10.28 School budget and number of students. Suppose that there is a linear relationship between the number of students x in a school system and the annual budget y. Write a population regression model to describe this relationship.

1. Which parameter in your model is the fixed cost in the budget (for example, the salary of the principals and some administrative costs) that does not change as x increases?

2. Which parameter in your model shows how total cost changes when there are more students in the system? Do you expect this number to be greater than 0 or less than 0?

3. Actual data from various school systems will not fit a straight line exactly. What term in your model allows variation among schools of the same size x?

10.29 Interpreting a residual plot. Figure 10.18 shows four plots of residuals versus x. For each plot, comment on the regression model conditions necessary for inference. Which plots suggest a reasonable fit to the linear regression model?

10.30 The relationship between cell phone use and academic performance. College students are the most rapid adopters of cell phone technology. They use the phone to surf the Internet, watch videos, listen to music, email, and play video games. Because a cell phone is almost always nearby, researchers have begun studying the relationship between cell phone use and various attitudes and behaviors. In one study, researchers assessed the relationship between cell phone use (CPU), cumulative GPA, anxiety, and general life satisfaction (GLS) among 496 students.¹⁶

1. Participants were undergraduates from a large midwestern university. They were recruited during class time from courses in sociology, general biology, American politics, human nutrition, and world history. The researchers argued that these courses attracted students from many majors. To participate, students had to consent to have their GPA retrieved. What do you think about this recruitment process? Can we feel comfortable assuming that this is an SRS from the population of undergraduates? Write a short summary of your opinions.

2. The following table summarizes the pairwise correlations among the four variables. For each pair of variables, test the null hypothesis that the correlation is zero. Make sure to state the test statistic, degrees of freedom, and P-value.

   	GPA	Anxiety	GLS
   CPU	−0.203	0.096	0.012
   GPA		0.004	0.207
   Anxiety			−0.221

3. Write a short paragraph that summarizes your findings.

10.31 Temperature and academic performance. Does temperature affect academic performance? If yes, does the relationship vary by sex? To study these questions, researchers from Berlin, Germany, divided 543 students into 24 sessions. In each session, students were presented with 50 similar arithmetic problems and given 5 minutes to complete as many as possible. (They were monetarily rewarded for the number of correct answers.) The sessions varied only in terms of the room temperature, which ranged from 16.19 to 32.57°C.¹⁷ Although the number of male and female students varied across sessions, let’s look at the relationship between each session’s room temperature (Temp) and the average number of correct answers by sex (Mave and Fave).

1. Make a scatterplot of Mave versus Temp. Describe the relationship.

2. Find the equation of the least-squares regression line for predicting Mave based on the room temperature and add this line to your scatterplot.

3. What is r2 for these data? Briefly explain what this tells you about the overall fit of the model to these data.

4. Check the conditions that must be approximately met for inference. Provide a set of plots and any concerns you have.

5. Assuming that inference is appropriate, is there significant evidence that temperature is associated with performance? State the hypotheses, give a test statistic and P-value, and summarize your conclusion.

10.32 Temperature and academic performance, continued. Refer to the previous exercise. Repeat parts (a)–(e) using the female average score, Fave, as the response variable.

10.33 Interpreting the results. Refer to the previous two exercises. You are a math teacher whose pay raise is based on your students’ academic performance. Suppose your class has 40 students, 20 of each sex. Explain how you might use the model results of the previous two exercises to determine an ideal room temperature for your classroom.

10.34 Public university tuition: 2014 versus 2018. TABLE 10.2 shows the in-state undergraduate tuition in 2014 and 2018 for 33 public universities.¹⁸

1. Plot the data with the 2014 tuition on the x axis and describe the relationship. Are there any outliers or unusual values? Does a linear relationship between the tuition in 2014 and 2018 seem reasonable?

2. Run the simple linear regression and give the least-squares regression line.

3. Obtain the residuals and plot them versus the 2014 tuition amount. Is there anything unusual in the plot?

4. Construct a Normal quantile plot. Do the residuals appear to be approximately Normal? Explain your answer.

5. Using the results of parts (c) and (d), identify and remove any unusual observations and repeat parts (b)–(d).

6. Compare the two sets of least-squares results. Describe any impact these unusual observations have on the results.

10.35 More on public university tuition. Refer to the previous exercise. We’ll now move forward with inference using the model fit to the data without the unusual observations identified in part (e) of the previous exercise.

