Regression Analysis: SCORE versus TEST, PERF, …

15Printouts for PART AMODEL 1:? ? ? ?? ?? ?? ?01 2 3 4 567 8E SCORE TEST PERF G G TEST G PERFR R TEST R PERF?? ? ? ? ??? ?? ? ? ? ? ? ?? ? ? ? ?? ?? ? ? ? ?Regression Analysis: SCORE versus TEST, PERF, …The regression equation isSCORE = – 11.7 + 0.373 TEST + 0.742 PERF + 8.9 G – 0.083 G*TEST – 0.112 G*PERF- 8.2 R + 0.083 R*TEST – 0.037 R*PERFPredictor Coef SE Coef T P VIFConstant -11.71 12.68 -0.92 0.357TEST 0.3732 0.1486 2.51 0.013 3.878PERF 0.7418 0.1902 3.90 0.000 3.896G 8.94 14.99 0.60 0.552 64.391G*TEST -0.0828 0.1816 -0.46 0.649 51.776G*PERF -0.1116 0.2386 -0.47 0.640 81.078R -8.21 15.29 -0.54 0.592 66.660R*TEST 0.0829 0.1761 0.47 0.638 46.949R*PERF -0.0372 0.2378 -0.16 0.876 77.507S = 13.1427 R-Sq = 38.0% R-Sq(adj) = 35.4%Analysis of VarianceSource DF SS MS F PRegression 8 20242.8 2530.4 14.65 0.000Residual Error 191 32991.4 172.7Total 199 53234.2Source DF Seq SSTEST 1 11742.2PERF 1 6235.8G 1 827.6G*TEST 1 174.0G*PERF 1 86.2R 1 1137.7R*TEST 1 35.1R*PERF 1 4.2Correlations: TEST, PERF, G, G*TEST, G*PERF, R, R*TEST, R*PERFTEST PERF G G*TEST G*PERF R R*TESTPERF 0.486G 0.066 0.066G*TEST 0.212 0.151 0.975G*PERF 0.136 0.190 0.983 0.982R -0.028 -0.029 -0.113 -0.108 -0.101R*TEST 0.131 0.036 -0.096 -0.073 -0.077 0.973R*PERF 0.025 0.079 -0.091 -0.079 -0.071 0.985 0.976Cell Contents: Pearson correlation16MODEL 2:? ? 01 2 E SCORE TEST PERF ? ?? ?? ?? ?Regression Analysis: SCORE versus TEST, PERFThe regression equation isSCORE = – 10.3 + 0.370 TEST + 0.663 PERFPredictor Coef SE Coef T P VIFConstant -10.339 7.245 -1.43 0.155TEST 0.36993 0.08789 4.21 0.000 1.309PERF 0.6625 0.1122 5.90 0.000 1.309S = 13.3778 R-Sq = 33.8% R-Sq(adj) = 33.1%Analysis of VarianceSource DF SS MS F PRegression 2 17978.0 8989.0 50.23 0.000Residual Error 197 35256.2 179.0Total 199 53234.2Best Subsets Regression: SCORE versus TEST, PERF, …Response is SCOREG G R R* * * *T P T P T PE E E E E EMallows S R S R S RVars R-Sq R-Sq(adj) Cp S T F G R T F T F1 27.8 27.5 26.5 13.931 X1 22.1 21.7 44.2 14.476 X2 33.8 33.1 10.1 13.378 X X2 29.8 29.1 22.4 13.775 X X3 35.6 34.6 6.4 13.222 X X X3 35.6 34.6 6.5 13.226 X X X4 37.8 36.5 1.7 13.030 X X X X4 37.8 36.5 1.8 13.034 X X X X5 37.9 36.3 3.4 13.053 X X X X X5 37.9 36.3 3.4 13.056 X X X X X6 38.0 36.0 5.2 13.082 X X X X X X6 38.0 36.0 5.2 13.082 X X X X X X7 38.0 35.8 7.0 13.109 X X X X X X X7 38.0 35.7 7.2 13.116 X X X X X X X8 38.0 35.4 9.0 13.143 X X X X X X X X17Stepwise Regression: SCORE2 versus TEST, PERF, …Forward selection. Alpha-to-Enter: 0.25Response is SCORE2 on 8 predictors, with N = 200Step 1 2 3 4Constant 11.8323 2.3926 0.6750 3.0635PERF 0.774 0.570 0.618 0.617T-Value 7.91 5.26 5.78 5.85P-Value 0.000 0.000 0.000 0.000TEST 0.329 0.343 0.340T-Value 3.88 4.13 4.16P-Value 0.000 0.000 0.000G*PERF -0.083 -0.089T-Value -3.18 -3.45P-Value 0.002 0.001R -4.5T-Value -2.51P-Value 0.013S 13.3 12.9 12.6 12.4R-Sq 24.00 29.41 32.88 34.99R-Sq(adj) 23.62 28.69 31.85 33.65Mallows Cp 28.3 14.4 6.1 1.9Stepwise Regression: SCORE2 versus TEST, PERF, …Backward elimination. Alpha-to-Remove: 0.1Response is SCORE2 on 8 predictors, with N = 200Step 1 2 3 4 5Constant -1.356 -1.344 -4.173 -4.201 1.132TEST 0.379 0.379 0.389 0.376 0.370T-Value 2.67 3.30 3.49 4.54 4.49P-Value 0.008 0.001 0.001 0.000 0.000PERF 0.64 0.64 0.67 0.68 0.61T-Value 3.53 3.68 4.34 5.20 5.81P-Value 0.001 0.000 0.000 0.000 0.000G 10 10 11 12T-Value 0.68 0.72 0.87 0.90P-Value 0.494 0.473 0.386 0.370R -5 -5T-Value -0.37 -0.37P-Value 0.712 0.711G*TEST -0.00T-Value -0.01P-Value 0.996G*PERF -0.228 -0.229 -0.249 -0.253 -0.088T-Value -1.00 -1.17 -1.34 -1.37 -3.44P-Value 0.318 0.242 0.183 0.173 0.00118R*TEST -0.068 -0.068 -0.087 -0.060 -0.062T-Value -0.41 -0.41 -0.56 -2.41 -2.50P-Value 0.685 0.680 0.576 0.017 0.013R*PERF 0.09 0.09 0.03T-Value 0.38 0.38 0.18P-Value 0.704 0.701 0.858S 12.5 12.5 12.5 12.4 12.4R-Sq 35.30 35.30 35.25 35.24 34.97R-Sq(adj) 32.59 32.94 33.24 33.57 33.64Mallows Cp 9.0 7.0 5.1 3.2 2.0MODEL 3:? ? ? ?? ? 