE 2.24

Does Television Extend Life?

Measure the number of television sets per person x and the average life expectancy y for the world’s nations. There is a high positive correlation: nations with many TV sets have higher life expectancies.

The basic meaning of causation is that by changing x, we can bring about a change in y. Could we lengthen the lives of people in Rwanda by shipping them TV sets? No. Rich nations have more TV sets than poor nations. Rich nations also have longer life expectancies because they offer better nutrition, clean water, and better health care. There is no cause-and-effect tie between TV sets and length of life.

Question 2.78

Question 2.79

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Question 2.80

An association between an explanatory variable x and a response variable y— even if it is very strong—is not, by itself, good evidence that changes in x actually cause changes in y.

The best way to get good evidence that x causes y is to do an experiment in which we change x and keep lurking variables under control. We will discuss experiments in Chapter 3. When experiments cannot be done, finding the explanation for an observed association is often difficult and controversial. Many of the sharpest disputes in which statistics plays a role involve questions of causation that cannot be settled by experiment. Does gun control reduce violent crime? Does cell phone usage cause brain tumors? Has increased free trade widened the gap between the incomes of more-educated and less-educated American workers? All of these questions have become public issues. All concern associations among variables. And all have this in common: they try to pinpoint cause and effect in a setting involving complex relations among many interacting variables.

data mining

For Exercises 2.77 to 2.79, see page 101; and for 2.80, see page 102.

Question 2.81

1. A negative relationship is always due to causation.
2. A lurking variable is always a quantitative variable.
3. If the residuals are all negative, this implies that there is a negative relationship between the response variable and the explanatory variable.

Question 2.82

1. An outlier will always have a large residual.
2. If we have data at values of x equal to 1, 2, 3, 4, and 5, and we try to predict the value of y at x = 2.5 using a least-squares regression line, we are extrapolating.
3. High correlation implies causation.

Question 2.83

Question 2.84

Question 2.85

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Question 2.86

Question 2.87

Question 2.88

Question 2.89

Question 2.90

CASE 2.2

Does the Right Music Sell the Product?

Market researchers know that background music can influence the mood and the purchasing behavior of customers. One study in a supermarket in Northern Ireland compared three treatments: no music, French accordion music, and Italian string music. Under each condition, the researchers recorded the numbers of bottles of French, Italian, and other wine purchased.¹⁴ Here is the two-way table that summarizes the data:

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The data table for Case 2.2 is a two-way table because it describes two categorical variables. The type of wine is the row variable because each row in the table describes the data for one type of wine. The type of music played is the column variable because each column describes the data for one type of music.The entries in the table are the counts of bottles of wine of the particular type sold while the given type of music was playing. The two variables in this example, wine and music, are both categorical variables.

This two-way table is a 3 × 3 table, to which we have added the marginal totals obtained by summing across rows and columns. For example, the first-rowtotal is 30 + 39 + 30 = 99. The grand total, the number of bottles of wine in the study, can be computed by summing the row totals, 99 + 31 + 113 = 243, or the column totals, 84 + 75 + 84 = 243. It is a good idea to do both as a check on your arithmetic.

marginal row totals

marginal column totals

marginal distribution

To find the marginal distribution for the row variable in a two-way table, divide each row total by the total number of entries in the table. Similarly, to find the marginal distribution for the column variable in a two-way table, divide each column total by the total number of entries in the table.

EXAMPLE 2.25

Calculating a Marginal Distribution

CASE 2.2 Let’s find the marginal distribution for the types of wine sold. The counts that we need for these calculations are in the margin at the right of the table:

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The percent of bottles of French wine sold is

Similar calculations for Italian wine and other wine give the following distribution in percents:

The total should be 100% because each bottle of wine sold is classified into exactly one of these three categories. In this case, the total is exactly 100%. Small deviations from 100% can occur due to roundoff error.

Question 2.91

Question 2.92

Question 2.93

1. Calculate the marginal totals, and add them to the table.
2. Find the marginal distribution of country, and give a graphical display of the distribution.
3. Do the same for the marginal distribution of field of study.

To find the conditional distribution of the column variable for a particular value of the row variable in a two-way table, divide each count in the row by the row total. Similarly, to find the conditional distribution of the row variable for a particular value of the column variable in a two-way table, divide each count in the column by the column total.

EXAMPLE 2.26

Wine Purchased When No Music Was Playing

CASE 2.2 What types of wine were purchased when no music was playing? To answer this question, we find the marginal distribution of wine type for the value of music equal to none. The counts we need are in the first column of our table:

What percent of French wine was sold when no music was playing? To answer this question, we divide the number of bottles of French wine sold when no music was playing by the total number of bottles of wine sold when no music was playing:

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In the same way, we calculate the percents for Italian and other types of wine. Here are the results:

Other wine was the most popular choice when no music was playing, but French wine has a reasonably large share. Notice that these percents sum to 100%. There is no roundoff error here. The distribution is displayed in Figure 2.22.

