11.15 Refining the GPA model using all six explanatory variables: Residual checks. Figure 11.11 (page 631) provides a list of the top models based on R². Let’s look more closely at the four models listed with p = 3 and p = 4. Fit each of these models to the data and obtain the residuals. Do the data, at least approximately, meet the conditions of the multiple regression model? Provide some plots to support your opinion.

11.16 Refining the GPA model using all six explanatory variables: Inference. Refer to the previous exercise. For each of the four models considered in the previous exercise, report the least-squares equation, estimated model standard deviation s, and the P-values for each of the individual coefficients. Based on these results and the residuals checks of the previous exercise, which model do you think provides the “best’’ fit? Explain your answer.

11.17 A mechanistic explanation of popularity. In Exercise 10.65 (page 605), correlations between an adolescent’s “popularity,’’ expression of a serotonin receptor gene, and rule-breaking behaviors were assessed. An additional portion of the analysis looked at the relationship between the gene expression level and popularity, after adjusting for rule-breaking (RB) behaviors. This adjustment was necessary because RB is positively associated both with this gene expression and with popularity in adolescents. The following summarizes these regression analyses using the composite (questionnaire and video) RB score. A total of 202 individuals were included in this analysis.

For all analysis use the 0.05 significance level.

Results such as these suggest not only that adolescents with high serotonin receptor gene expression are predisposed to increased RB behaviors, but also that such behaviors are socially advantageous.

11.18 Predicting college debt: Multiple regression. Refer to Exercises 10.12 (page 579) and 10.17 (page 580) for a description of the problem. Let’s now consider fitting a model using Admit, GradRate, InCostAid, and OutCostAid as the explanatory variables.

11.19 Predicting college debt: Inference. Refer to the previous exercise. Let’s proceed using the data set without Baruch College.

11.20 Testing a collection of variables. Refer to the previous exercise. For the model that included all p = 4 explanatory variables, only InCostAid is found significant using the individual parameter t tests. This raises the question whether these other three variables further contribute to the prediction of average debt given in-state cost is in the model.

In this chapter, we discussed the F test for a collection of regression coefficients. In most cases, this capability is provided by the software. When it is not, the test can be performed using the R²-values from the larger (full) and smaller (reduced) models. The test statistic is

with q and n − p − 1 degrees of freedom. is the value for the full model, and is the value for the reduced model. Here n = 24 schools, p = 4 variables in the full model, and q = 3 variables were removed to form the reduced model. Plug in the values of R² from part (b) of the previous exercise and the R² value from Figure 10.13 (page 580). Compute the test statistic and P-value. Do Admit, GradRate, and OutCostAid combined add any significant predictive information beyond what is already contained in InCostAid?

11.21 Comparison of prediction intervals. Refer to the previous exercise. Another way to compare these two models is in terms of prediction. The Ohio State University has Admit = 56, GradRate = 59, InCostAid = 12,103, and OutCostAid = 28,603. Use statistical software to construct.

11.22 Consider the sex of the students. Refer to Exercises 11.15 and 11.16. The seventh explanatory variable provided in the GPA data set is a sex indicator variable. This variable (SEX) takes the value 0 for males and 1 for females. If we include it in our model involving the other six variables, it allows the intercept to differ for the two genders. Using b₇ to represent the fitted coefficient for the SEX variable, the estimated male intercept is b₀ + b₇(0) = b₀ and the estimated female intercept is b₀ + b₇(1) = b₀ + b₇. The difference between these two intercept estimates is (b₀ + b₇) − b₀ = b₇, so the coefficient is also an estimate of the difference in intercepts.

11.23 Predicting energy-drink consumption. Energy-drink advertising consistently emphasizes a physically active lifestyle and often features extreme sports and risk taking. Are these typical characteristics of an energy-drink consumer? A researcher decided to examine the links between energy-drink consumption, sport-related (jock) identity, and risk taking.⁵ She invited more than 1500 undergraduate students enrolled in large introductory-level courses at a public university to participate. Each participant had to complete a 45-minute anonymous questionnaire. From this questionnaire, jock identity and risk-taking scores were obtained, where the higher the score, the stronger the trait. She ended up with 795 respondents. The following table summarizes the results of a multiple regression analysis using the frequency of energy-drink consumption in the past 30 days as the response variable:

A superscript of ** means that the individual coefficient t test had a P-value less than 0.01, and a superscript of *** means that the test had a P-value less than 0.001. All other P-values were greater than 0.05.

11.24 Is the number of tornadoes increasing? In Exercise 10.19, data on the number of tornadoes in the United States between 1953 and 2014 were analyzed to see if there was a linear trend over time. Some argue that it’s not the number of tornadoes increasing over time, but rather the probability of sighting them because there are more people living in the United States. Let’s investigate this by including the U.S. census count as an additional explanatory variable.