Are high- achieving high school students more inclined to take Advanced Placement exams? In a recent student survey, the co-author of this book asked students in his statistics courses to report the number of Advanced Placement (AP) exams they had taken in high school. In addition, information on their self-reported cumulative high school grade point average (HSGPA) and gender was recorded. Assume that students attended high school for four years.
a. What is the amount of exposure related to each of the observed responses? Is the exposure for any of the subjects different from the exposure for the others?
b. Is it necessary to include an offset for these data in the generalized linear model, as in Equations (8.7) and (8.12)? Why or why not?
c. Fit a Poisson regression model to the data with Y = number of AP exams taken while attending high school as the response and HSGPA as the explanatory variable.
d. Based on the computer output, interpret the estimated coefficient corresponding to HSGPA.
e. What is the estimated multiplicative increase in the rate of taking AP exams for any half-unit increase in HSGPA?
f. Investigate the adequacy of the model, as well as any departures from the model assumptions, such as zero counts and overdispersion. Should the results of Wald’s test be trusted? Why or why not?