Question 2.94

1. Write down the column of counts that you need to compute the conditional distribution of the type of wine sold when French music was playing.
2. Compute this conditional distribution.
3. Display this distribution graphically.
4. Compare this distribution with the one in Example 2.26. Was there an increase in sales of French wine when French music was playing rather than no music?

Question 2.95

1. Write down the column of counts that you need to compute the conditional distribution of the type of wine sold when Italian music was playing.
2. Compute this conditional distribution.
3. Display this distribution graphically.
4. Compare this distribution with the one in Example 2.26. Was there an increase in sales of Italian wine when Italian music was playing rather than no music?

Question 2.96

Question 2.97

1. Find the seven conditional distributions giving the distribution of graduates in the different fields of study for each country.
2. Display the conditional distributions graphically.
3. Write a paragraph summarizing the relationship between field of study and country.

Question 2.98

Question 2.99

1. Compare these two approaches.
2. Which do you prefer? Give a reason for your answer.
3. What kinds of questions are most easily answered by each of the two approaches? Explain your answer.

mosaic plot

EXAMPLE 2.27

Software Output for Wine and Music

CASE 2.2 Output from JMP statistical software for the wine and music data is given in Figure 2.23. The mosaic plot is given in the top part of the display. Here, we think of music as the explanatory variable and wine as the response variable, so music is displayed across the x axis in the plot. The conditional distributions of wine for each type of music are displayed in the three columns. Note that when French is playing, 52% of the wine sold is French wine. The red bars display the percents of French wine sold for each type of music. Similarly, the green and blue bars display the correspondence to Italian wine and other wine, respectively. The widths of the three sets of bars display the marginal distribution of music. We can see that the proportions are approximately equal, but the French wine sold a little less than the other two categories of wine.

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EXAMPLE 2.28

Which Customer Service Representative Is Better?

A customer service center has a goal of resolving customer questions in 10 minutes or less. Here are the records for two representatives:

Ashley has met the goal 172 times out of 200, a success rate of 86%. For Joshua, the success rate is 118 out of 200, or 59%. Ashley clearly has the better success rate.

EXAMPLE 2.29

Let’s Look at the Data More Carefully

Here are the counts broken down by week:

For Week 1, Ashley met the goal 90% of the time (162/180), while Joshua met the goal 95% of the time (19/20). Joshua had the better performance in Week 1. What about Week 2? Here, Ashley met the goal 50% of the time (10/20), while the success rate for Joshua was 55% (99/180). Joshua again had the better performance. How does this analysis compare with the analysis that combined the counts for the two weeks? That analysis clearly showed that Ashley had the better performance, 86% versus 59%.

An association or comparison that holds for all of several groups can reverse direction when the data are combined to form a single group. This reversal is called Simpson’s paradox.

Question 2.100

Question 2.101

1. Find the percent of Hospital A patients who died who were classified as “poor” before surgery. Do the same for Hospital B. In which hospital do “poor” patients fare better?
2. Repeat part (a) for patients classified as “good” before surgery.
3. What is your recommendation to someone facing surgery and choosing between these two hospitals?
4. How can Hospital A do better in both groups, yet do worse overall? Look at the data and carefully explain how this can happen.

The data in Example 2.28 can be given in a three-way table that reports counts for each combination of three categorical variables: week, representative, and whether or not the goal was met. In Example 2.29, we constructed two two-way tables for representative by goal, one for each week. The original table, the one that we showed in Example 2.28, can be obtained by adding the corresponding counts for the two tables in Example 2.29. This process is called aggregating the data. When we aggregated data in Example 2.28, we ignored the variable week, which then became a lurking variable. Conclusions that seem obvious when we look only at aggregated data can become quite different when the data are examined in more detail.

For Exercise 2.91, see page 106; for 2.92 and 2.93, see pages 106–107; for 2.94 to 2.96, see page 108; for 2.97 to 2.99, see page 109; and for 2.100 and 2.101, see pages 111–112.

Question 2.102

Question 2.103

Question 2.104

1. Find the distribution of adequate sleep for the high exercisers.
2. Do the same for the low exercisers.
3. If you have the appropriate software, use a mosaic plot to illustrate the marginal distribution of exercise and your results in parts (a) and (b).
4. Summarize the relationship between adequate sleep and exercise using the results of parts (a) and (b).

Question 2.105

1. Find the distribution of exercise for those who get adequate sleep.
2. Do the same for those who do not get adequate sleep.
3. Write a short summary of the relationship between adequate sleep and exercise using the results of parts (a) and (b).
4. Compare this summary with the summary that you obtained in part (c) of the previous exercise. Which do you prefer? Give a reason for your answer.

