Lecture 12 More Examples for SLR More Examples for MLR 9/19/2018 ST3131, Lecture 12

More Examples for SLR Example 1(3.10 Page 77) One may wonder if people of similar heights tend to marry each other. For this purpose, a sample of newly married couples was selected. Let X=height of a husband, Y=height of the wife. The data can be downloadable from the class website. Let Wife=beta0+beta1 husband. Test both the intercept and the slope are zero. Is it true that in general, a tall man expects a tall woman as his wife? Can you verify this statistically? 9/19/2018 ST3131, Lecture 12

9/19/2018 ST3131, Lecture 12 Results for: P049.txt Regression Analysis: Wife versus Husband The regression equation is Wife = 41.9 + 0.700 Husband Predictor Coef SE Coef T P Constant 41.93 10.66 3.93 0.000 Husband 0.69965 0.06106 11.46 0.000 S = 5.928 R-Sq = 58.3% R-Sq(adj) = 57.8% Analysis of Variance Source DF SS MS F P Regression 1 4613.7 4613.7 131.29 0.000 Residual Error 94 3303.3 35.1 Total 95 7917.0 Descriptive Statistics: Wife, Husband Variable N Mean Median TrMean StDev SE Mean Wife 96 163.90 164.50 164.10 9.13 0.93 Husband 96 174.32 175.50 174.49 9.96 1.02 Variable Minimum Maximum Q1 Q3 Wife 141.00 181.00 158.00 170.75 Husband 152.00 192.00 166.25 182.75 9/19/2018 ST3131, Lecture 12

(c) Suppose that Tom is 165cm tall, what is the estimated height of his future wife? Give a 95% prediction interval for his future wife height. (d) Suppose that Bill is 270cm tall, expectedly, how tall is his future wife? Give a 95% confidence interval for the expected height of his wife. 9/19/2018 ST3131, Lecture 12

More Examples for MLR Example 1 (3.3, page 75) table 3.10 shows the scores in the final examination F and the scores in two preliminary examinations P1 and P2 for 22 students in a statistics course. The data can be found in the class website. (a). Fit each of the following models to the data: 9/19/2018 ST3131, Lecture 12

9/19/2018 ST3131, Lecture 12 Results for: P076.txt Regression Analysis: F versus P1 The regression equation is F = - 22.3 + 1.26 P1 Predictor Coef SE Coef T P Constant -22.34 11.56 -1.93 0.068 P1 1.2605 0.1399 9.01 0.000 S = 5.081 R-Sq = 80.2% R-Sq(adj) = 79.2% Analysis of Variance Source DF SS MS F P Regression 1 2094.7 2094.7 81.14 0.000 Residual Error 20 516.3 25.8 Total 21 2611.1 9/19/2018 ST3131, Lecture 12

9/19/2018 ST3131, Lecture 12 Regression Analysis: F versus P2 The regression equation is F = - 1.85 + 1.00 P2 Predictor Coef SE Coef T P Constant -1.854 7.562 -0.25 0.809 P2 1.00427 0.09059 11.09 0.000 S = 4.275 R-Sq = 86.0% R-Sq(adj) = 85.3% Analysis of Variance Source DF SS MS F P Regression 1 2245.6 2245.6 122.89 0.000 Residual Error 20 365.5 18.3 Total 21 2611.1 9/19/2018 ST3131, Lecture 12

9/19/2018 ST3131, Lecture 12 Regression Analysis: F versus P1, P2 The regression equation is F = - 14.5 + 0.488 P1 + 0.672 P2 Predictor Coef SE Coef T P Constant -14.501 9.236 -1.57 0.133 P1 0.4883 0.2330 2.10 0.050 P2 0.6720 0.1793 3.75 0.001 S = 3.953 R-Sq = 88.6% R-Sq(adj) = 87.4% Analysis of Variance Source DF SS MS F P Regression 2 2314.3 1157.1 74.07 0.000 Residual Error 19 296.8 15.6 Total 21 2611.1 9/19/2018 ST3131, Lecture 12

(c ) Which variable individually , P1 or P2 is a better predictor of F? (d) Which of the 3 models would you use to predict the final examination scores for a student who scored 78 and 85 on the first and second preliminary exams, respectively? What is your prediction in this case? 9/19/2018 ST3131, Lecture 12

Salary=annual salary in thousands of dollars Example 2 (3.11, page 79) To decide whether a company is discriminating against women, the following data were collected from the company’s records: Salary=annual salary in thousands of dollars Qualification=an index of employee qualification Sex= 1 for male employee 0 for female employee. Two linear models were fit to the data and the regression outputs are: Model 1: Salary=Constant+Beta1 Qualification+Beta2 Sex+ error Variable Coefficient s.e. T-test P-value constant 20009.5 .8244 24271 <.0001 Qualification .935253 .0500 18.7 Sex .224337 .4681 .479 .6329 Model 2: Qualification=Constant+Beta1 Sex+Beta2 Salary+ error Variable Coefficient s.e. T-test P-value constant -16744.4 896.4 -18.7 <.0001 Qualification .850979 .4349 1.96 .0532 Sex .836991 .0448 18.7 9/19/2018 ST3131, Lecture 12

Suppose that the usual regression assumptions hold. Are men paid more than equally qualified women? Are men less qualified than equally paid women? Do you detect any inconsistency in the above results? Explain. Which model would you advocate if you were the defense lawyer? Explain. 9/19/2018 ST3131, Lecture 12

Example 3 (3.15, page 80) Consider the two models: Develop an F-test for testing the above hypotheses. Let p=1 (SLR) and construct a data set Y and X1 such that H0 is not rejected at the 5% significance level. 9/19/2018 ST3131, Lecture 12

(c ) What does the null hypothesis indicate in this case? (d) Compute the appropriate value of R_square that relates the above two models. 9/19/2018 ST3131, Lecture 12