1. Give the null and alternative hypotheses for examining if there is a linear relationship between 2014 and 2018 tuition amounts.

2. Write down the test statistic and P-value for the hypotheses stated in part (a). State your conclusions.

3. What percent of the variability in 2018 tuition is explained by a linear regression model using the 2014 tuition?

4. Construct a 95% confidence interval for the slope. Is there evidence that the slope is significantly different from 1? Explain your answer. (Note: If β1=1, the model implies the tuition increases $β0 on average.)

10.36 Even more on public university tuition. Refer to the previous two exercises.

1. The tuition at Skinflint U was $9800 in 2014. What is the predicted tuition in 2018?

2. The tuition at I.O.U. was $17,800 in 2014. What is the predicted tuition in 2018?

3. Discuss the appropriateness of using the fitted equation to predict tuition for each of these universities.

4. If you were to construct 95% prediction intervals for each of these universities, which interval would be wider, and why?

10.37 Predicting public university tuition: 2008 versus 2018. Refer to Exercise 10.35. The data file also includes the in-state undergraduate tuition for the year 2008.

1. Run the simple linear regression using year 2008 in place of year 2014. What is the least-squares line?

2. Obtain the residuals and check model assumptions.

3. If you had to choose between the model using 2008 tuition and the model using 2014 tuition, which would you choose? Give reasons for your answers.

10.38 Draw the fitted line. Suppose you fit 10 pairs of (x, y) data using least squares. Draw the fitted line if x¯=4, y¯=9, and the residual for the pair (2, 4) is -1.

10.39 Incentive pay and job performance. In the National Football League (NFL), performance bonuses now account for roughly 25% of player compensation.¹⁹ Does tying a player’s salary to performance bonuses result in better individual or team success on the field? Focusing on linebackers, let’s look at the relationship between a player’s end-of-year production rating and the percent of his salary devoted to incentive payments in that same year.

1. Use numerical and graphical methods to describe the two variables. Summarize your results.

2. Both variable distributions are non-Normal. Does this necessarily pose a problem for performing linear regression? Explain.

3. Construct a scatterplot of the data and describe the relationship. Are there any outliers or unusual values? Does a linear relationship between the percent of salary and the player rating seem reasonable? Is it a very strong relationship? Explain your answers.

4. Run a simple linear regression and state the least-squares regression line.

5. Obtain the residuals and assess whether the assumptions for the linear regression analysis are reasonable. Include all plots and numerical summaries used in doing this assessment.

10.40 Performance bonuses, continued. Refer to the previous exercise.

1. Now run the simple linear regression for the variable’s square root of the performance rating and percent of salary devoted to incentive payments.

2. Obtain the residuals and assess whether the assumptions for the linear regression analysis are reasonable. Include all plots and numerical summaries used in doing this assessment.

3. Construct a 95% confidence interval for the square root increase in rating, given a 1% increase in the percent of salary devoted to incentive payments.

4. Consider the values 0%, 20%, 40%, 60%, and 80% salary devoted to incentives. Compute the predicted rating for this model and for the one in the previous exercise. For the model in this problem, you will need to square the predicted value to get back to the original units.

5. Plot the predicted values versus the percent and connect those values from the same model. For which regions of percent do the predicted values from the two models differ the most?

6. Based on the comparison of regression models (both predicted values and residuals), which model do you prefer? Explain.

10.41 Studying the residuals. Refer to the previous two exercises. Using the residuals from the model fits in Exercise 10.39 and 10.40, who are the top three players to outperform their bonus percent, and who are the top three players to underperform their bonus percent? Does the choice of response variable, untransformed or transformed, impact this list? If so, which model list do you trust more?

10.42 Are female CEOs older? A pair of researchers looked at the age and sex of a large sample of CEOs.²⁰ To investigate the relationship between these two variables, they fit a regression model with age as the response variable and sex as the explanatory variable. The explanatory variable was coded x=0 for males and x=1 for females. The resulting least-squares regression line was

1. What is the expected age for a male CEO (x=0)?

2. What is the expected age for a female CEO (x=1)?

3. What is the difference in the expected age of female and male CEOs?

4. Relate your answers to parts (a) and (c) to the least-squares estimates b0 and b1.

5. The t statistic for testing H0: β1=0 was reported as −6.474. Based on this result, what can you conclude about the average ages of female and male CEOs?

