Chapter 9.4, Problem 4P

For Problems 3-6, use appropriate multiple regression software of your choice and enter the data. Note that the data are also available for download at the Companion Sites for this text.

4. Education: Exam Scores Professor Gill has taught general psychology for many years. During the semester, she gives three multiple-choice exams, each worth 100 points. At the end of the course, Dr. Gill gives a comprehensive final worth 200 points. Let x₁, x₂, and x₃ represent a student’s scores on exams 1, 2, and 3, respectively. Let x₄ represent the student’s score on the final exam. Last semester Dr. Gill had 25 students in her class. The student exam scores are shown on the next page.

Since Professor Gill has not changed the course much from last semester to the present semester, the preceding data should be useful for constructing a regression model that describes this semester as well.

1. (a) Generate summary statistics, including the mean and standard deviation of each variable. Compute the coefficient of variation (see Section 3.2) for each variable. Relative to its mean, would you say that each exam had about the same spread of scores? Most professors do not wish to give an exam that is extremely easy or extremely hard. Would you say that all of the exams were about the same level of difficulty? (Consider both means and spread of test scores.)
2. (b) For each pair of variables, generate the sample correlation coefficient r. Compute the corresponding coefficient of determination r² Of the three exams 1,2, and 3, which do you think had the most influence on the final exam 4? Although one exam had more influence on the final exam, did the other two exams still have a lot of influence on the final? Explain each answer.
3. (c) Perform a regression analysis with x₄ as the response variable. Use x₁ x₂, and x₃ as explanatory variables. Look at the coefficient of multiple deter­mination. What percentage of the variation in x₄ can be explained by the corresponding variations in x₁ x₂, and x₃ taken together?
4. (d) Write out the regression equation. Explain how each coefficient can be thought of as a slope. If a student were to study "extra hard" for exam 3 and increase his or her score on that exam by 10 points, what correspond­ing change would you expect on the final exam? (Assume that exams I and 2 remain "fixed" in their scores.)
5. (e) Test each coefficient in the regression equation to determine if it is zero or not zero. Use level of significance 5%. Why would the outcome of each hypothesis test help us decide whether or not a given variable should be used in the regression equation?
6. (f) Find a 90% confidence interval for each coefficient.
7. (g) This semester Susan has scores of 68, 72, and 75 on exams 1, 2, and 3, respectively. Make a prediction for Susan’s score on the final exam and find a 90% confidence interval for your prediction (if your software supports prediction intervals).