Question 2.106

1. Find the distribution of age for full-time students.
2. Do the same for the part-time students.
3. Use the summaries in parts (a) and (b) to describe the relationship between full- or part-time status and age. Write a brief summary of your conclusions.

Question 2.107

1. For each age group, compute the percent of students who are full-time and the percent of students who are part-time.
2. Make a graphical display of the results that you found in part (a).
3. If you have the appropriate software, make a mosaic plot.
4. In a short paragraph, describe the relationship between age and full- or part-time status using your numerical and graphical summaries.
5. Explain why you need only the percents of students who are full-time for your summary in part (b).
6. Compare this way of summarizing the relationship between these two variables with what you presented in part (c) of the previous exercise.

Question 2.108

1. Add the marginal totals to the table.
2. Calculate appropriate percents to describe the results of this question.
3. Summarize your findings in a short paragraph.

Question 2.109

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Question 2.110

1. Fill in the marginal totals in the table.
2. Find the marginal distribution for the variable course level.
3. Do the same for the variable class size.
4. For each course level, find the conditional distribution of class size.
5. Summarize your findings in a short paragraph.

Question 2.111

1. Find the two conditional distributions of hired/not hired—one for applicants who are less than 40 years old and one for applicants who are not less than 40 years old.
2. Based on your calculations, make a graph to show the differences in distribution for the two age categories.
3. Describe the company’s hiring record in words. Does the company appear to discriminate on the basis of age?
4. What lurking variables might be involved here?

Question 2.112

1. What was the overall percent of nonresponse?
2. Describe how nonresponse is related to the size of the business. (Use percents to make your statements precise.)
3. Draw a bar graph to compare the nonresponse percents for the three size categories.

Question 2.113

1. Find the sum of the entries for each column.
2. Find the marginal distributions.
3. Find the conditional distributions.
4. If you have the appropriate software, make a mosaic plot.
5. Write a short description of the relationship between marital status and age for women.

Question 2.114

1. Using the data in the previous exercise, compare the conditional distributions of marital status for women aged 18 to 24 and women aged 40 to 64. Briefly describe the most important differences between the two groups of women, and back up your description with percents.
2. Your company is planning a magazine aimed at women who have never been married. Find the conditional distribution of age among never-married women, and display it in a bar graph. What age group or groups should your magazine aim to attract?

Question 2.115

Question 2.116

1. Make a two-way table of gender by admission decision for the two professional schools together by summing entries in these tables.
2. From the two-way table, calculate the percent of male applicants who are admitted and the percent of female applicants who are admitted. Wabash admits a higher percent of male applicants.
3. Now compute separately the percents of male and female applicants admitted by the business school and by the law school. Each school admits a higher percent of female applicants.
4. This is Simpson’s paradox: both schools admit a higher percent of the women who apply, but overall, Wabash admits a lower percent of female applicants than of male applicants. Explain carefully, as if speaking to a skeptical reporter, how it can happen that Wabash appears to favor males when each school individually favors females.

Question 2.117

• Overweight smokers and overweight nonsmokers both tend to die earlier than those not overweight.
• But when smokers and nonsmokers are combined into a two-way table of overweight by early death, persons who are not overweight tend to die earlier.

Question 2.118

Question 2.119

1. Which variable do you choose to be the explanatory variable, and which do you choose to be the response variable? Explain your answer.
2. Plot the data with the least-squares regression line. Summarize the major features of your plot.
3. Give the equation of the least-squares regression line.
4. Find the predicted value and the residual for Sweden.
5. Find the correlation between the two variables.
6. Compare the results found in this exercise with those you found in Exercises 2.10, 2.27, and 2.58. Do you prefer the analysis with the original data or the analysis using logs? Give reasons for your answer.

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Question 2.120

1. Use a histogram to examine the distribution of the residuals.
2. Make a Normal quantile plot of the residuals.
3. Summarize the distribution of the residuals using the graphical displays that you created in parts (a) and (b).
4. Repeat parts (a), (b), and (c) for the original data, and compare these results with those you found in parts (a), (b), and (c). Which do you prefer? Give reasons for your answer.

Question 2.121

1. Make a scatterplot with sales as the response variable and permits issued for new dwellings as the explanatory variable. Describe the relationship. Are there any outliers or influential observations?
2. Find the least-squares regression line and add it to your plot.
3. What is the predicted value of sales for a country that has an index of 160 for dwelling permits?
4. The Netherlands has an index of 160 for dwelling permits. Find the residual for this country.
5. What percent of the variation in sales is explained by dwelling permits?