6. To compare the average age of male and female CEOs, the researchers could have instead performed a two-sample t test (Chapter 7). Will this regression approach provide the same result? Explain your answer.

10.43 Gambling and alcohol use by first-year college students. Gambling and alcohol use are problematic behaviors for many college students. One study looked at 908 first-year students from a large northeastern university.²¹ Each participant was asked to fill out the 10-item Alcohol Use Disorders Identification Test (AUDIT) and a 7-item inventory used in prior gambling research among college students. AUDIT assesses alcohol consumption and other alcohol-related risks and problems. (A higher score means more risks.) A correlation of 0.29 was reported between the frequency of gambling and the AUDIT score.

1. What percent of the variability in AUDIT score is explained by frequency of gambling?

2. Test the null hypothesis that the correlation between the gambling frequency and the AUDIT score is zero.

3. The sample in this study represents 45% of the students contacted for the online study. To what extent do you think these results apply to all first-year students at this university? To what extent do you think these results apply to all first-year students? Give reasons for your answers.

10.44 Predicting water quality. The index of biotic integrity (IBI) is a measure of the water quality in streams. IBI and land use measures for a collection of streams in the Ozark Highland ecoregion of Arkansas were collected as part of a study.²² TABLE 10.3 gives the data for IBI, the percent of the watershed that was forest, and the area of the watershed, in square kilometers, for streams in the original sample with watershed area less than or equal to 70 km2.

1. Use numerical and graphical methods to describe the variable IBI. Do the same for area. Summarize your results.

2. Plot the data and describe the relationship between IBI and area. Are there any outliers or unusual patterns?

3. Give the statistical model for simple linear regression for this problem.

4. State the null and alternative hypotheses for examining the relationship between IBI and area.

5. Run the simple linear regression and summarize the results.

6. Obtain the residuals and plot them versus area. Is there anything unusual in the plot?

7. Do the residuals appear to be approximately Normal? Give reasons for your answer.

8. Do the assumptions for the analysis of these data using the model you gave in part (c) appear to be reasonable? Explain your answer.

10.45 More on predicting water quality. The researchers who conducted the study described in the previous exercise also recorded the percent of the watershed area that was forest for each of the streams. These data are also given in Table 10.3. Analyze these data using the questions in the previous exercise as a guide.

10.46 Comparing the analyses. In Exercises 10.44 and 10.45, you used two different explanatory variables to predict IBI. Summarize the two analyses and compare the results. If you had to choose between the two explanatory variables for predicting IBI, which one would you prefer? Give reasons for your answer.

10.47 How an outlier can affect statistical significance. Consider the data in Table 10.3 and the relationship between IBI and the percent of watershed area that was forest. The relationship between these two variables is almost significant at the 0.05 level. In this exercise you will demonstrate the potential effect of an outlier on statistical significance. Investigate what happens when you decrease the IBI to 0.0 for (1) an observation with 0% forest and (2) an observation with 100% forest. Write a short summary of what you have learned from this exercise.

10.48 Predicting water quality for an area of 40 km2. Refer to Exercise 10.44.

1. Find a 95% confidence interval for the mean response corresponding to an area of 40 km2.

2. Find a 95% prediction interval for a future response corresponding to an area of 40 km2.

3. Write a short paragraph interpreting the meaning of the intervals in terms of Ozark Highland streams.

4. Do you think that these results can be applied to other streams in Arkansas or in other states? Explain why or why not.

10.49 Compare the predictions. Refer to Exercise 10.46. Another way to compare analyses is to compare predictions. Consider Case 37 in Table 10.3 (8th row, 2nd column). For this case, the area is 10 km2, and the percent forest is 63%. Calculate the predicted index of biotic integrity based on area and the predicted index of biotic integrity based on percent forest. Compare these two predictions and explain why they differ. Use the idea of a prediction interval to interpret these results.

10.50 CEO pay and gross profits. Publicly traded companies must disclose their workers’ median pay and the compensation ratio between a worker and the company’s CEO. Does this ratio say something about the performance of the company? CNBC collected this ratio and the gross profits per employee from a variety of companies.²³

1. Generate a scatterplot of the gross profit per employee (Profit) versus the CEO pay ratio (Ratio). Describe the relationship.

2. To compensate for the severe right-skewness of both variables, take the logarithm of each variable. Generate a scatterplot and describe the relationship between these transformed variables.