01 2 3 4 E SCORE TEST PERF G TEST R TEST ? ?? ?? ?? ? ? ? ?? ? ? ?Regression Analysis: SCORE versus TEST, PERF, G*TEST, R*TESTThe regression equation isSCORE = – 11.2 + 0.435 TEST + 0.670 PERF – 0.0669 G*TEST – 0.0656 R*TESTPredictor Coef SE Coef T P VIFConstant -11.232 7.095 -1.58 0.115TEST 0.43492 0.08781 4.95 0.000 1.373PERF 0.6695 0.1097 6.10 0.000 1.314G*TEST -0.06691 0.02583 -2.59 0.010 1.062R*TEST -0.06565 0.02589 -2.54 0.012 1.029S = 13.0533 R-Sq = 37.6% R-Sq(adj) = 36.3%Analysis of VarianceSource DF SS MS F PRegression 4 20008.7 5002.2 29.36 0.000Residual Error 195 33225.6 170.4Total 199 53234.2Predicted Values for New ObservationsNewObs Fit SE Fit 95% CI 95% PI1 78.298 2.325 (73.713, 82.884) (52.150, 104.447)2 73.280 2.236 (68.871, 77.690) (47.162, 99.399)3 73.375 2.389 (68.664, 78.086) (47.204, 99.546)4 68.357 2.464 (63.497, 73.216) (42.158, 94.555)Values of Predictors for New ObservationsNewObs TEST PERF G*TEST R*TEST1 75.0 85.0 0 02 75.0 85.0 75.0 03 75.0 85.0 0 75.04 75.0 85.0 75.0 75.01Part A. [76 points] The workers in a large, multi-national corporation undergo annualevaluations by their immediate supervisors. A key element of the evaluation isthe summary score(on a scale from 1 to 100 ) that each worker receives; these scores are essential ingredients forsalary and promotion reviews. It is crucial, then, that the scores be assigned fairly; in particular,they must be free of any gender or racial bias. The corporation hires consultants in the humanresource management field to investigate the objectivity of the scoring system. As part of theirstudy, the consultants take a random sample of workers. Each worker in the sample is given awritten test that assesses the worker’s command of the knowledge their job requires;furthermore, the consultants measure how well each worker has performed recently on the job.Theconsultants considerregressionanalysesusingthefollowingvariables:SCORE , the most recent summary score received from the immediate supervisor;TEST , the test score (on a scale of 1 to 100 );PERF , the job performance measure (on a scale of 1 to 100 );G, gender ( 0 for male and 1 for female);R , race ( 0 for white and 1 for non-white).Some ofthe analyses performed by the consultants are given in the printouts; base your answersto the following questions on MODEL 1.1. [2 points] How many workers are included in the random sample?2. [3 points] What isthe fitted regression equation for white males?3. [3 points] What isthe fitted regression equation for non-white females?24. [2 points] What is the interpretation of ?1, the coefficient of TEST ?5. [1 point] What is the least squares estimate of?1?6. [1 point] What isthe estimate ofthe standard deviation of the leastsquares estimate of ?1?7. [1 point] What isthe value ofthe teststatistic fortesting that ?1is equalto zero, i.e., fortesting the null hypothesis H0:?1?0?8. [1 point] What isthe p-value fortesting the null hypothesis H0:?1?0 against the alternativehypothesis Ha:?1?0 ?9. [2 points] What distribution doesthe computer use to compute the p-valuesin question 8?10. [2 points] What isthe p-value fortesting the null hypothesis H0:?1?0 againstthe alternativehypothesis Ha:?1?0 ?311. [2 points] Would the null hypothesis H0: ?1? 