Question 2.122

1. Make a scatterplot with production as the response variable and permits issued for new dwellings as the explanatory variable. Describe the relationship. Are there any outliers or influential observations?
2. Find the least-squares regression line and add it to your plot.
3. What is the predicted value of production for a country that has an index of 160 for dwelling permits?
4. The Netherlands has an index of 160 for dwelling permits. Find the residual for this country.
5. What percent of the variation in production is explained by dwelling permits? How does this value compare with the value you found in the previous exercise for the percent of variation in sales that is explained by building permits?

Question 2.123

1. Make a scatterplot with sales as the response variable and production as the explanatory variable. Describe the relationship. Are there any outliers or influential observations?
2. Find the least-squares regression line and add it to your plot.
3. What is the predicted value of sales for a country that has an index of 125 for production?
4. Finland has an index of 125 for production. Find the residual for this country.
5. What percent of the variation in sales is explained by production? How does this value compare with the percents of variation that you calculated in the two previous exercises?

Question 2.124

1. Figure 2.24 is a scatterplot of salary versus year with the least-squares regression line. Describe the relationship between salary and year for this person.
2. The value of r² for these data is 0.9832. What percent of the variation in salary is explained by year? Would you say that this is an indication of a strong linear relationship? Explain your answer.

Figure 2.24: FIGURE 2.24 Plot of salary versus year, with the least-squares regression line, for an individual who receives approximately a 5% raise each year for 20 years, Exercise 2.124.

Question 2.125

1. Interpret the residual plot.
2. Explain how this plot highlights the deviations from the least-squares regression line that you can see in Figure 2.24.

Figure 2.25: FIGURE 2.25 Plot of residuals versus year for an individual who receives approximately a 5% raise each year for 20 years, Exercise 2.125.

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Question 2.126

1. Compare this plot with Figure 2.24. Write a short summary of the similarities and the differences.
2. Figure 2.27 is a plot of the residuals for the model using year to predict log salary. Compare this plot with Figure 2.25 and summarize your findings.

Figure 2.26: FIGURE 2.26 Plot of log salary versus year, with the least-squares regression line, for an individual who receives approximately a 5% raise each year for 20 years, Exercise 2.126.

Figure 2.27: FIGURE 2.27 Plot of residuals, based on log salary, versus year for an individual who receives approximately a 5% raise each year for 20 years, Exercise 2.126.

Question 2.127

1. Use the least-squares regression equation constructed to predict salary from year to predict her salary for Year 25.
2. Use the least-squares regression equation constructed to predict log salary from year to predict her salary for Year 25. Note that you will need to convert the predicted log salary back to the predicted salary. Many calculators have a function that will perform this operation.
3. Which prediction do you prefer? Explain your answer.
4. Someone looking at the numerical summaries, and not the plots, for these analyses says that because both models have very high values of r², they should perform equally well in doing this prediction. Write a response to this comment.
5. Write a short paragraph about the value of graphical summaries and the problems of extrapolation using what you have learned from studying these salary data.

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Question 2.128

1. Construct a scatterplot with the 2015–2016 salaries on the vertical axis and the 2014–2015 salaries on the horizontal axis.
2. Comment on the form, direction, and strength of the relationship in your scatterplot.
3. What proportion of the variation in 2015–2016 salaries is explained by 2014–2015 salaries?

Question 2.129

1. Find the least-squares regression line for predicting 2015–2016 salaries from 2014–2015 salaries.
2. Analyze the residuals, paying attention to any outliers or influential observations. Write a summary of your findings.

Question 2.130

1. Compute the percent raise for each faculty member. Take the difference between the 2015–2016 salary and the 2014–2015 salary, divide by the 2014–2015 salary, and then multiply by 100. Make a scatterplot with the raise as the response variable and the 2014–2015 salary as the explanatory variable. Describe the relationship that you see in your plot.
2. Find the least-squares regression line and add it to your plot.
3. Analyze the residuals. Are there any outliers or influential cases? Make a graphical display and include it in a short summary of what you conclude.
4. Is there evidence in the data to support the idea that greater percentage raises are given to those with lower salaries? Summarize your findings and include numerical and graphical summaries to support your conclusion.

Question 2.131

Question 2.132

1. Plot the data and describe the overall pattern.
2. Find the equation of the least-squares regression line, and add the line to your plot.
3. Find the predicted number of pages for the next edition if the number of LaTeX pages for a chapter is 62.
4. Write a short report for the editor explaining to her how you constructed the regression equation and how she could use it to estimate the number of pages in the next edition of the text.

