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

2. 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

3. 9/19/2018 ST3131, Lecture 12 Results for: P049.txt
Regression Analysis: Wife versus Husband
The regression equation is Wife = Husband
Predictor Coef SE Coef T P
Constant
Husband
S = R-Sq = 58.3% R-Sq(adj) = 57.8%
Analysis of Variance
Source DF SS MS F P
Regression
Residual Error
Total
Descriptive Statistics: Wife, Husband
Variable N Mean Median TrMean StDev SE Mean
Wife
Husband
Variable Minimum Maximum Q Q3
Wife
Husband
9/19/2018
ST3131, Lecture 12

4. (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

5. 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

6. 9/19/2018 ST3131, Lecture 12 Results for: P076.txt
Regression Analysis: F versus P1
The regression equation is F = P1
Predictor Coef SE Coef T P
Constant P
S = R-Sq = 80.2% R-Sq(adj) = 79.2%
Analysis of Variance
Source DF SS MS F P
Regression
Residual Error
Total
9/19/2018
ST3131, Lecture 12

7. 9/19/2018 ST3131, Lecture 12 Regression Analysis: F versus P2
The regression equation is
F = P2
Predictor Coef SE Coef T P
Constant P
S = R-Sq = 86.0% R-Sq(adj) = 85.3%
Analysis of Variance
Source DF SS MS F P
Regression
Residual Error
Total
9/19/2018
ST3131, Lecture 12

8. 9/19/2018 ST3131, Lecture 12 Regression Analysis: F versus P1, P2
The regression equation is
F = P P2
Predictor Coef SE Coef T P
Constant P P
S = R-Sq = 88.6% R-Sq(adj) = 87.4%
Analysis of Variance
Source DF SS MS F P
Regression
Residual Error
Total
9/19/2018
ST3131, Lecture 12

9. (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

10. 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
.8244
24271
<.0001
Qualification
.0500
18.7
Sex
.4681
.479
.6329
Model 2: Qualification=Constant+Beta1 Sex+Beta2 Salary+ error
Variable
Coefficient
s.e.
T-test
P-value
constant
896.4
-18.7
<.0001
Qualification
.4349
1.96
.0532
Sex
.0448
18.7
9/19/2018
ST3131, Lecture 12

11. 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

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

13. (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