3. Fit a simple linear regression for log Profits versus log Ratio.

4. Examine the residuals. Are the model conditions approximately satisfied? Explain your answer.

5. Construct a 95% confidence interval for β1 and interpret the result (Note: In business and economics, we often encounter models in which both variables are on the log scale. In these cases, the slope approximates the percent change in y for a 1% change in x. This relationship is known as elasticity, a very important concept in economic theory.)

10.51 Leaning Tower of Pisa. The Leaning Tower of Pisa is an architectural wonder. Engineers concerned about the tower’s stability have done extensive studies of its increasing tilt. Measurements of the lean of the tower over time provide much useful information. The following table gives measurements for the years 1975 to 1987. The variable Lean represents the difference between where a point on the tower would be if the tower were straight and where it actually is. The data are coded as tenths of a millimeter in excess of 2.9 meters, so that the 1975 lean, which was 2.9642 meters, appears in the table as 642. Only the last two digits of the year were entered into the computer.²⁴

1. Plot the data. Does the trend in lean over time appear to be linear?

2. What is the equation of the least-squares line? What percent of the variation in lean is explained by this line?

3. Give a 99% confidence interval for the average rate of change (tenths of a millimeter per year) of the lean.

10.52 More on the Leaning Tower of Pisa. Refer to the previous exercise.

1. In 1918 the lean was 2.9071 meters. (The coded value is 71.) Using the least-squares equation for the years 1975 to 1987, calculate a predicted value for the lean in 1918. (Note that you must use the coded value 18 for year.)

2. Although the least-squares line gives an excellent fit to the data for 1975 to 1987, this pattern did not extend back to 1918. Write a short statement explaining why this conclusion follows from the information available. Use numerical and graphical summaries to support your explanation.

10.53 Predicting the lean in 2021. Refer to the previous two exercises.

1. How would you code the explanatory variable for the year 2021?

2. The engineers working on the Leaning Tower of Pisa were most interested in how much the tower would lean if no corrective action were taken. Use the least-squares equation to predict the tower’s lean in the year 2021. (NOTE: The tower was renovated in 2001 to make sure it would not fall down.)

3. To give a margin of error for the lean in 2021, would you use a confidence interval for a mean response or a prediction interval? Explain your choice.

10.54 Does a math pretest predict success? Can a pretest on mathematics skills predict success in a statistics course? The 62 students in an introductory statistics class took a pretest at the beginning of the semester. The least-squares regression line for predicting the score y on the final exam from the pretest score x was y^=13.8+0.81x. The standard error of b1 was 0.43.

1. Test the null hypothesis that there is no linear relationship between the pretest score and the score on the final exam against the two-sided alternative.

2. Would you reject this null hypothesis versus the one-sided alternative that the slope is positive? Explain your answer.

10.55 Significance test of the correlation. A study reported a correlation r=0.5 based on a sample size of n=15; another reported the same correlation based on a sample size of n=25. For each, perform the test of the null hypothesis that ρ=0. Describe the results and explain why the conclusions are different.

10.56 State and college binge drinking. Excessive consumption of alcohol is associated with numerous adverse consequences. In one study, researchers analyzed binge-drinking rates from two national surveys, the Harvard School of Public Health College Alcohol Study (CAS) and the Centers for Disease Control and Prevention’s Behavioral Risk Factor Surveillance System (BRFSS).²⁵ The CAS survey was used to provide an estimate of the college binge-drinking rate in each state, and the BRFSS was used to determine the adult binge-drinking rate in each state. A correlation of 0.43 was reported between these two rates for their sample of n=40 states. The college binge-drinking rate had a mean of 46.5% and standard deviation 13.5%. The adult binge-drinking rate had a mean of 14.88% and standard deviation 3.8%.

1. Find the equation of the least-squares line for predicting the college binge-drinking rate from the adult binge-drinking rate.

2. Give the results of the significance test for the null hypothesis that the slope is 0. (Hint: What is the relation between this test and the test for a zero correlation?)

10.57 SAT versus ACT. The SAT and the ACT are the two major standardized tests that colleges use to evaluate candidates. Most students take just one of these tests. However, some students take both. Consider the scores of 60 students who did this. How can we relate the two tests?

1. Plot the data with SAT on the x axis and ACT on the y axis. Describe the overall pattern and any unusual observations.

2. Find the least-squares regression line and draw it on your plot. Give the results of the significance test for the slope.