Question 2.133

Question 2.134

Question 2.135

1. We will use amps as the explanatory variable and weight as the response variable. Give a reason for this choice.
2. Make a scatterplot of the data. What do you notice about the weight and amp values?
3. Report the equation of the least-squares regression line along with the value of r².
4. Interpret the value of the estimated slope.
5. How much of an increase in amps would you expect to correspond to a one-pound increase in the weight of a saw, on average, when comparing two saws?
6. Create a residual plot for the model in part (b). Does the model indicate curvature in the data?

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Question 2.136

1. Calculate the correlation between the weights and the amps of the 19 saws.
2. Calculate the average weight of the saws for each of the five amp ratings.
3. Calculate the correlation between the average weights and the amps. Is the correlation between average weights and amps greater than, less than, or equal to the correlation between individual weights and amps?

Question 2.137

Question 2.138

1. Make a scatterplot of the data for Group 1. Find the least-squares regression line and add it to your plot. Describe the relationship between y and x for Group 1.
2. Do the same for Group 2.
3. Make a scatterplot using all 10 observations. Find the least-squares line and add it to your plot.
4. Make a plot with all of the data using different symbols for the two groups. Include the three regression lines on the plot. Write a paragraph about Simpson’s paradox for regression using this graphical display to illustrate your description.

Question 2.139

1. Make a scatterplot that shows how the length of a beam affects its strength.
2. Describe the overall pattern of the plot. Are there any outliers?
3. Fit a least-squares line to the entire set of data. Graph the line on your scatterplot. Does a straight line adequately describe these data?
4. The scatterplot suggests that the relation between length and strength can be described by two straight lines, one for lengths of 5 to 9 inches and another for lengths of 9 to 14 inches. Fit least-squares lines to these two subsets of the data, and draw the lines on your plot. Do they describe the data adequately? What question would you now ask the wood experts?

Question 2.140

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Question 2.141

1. From these data, make a two-way table of smoking (yes or no) by dead or alive. What percent of the smokers stayed alive for 20 years? What percent of the nonsmokers survived? It seems surprising that a higher percent of smokers stayed alive.
2. The age of the women at the time of the study is a lurking variable. Show that within each of the three age groups in the data, a higher percent of nonsmokers remained alive 20 years later. This is another example of Simpson’s paradox.
3. The study authors give this explanation: “Few of the older women (over 65 at the original survey) were smokers, but many of them had died by the time of follow-up.” Compare the percent of smokers in the three age groups to verify the explanation.

Question 2.142

1. Find the marginal distribution of opinion about quality. Assuming that these people represent all users of coffee filters, what does this distribution tell us?
2. How do the opinions of buyers and nonbuyers differ? Use conditional distributions as a basis for your answer. Include a mosaic plot if you have access to the needed software. Can you conclude that using recycled filters causes more favorable opinions? If so, giving away samples might increase sales.

CASE 2.1

Education Expenditures and Population: Benchmarking

We expect that states with larger populations would spend more on education than states with smaller populations.¹ What is the nature of this relationship? Can we use this relationship to evaluate whether some states are spending more than we expect or less than we expect? This type of exercise is called benchmarking. The basic idea is to compare processes or procedures of an organization with those of similar organizations.

The data file EDSPEND gives

for each of the 50 states in the United States.

Question 2.3

Question 2.4

1. Use graphical and numerical summaries to describe the distribution of spending.
2. Do the same for population.
3. Write a short paragraph summarizing your work in parts (a) and (b).

EXAMPLE 2.3

Spending and population

CASE 2.1 A state with a larger number of people needs to spend more money on education. Therefore, we think of population as an explanatory variable and spending on education as a response variable. We begin our study of this relationship with a graphical display of the two variables.

Figure 2.1 is a scatterplot that displays the relationship between the response variable, spending, and the explanatory variable, population. The data appear to cluster around a line with relatively small variation about this pattern. The relationship is positive: states with larger populations generally spend more on education than states with smaller populations. There are three or four states that are somewhat extreme in both population and spending on education, but their values still appear to be consistent with the overall pattern.

A scatterplot shows the relationship between two quantitative variables measured on the same cases. The values of one variable appear on the horizontal axis, and the values of the other variable appear on the vertical axis. Each case in the data appears as the point in the plot fixed by the values of both variables for that case.

Question 2.5

1. Make a scatterplot similar to Figure 2.1 for the education spending data.
2. Label the four points with high population and high spending with the names of these states.

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Question 2.6

1. Create a spreadsheet with the education spending data with education spending expressed in millions of dollars and population in thousands. In other words, multiply education spending by 1000 and multiply population by 1000.
2. Make a scatterplot for the data coded in this way.
3. Describe how this scatterplot differs from Figure 2.1.

examining a
distribution, p. 18

In any graph of data, look for the overall pattern and for striking deviations from that pattern.

You can describe the overall pattern of a scatterplot by the form, direction, and strength of the relationship.