0 be rejected in favor of the alternativehypothesis Ha: ?1? 0 in a test at the 1% level?12. [2 points] Isthere evidence that mean supervisorscore increases as workerjob knowledgeincreases for white males?13. [4 points] Construct a 95% confidence interval for ?1.14. [3 points] What isthe value ofthe teststatistic fortesting the null hypothesis H0:?1?0.6 ?415. [4 points] Whatisthe p-value fortesting the null hypothesis ?1?0.6 againstthe alternativehypothesis Ha:?1?0.6 ?16. [4 points] What is the value of the test statistic for testing the null hypothesis that thesupervisorscores exhibit no racialbias on average, i.e.,fortesting the null hypothesisthatthemean supervisorscore does not depend on any of the predictorsinvolving the variable R ?17. [4 points] Consider the p-value for testing the null hypothesis that the supervisor scoresexhibit no racial bias on average againstthe alternative hypothesisthatthey do exhibitsuch bias.Isthe p-value lessthan 5% ?518. [4 points] By using MODEL 2, compute the value of the test statistic for testing the nullhypothesis in MODEL 1 that the supervisor scores exhibit neither gender nor racial bias onaverage, i.e., fortesting the null hypothesisthatthe mean supervisorscore does not depend onany of the predictorsinvolving either of the variables G or R .19. [4 points] Is the p-value for testing the null hypothesis that the supervisor scores exhibitneither gender norracial bias on average againstthe alternative hypothesisthatthey do exhibitsuch biaslessthan 5% ?20. [1 point] What proportion of the variability in the supervisor scores is explained by MODEL1?21. [2 points] Based on the information given in the printout for MODEL 1, would you judgemulticollinearity tobe present?622. [4 points] Based on the given correlations, describe the nature of anymulticollinearity thatmight bepresent.23. [4 points] Based on the given printout for bestsubsetsregression, which predictor variablesshould be included in themodel? (Be sure to justify your answer.)24. [3 points] Based on the stepwise regression printout, what final model doesthe forwardselection processfitto the model? (Simply report the fitted model.)25. [3 points] Based on the stepwise regression printout, whatfinal model doesthe backwardselimination processfitto the model? (Simply report the fitted model.)7The consultants decide to use MODEL 3 .26. [1 point] Based on MODEL 3, what isthe estimate of the mean supervisorscore for nonwhitemaleworkers whose testscore is 75 and whose performance score is 85?27. [1 point] Based on MODEL 3 , what is the estimate of the standard deviation of thesupervisorscoresfor white female workers whose testscore is 75 and whose performance scoreis 85 ?28. [1 point] Based on MODEL 3 , give a 95% confidence interval for the mean supervisor scorefor white male workers whose testscore is 75 and whose performance score is 85.29. [3 points] Let ? denote the mean supervisorscore for white male workers whose testscoreis 75 and whose performance score is 85. Based on MODEL 3, what isthe value of the teststatistic fortesting the null hypothesis H0:? ?72?30. [1 point] Based on MODEL 3 , give a 95% prediction interval for the supervisor score of arandomly chosen non-white female worker whose testscore is 75 and whose performance scoreis 85 .831. [3 points] Compute the standard error of prediction that is used to compute the predictioninterval in question 30.