3. What is the correlation between the two tests?

10.58 SAT versus ACT, continued. Refer to the previous exercise. Find the predicted value of ACT for each observation in the data set.

1. What is the mean of these predicted values? Compare it with the mean of the ACT scores.

2. Compare the standard deviation of the predicted values with the standard deviation of the actual ACT scores. If least-squares regression is used to predict ACT scores for a large number of students such as these, the average predicted value will be accurate, but the variability of the predicted scores will be too small.

3. Find the SAT score for a student who is 1 standard deviation above the mean (z=(x-x¯)/sx=1). Find the predicted ACT score and standardize this score. (Use the means and standard deviations from this set of data for these calculations.)

4. Repeat part (c) for a student whose SAT score is 1 standard deviation below the mean (z=−1).

5. What do you conclude from parts (c) and (d)? Perform additional calculations for different z’s, if needed.

10.59 Matching standardized scores. Refer to the previous two exercises. An alternative to the least-squares method is based on matching standardized scores. Specifically, we set

and solve for y. Let’s use the notation y=a0+a1x for this line. The slope is a1=sy/sx, and the intercept is a0=y¯−a1x¯. Compare these expressions with the formulas for the least-squares slope and intercept (page 520).

1. Using the data in the previous exercise, find the values of a0 and a1.

2. Plot the data with the least-squares line and the new prediction line.

3. Use the new line to find predicted ACT scores. Find the mean and the standard deviation of these scores. How do they compare with the mean and standard deviation of the ACT scores?

10.60 Are the results consistent? A researcher surveyed n=214 hotel managers to assess the relationship between customer-relationship management (CRM) and organizational culture.²⁶ Each variable was an average of more than 25 5-point Likert survey responses and, therefore, was treated as a quantitative variable. The researcher reports a sample correlation of r=0.74 and an ANOVA F statistic of 60.35 for a simple linear regression of CRM on organizational culture. Using the relationship between testing H0: ρ=0 and testing H0: β1=0, show that these two results are not consistent. (Hint: It’s far more likely that there was a typo and r=0.47.)

10.61 A mechanistic explanation of popularity. Previous experimental work has suggested that the serotonin system plays an important and causal role in social status. In other words, genes may predispose individuals to be popular/likable. As part of a recent study on adolescents, an experimenter looked at the relationship between the expression of a particular serotonin receptor gene, a person’s “popularity,” and the person’s rule-breaking (RB) behaviors.²⁷ RB was measured using both a questionnaire and video observation. The composite score is an equal combination of these two assessments. Here is a table of the correlations:

For each correlation, test the null hypothesis that the corresponding true correlation is zero. Reproduce the table and mark the correlations that have P<0.001 with ***, those that have P<0.01 with **, and those that have P<0.05 with *. Write a summary of the results of your significance tests.

10.62 Resting metabolic rate and exercise. Metabolic rate, the rate at which the body consumes energy, is important in studies of weight gain, dieting, and exercise. The following table gives data on the lean body mass and resting metabolic rate for 12 women and 7 men who are subjects in a study of dieting. Lean body mass, given in kilograms, is a person’s weight, leaving out all fat. Metabolic rate is measured in calories burned per 24 hours, the same calories used to describe the energy content of foods. The researchers believe that lean body mass is an important influence on metabolic rate.

1. Make a scatterplot of the data, using different symbols or colors for men and women. Summarize what you see in the plot.

2. Run the regression to predict metabolic rate from lean body mass for the women in the sample and summarize the results. Do the same for the men.

10.63 Resting metabolic rate and exercise, continued. Refer to the previous exercise. It is tempting to conclude that there is a strong linear relationship for the women but no relationship for the men. Let’s look at this issue a little more carefully.

1. Find the confidence interval for the slope in the regression equation that you ran for the females. Do the same for the males. What do these suggest about the possibility that these two slopes are the same? (The formal method for making this comparison is a bit complicated and is beyond the scope of this chapter.)

2. Examine the formula for the standard error of the regression slope given on page 548. The term in the denominator is ∑(xi-x¯)2. Find this quantity for the females; do the same for the males. How do these calculations help to explain the results of the significance tests?

3. Suppose that you were able to collect additional data for males. How would you use lean body mass in deciding which subjects to choose?

10.64 Significance tests and confidence intervals. The significance test for the slope in a simple linear regression gave a value t=2.08 with 18 degrees of freedom. Would the 95% confidence interval for the slope include the value zero? Give a reason for your answer.