An important kind of deviation is an outlier, an individual value that falls outside the overall pattern of the relationship.

linear relationship

EXAMPLE 2.4

Scatterplot with a Straight Line

CASE 2.1 Figure 2.2 plots the education spending data along with a fitted straight line. This plot confirms our initial impression about these data. The overall pattern is approximately linear and there are a few states with relatively high values for both variables.

Two variables are positively associated when above-average values of one tend to accompany above-average values of the other, and below-average values also tend to occur together.

Two variables are negatively associated when above-average values of one tend to accompany below-average values of the other, and vice versa.

EXAMPLE 2.5

Smooth Relationship for Education Spending

Figure 2.3 is a scatterplot of the population versus education spending for the 50 states in the United States with a smooth curve generated by software. The smooth curve follows the data very closely and is somewhat bumpy. We can adjust the extent to which the relationship is smoothed by changing the smoothing parameter. Figure 2.4 is the result. Here we see that the smooth curve is very close to our plot with the line in Figure 2.2. In this way, we have confirmed our view that we can summarize this relationship with a line.

smoothing parameter

log transformation

Question 2.7

EXAMPLE 2.6

Education Spending and Population with Logarithms

Figure 2.5 is a scatterplot of the log of education spending versus the log of education for the 50 states in the United States. The line on the plot fits the data well, and we conclude that the relationship is linear in the transformed variables.

Notice how the data are more evenly spread throughout the range of the possible values. The three or four high values no longer appear to be extreme. We now see them as the high end of a distribution.

EXAMPLE 2.7

Fuel Efficiency and CO_2 Emissions

Let’s look at the relationship between highway MPG and CO2 emissions, two quantitative variables, while also taking into account the type of fuel, a categorical variable. The JMP statistical software was used to produce the plot in Figure 2.6. We see that there is a negative relationship between the two quantitative variables. Better (higher) MPG is associated with lower CO2 emissions. The relationship is curved, however, not linear.

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The legend on the right side of the figure identifies the colors used to plot the four types of fuel, our categorical variable. The vehicles that use regular gasoline (green) and premium gasoline (purple) appear to be mixed together. The diesel-burning vehicles (blue) are close to the the gasoline-burning vehicles, but they tend to have higher values for both MPG and emissions. On the other hand, the vehicles that burn ethanol (red) are clearly separated from the other vehicles.

For Exercises 2.1 and 2.2, pages 64–65; for 2.3 and 2.4, see page 65; for 2.5 and 2.6, pages 66–67; and for 2.7, see page 68.

Question 2.8

1. If two variables are negatively associated, then low values of one variable are associated with low values of the other variable.
2. A stemplot can be used to examine the relationship between two variables.
3. In a scatterplot, we put the response variable on the x axis and the explanatory variable on the y axis.

Question 2.9

1. No apparent relationship.
2. A weak negative linear relationship.
3. A strong positive relationship that is not linear.
4. A more complicated relationship. Explain the relationship.

Question 2.10

1. Which variable would you choose as the explanatory variable, and which would you choose as the response variable. Give reasons for your answers.
2. Make a scatterplot of the data.
3. Describe the form, the direction, and the strength of the relationship.
4. Are there any outliers? If yes, identify them by name.

Question 2.11

Question 2.12

1. Give a brief summary of what you learned about these variables in those exercises. (If you did not do them when you studied Chapter 1, do them now.)
2. Make a scatterplot of calories versus percent alcohol.
3. Describe the form, direction, and strength of the relationship.
4. Are there any outliers? If yes, identify them by name.

Question 2.13

Question 2.14

1. Make a scatterplot of the percent of the population over 65 versus the percent of the population under 15.
2. Describe the form, direction, and strength of the relationship.

Question 2.15

1. Generate a scatterplot of the Canadian demographic data similar to the one that you made in the previous exercise but with the points labeled “P” for provinces and “T” for territories (or some other way if that is easier to do with your software.)
2. Use your new scatterplot to write a new summary of the demographics for the 13 Canadian provinces and territories.

Question 2.16

1. What is the explanatory variable? What is the response variable? Explain your answers.
2. Are these variables categorical or quantitative?
3. Do you expect a positive or negative association between these variables? Why?
4. How strong do you expect the relationship to be? Give reasons for your answer.

Question 2.17

1. Make a scatterplot of the data. Give reasons for the choice of which variables to use on the x and y axes.
2. Describe the overall pattern in the scatterplot.
3. Describe the form, direction, and strength of the relationship.
4. Identify any outliers.
5. Is the relationship approximately linear? Explain your answer.

Question 2.18

Question 2.19

1. Why should you use the time for 2008 as the explanatory variable and the time for 2013 as the response variable?
2. Make a scatterplot of the two variables.
3. How many points are in your plot? Explain why there are not 189 points.
4. Describe the form, direction, and strength of the relationship.
5. Identify any outliers.
6. Is the relationship approximately linear? Explain your answer.

