Homework answers / question archive / STAT5131&6031          Homework 6 (Due: Monday 10/31/2016)      Fall 2016 Attach to front of homework paper STAT :   Name (LAST, First):   Reminder: DO NOT hand in unedited SAS output/code

STAT5131&6031          Homework 6 (Due: Monday 10/31/2016)      Fall 2016 Attach to front of homework paper STAT :   Name (LAST, First):   Reminder: DO NOT hand in unedited SAS output/code

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STAT5131&6031          Homework 6 (Due: Monday 10/31/2016)      Fall 2016

Attach to front of homework paper

STAT :

 

Name (LAST, First):

 

Reminder:

• DO NOT hand in unedited SAS output/code.
• ONLY include the results required
• Answer questions IN ORDER; Include question number; Label graphs and tables
• Include SAS input for questions at the end of the entire homework as Appendix
• SAVE your SAS code, since you may be asked to continue problems on successive homeworks

Assigned problems:

1. Based on the following small data set, construct the design matrix, X, its transpose X⁰, and the matrices X⁰X, (X⁰X)⁻¹, X⁰Y, and the vector b = (X⁰X)⁻¹X⁰Y. (KNNL Chapter 5 discusses finding the inverse of a matrix.)

 

X

Y

2

1

4

2

6

3

8

7

10

9

 

For the following 5 problems, consider the data given in the file CH06PR18.DAT, which describes a data set (n = 24) used to evaluate the relation between intermediate and senior level annual salaries of bachelor’s and master’s level mathematicians (Y , in thousand dollars) and an index of work quality (X₁), number of years of experience (X₂), and an index of publication success (X₃).

2. Run the multiple linear regression with quality, experience, and publications as the explanatory variables and salary as the response variable. Summarize the regression results by giving
   • the fitted regression equation
   • the value of R²
   • the results of the significance test for the null hypothesis that the three regression coefficients for the explanatory variables are all zero (give null and alternative hypotheses, test statistic with degrees of freedom, p-value, and brief conclusion in words)
3. Give 95% confidence intervals respectively (do not use a Bonferroni correction) for regression coefficients of quality, experience, and publications based on the multiple regression. Describe the results of the hypothesis tests for the individual regression coefficients ( give null and alternative hypotheses, test statistic with degrees of freedom, p-value, and a brief conclusion in words). What is the relationship between these results and the confidence intervals?
4. Plot the residuals versus the predicted salary and each of the explanatory variables (i.e., 4 residual plots). Are there any unusual patterns?
5. Examine the assumption of normality for the residuals using a qqplot and histogram. State your conclusions.
6. Predict the salary for a mathematician with quality index equal to 2, 8 years of experience, and publication index equal to 5.9 . Provide a 95% prediction interval with your prediction.

For the following problems use the computer science data. You can download the data set csdata.dat from Blackboard. The variables are: id, a numerical identifier for each student; GPA, the grade point average after three semesters; HSM; HSS; HSE; SATM; SATV, which were all explained in class; and GENDER, coded as 1 for men and 2 for women.

7. In this exercise you will illustrate some of the ideas described in Chapter 7 related to the extra sums of squares.

a.       Create a new variable called SAT which equals SATM + SATV and run the following two regressions:

• • Predict GPA using HSM, HSS, and HSE
  • Predict GPA using SAT, HSM, HSS, and HSE

Calculate the extra sum of squares for the comparison of these two analyses. Use it to construct the (partial) F-statistic (i.e., the general linear test statistic) for testing the null hypothesis that the coefficient of the SAT variable is zero in the model with all four predictors. What are the degrees of freedom for this test statistic?

7. 1. 2. Use the test statement in proc reg to obtain the same test statistic. Give the statistic, degrees of freedom, p-value and conclusion.
      3. Compare the test statistic and p-value from the test statement with the individual t-test for the coefficient of the SAT variable in the full model. Explain the relationship.
8. Run the regression to predict GPA using SATM, SATV, HSM, HSE, and HSS. Put the variables in the order given above on the model statement. Use the SS1 and SS2 options on the model statement.
   1. 1. Add the Type I sums of squares for the five predictor variables. Do the same for the Type II sums of squares. Do either of these sum to the model sum of squares? Are there any predictors for which the two sums of squares (Type I and Type II) are the same? Explain why.
      2. Verify (by running additional regressions and doing some arithmetic with the results) that the Type I sum of squares for the variable SATV is the difference in the model sum of squares (or error sum of squares) for the following two analyses:
   2. Predict GPA using SATM, SATV
   3. Predict GPA using SATM
9. Create an additional variable called HS that is the sum of the three high school scores (HSE + HSS + HSM). Run the regression to predict GPA using a variety of variables, including HS and SAT, as described below. Summarize the results by making a table giving the percentage of variation explained (R²) by each of the following models:

(Please DO NOT include SAS output for these models. Only the R² value is needed. You can run proc reg with multiple model statements to save typing.)

1. 1. SATM as the explanatory variable
   2. SATV as the explanatory variable
   3. HSM as the explanatory variable
   4. HSS as the explanatory variable
   5. HSE as the explanatory variable
   6. SATM and SATV as the explanatory variables
   7. SAT (=SATM+SATV) as the explanatory variable
   8. HSM, HSS, and HSE as the explanatory variables
   1. HS (=HSM+HSS+HSE) as the explanatory variable
   10. SATM, SATV, HSM, HSS, and HSE as the explanatory variables
   11. SAT and HS as the explanatory variables