Question 2.20

1. Answer the questions in the previous exercise for this setting.
2. Compare the strength of this relationship (between the 2013 times and the 2003 times) with the strength of the relationship in the previous exercise (between the 2013 times and the 2008 times). Interpret this finding.

Question 2.21

Question 2.22

The correlation measures the direction and strength of the linear relationship between two quantitative variables. Correlation is usually written as r.

Suppose that we have data on variables x and y for n cases. The values for the first case are x₁ and y₁, the values for the second case are x₂ and y₂, and so on. The means and standard deviations of the two variables are and s_x for the x-values, and and s_y for the y-values. The correlation r between x and y is

standardizing, p. 45

Question 2.23

Question 2.24

1. Find the correlation between spending on education and population using the new units.
2. Compare this correlation with the one that you computed in the previous exercise.
3. Generally speaking, what effect, if any, did changing the units in this way have on the correlation?

resistant, p. 25

EXAMPLE 2.8

Forecasting Earnings

Stock analysts regularly forecast the earnings per share (EPS) of companies they follow. EPS is calculated by dividing a company’s net income for a given time period by the number of common stock shares outstanding. We have two analysts’ EPS forecasts for a computer manufacturer for the next six quarters. How well do the two forecasts agree? The correlation between them is r = 0.9, but the mean of the first analyst’s forecasts is $3 per share lower than the second analyst’s mean.

These facts do not contradict each other. They are simply different kinds of information. The means show that the first analyst predicts lower EPS than the second. But because the first analyst’s EPS predictions are about $3 per share lower than the second analyst’s for every quarter, the correlation remains high. Adding or subtracting the same number to all values of either x or y does not change the correlation. The two analysts agree on which quarters will see higher EPS values. The high r shows this agreement, despite the fact that the actual predicted values differ by $3 per share.

Question 2.25

1. Make a scatterplot of y versus x.
2. Describe the relationship between y and x. Is it weak or strong? Is it linear?
3. Find the correlation between y and x.
4. What important point about correlation does this exercise illustrate?

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Question 2.26

1. If a store always prices its generic “store brand” products at exactly 90% of the brand name products’ prices, what would be the correlation between these two prices? (Hint: Draw a scatterplot for several prices.)
2. If the store always prices its generic products $1 less than the corresponding brand name products, then what would be the correlation between the prices of the brand name products and the store brand products?

A regression line is a straight line that describes how a response variable y changes as an explanatory variable x changes. We often use a regression line to predict the value of y for a given value of x.

EXAMPLE 2.9

World Financial Markets

The World Economic Forum studies data on many variables related to financial development in the countries of the world. They rank countries on their financial development based on a collection of factors related to economic growth.9 Two of the variables studied are gross domestic product per capita and net assets per capita. Here are the data for 15 countries that ranked high on financial development:

In this table, GDP is gross domestic product per capita in thousands of dollars and assets is net assets per capita in thousands of dollars. Figure 2.9 is a scatterplot of the data. The correlation is r = 0.76. The scatterplot includes a regression line drawn through the points.

prediction

Question 2.44

Question 2.45

EXAMPLE 2.10

The Least-Squares Idea

Figure 2.10 illustrates the idea. This plot shows the data, along with a line. The vertical distances of the data points from the line appear as vertical line segments.

The least-squares regression line of y on x is the line that makes the sum of the squares of the vertical distances of the data points from the line as small as possible.

We have data on an explanatory variable x and a response variable y for n cases. From the data, calculate the means and and the standard deviations s_x and s_y of the two variables and their correlation r. The least-squares regression line is the line

with slope

and intercept

EXAMPLE 2.11

The Equation for Predicting Net Assets

The line in Figure 2.9 is in fact the least-squares regression line for predicting net assets per capita from GDP per capita. The equation of this line is

slope

intercept

EXAMPLE 2.12

Predict Net Assets

The equation of the regression line makes prediction easy. Just substitute a value of x into the equation. To predict the net assets per capita for a country that has a GDP per capita of $50,000, we use x = 50:

The predicted net assets per capita is $198,000.

plot the line

Question 2.46

1. What is the slope of the regression line?
2. What is the intercept of the regression line?
3. Find the predicted values of y for x = 10, for x = 20, and for x = 30.
4. Plot the regression line for values of x between 0 and 50.

EXAMPLE 2.13

GDP and Assets Results Using Software

Figure 2.11 displays the selected regression output for the world financial markets data from JMP, Minitab, and Excel. The complete outputs contain many other items that we will study in Chapter 10.

Let’s look at the Minitab output first. A table gives the regression intercept and slope under the heading “Coefficients.” Coefficient is a generic term that refers to the quantities that define a regression equation. Note that the intercept is labeled “Constant,” and the slope is labeled with the name of the explanatory variable. In the table, Minitab reports the intercept as −27.2 and the slope as 4.50 followed by the regression equation.

85

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Excel provides the same information in a slightly different format. Here the intercept is reported as −27.16823305, and the slope is reported as 4.4998956. Check the JMP output to see how the regression coefficients are reported there.

Question 2.47

1. Use software to compute the coefficients of the regression equation. Indicate where to find the slope and the intercept on the output, and report these values.
2. Make a scatterplot of the data with the least-squares line.
3. Find the predicted value of assets for each country.
4. Find the difference between the actual value and the predicted value for each country.

EXAMPLE 2.14

Education Spending and Population

CASE 2.1 Figure 2.12 is a scatterplot of the education spending data described in Case 2.1 (page 65). There is a positive linear relationship.

The two lines on the plot are the two least-squares regression lines. The regression line for using population to predict education spending is solid. The regression line for using education spending to predict population is dashed. The two regressions give different lines. In the regression setting, you must choose one variable to be explanatory.

To find the percent of variation explained by the least-squares equation, square the value of the correlation and express the result as a percent.

EXAMPLE 2.15Using r²

The correlation between GDP per capita and net assets per capita in Example 2.12 (pages 83–84) is r = 0.76312, so r² = 0.58234. GDP per capita explains about 58% of the variability in net assets per capita.

Question 2.48

1. What percent of the observed variation in yearly changes in the index is explained by a straight-line relationship with the change during January?
2. What is the equation of the least-squares line for predicting the full-year change from the January change?
3. The mean change in January is . Use your regression line to predict the change in the index in a year in which the index rises 1.75% in January. Why could you have given this result (up to roundoff error) without doing the calculation?

Question 2.49

A residual is the difference between an observed value of the response variable and the value predicted by the regression line. That is,

EXAMPLE 2.16

Education Spending and Population

CASE 2.1 Figure 2.13 is a scatterplot showing education spending versus the population for the 50 states that we studied in Case 2.1 (page 65). Included on the scatterplot is the least-squares line. The points for the states with large values for both variables—California, Texas, Florida, and New York—are marked individually.

The equation of the least-squares line is where represents education spending and x represents the population of the state.

Let’s look carefully at the data for California, y = 110.1 and x = 38.7. The predicted education spending for a state with 38.7 million people is

The residual for California is the difference between the observed spending (y) and this predicted value.

Question 2.50

1. Find the predicted education spending for Texas.
2. Find the residual for Texas.
3. Which state, California or Texas, has a greater deviation from the regression line?

Question 2.51

A residual plot is a scatterplot of the regression residuals against the explanatory variable. Residual plots help us assess the fit of a regression line.

EXAMPLE 2.17

Residual Plot for Education Spending

CASE 2.1 Figure 2.14 gives the residual plot for the education spending data. The horizontal line at zero in the plot helps orient us.

Question 2.52

1. Sketch a version of Figure 2.14 or generate your own plot using the EDSPEND data file. Write in the names of the states California,Texas, Florida, and New York on your plot.
2. Explain how you were able to identify these four points on your sketch.

Normal quantile
plots, p. 51

EXAMPLE 2.18

Are the Residuals Approximately Normal?

CASE 2.1 Figure 2.16 gives the Normal quantile plot for the residuals in our education spending example. The distribution of the residuals is not Normal. Most of the points are close to a line in the center of the plot, but there appear to be five outliers—one with a negative residual and four with positive residuals.

influential

EXAMPLE 2.19

Is California Influential?

CASE 2.1 To answer this question, we compare the regression lines with and without California. The result is in Figure 2.17. The two lines are very close, so we conclude that California is not influential with respect to the least-squares slope and intercept.

EXAMPLE 2.20

Suppose California Spent Half as Much?

CASE 2.1 What would happen if California spent about half of what was actually spent, say, $55 million. Figure 2.18 shows the two regression lines, with and without California. Here we see that the regression line changes substantially when California is removed. Therefore, in this setting we would conclude that California is very influential.

An outlier is an observation that lies outside the overall pattern of the other observations. Points that are outliers in the y direction of a scatterplot have large regression residuals, but other outliers need not have large residuals.

A case is influential for a statistical calculation if removing it would markedly change the result of the calculation. Points that are extreme in the x direction of a scatterplot are often influential for the least-squares regression line.

Question 2.53

EXAMPLE 2.21

Effects on the Correlation

CASE 2.1 The correlation between the spending on education and population for the 50 states is r = 0.98. If we drop California, it decreases to 0.97. We conclude that California is not influential on the correlation.