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Problem (multiple regression).  Following data are taken from textbook (p.614) Student Math-proficiency   X1 SAT-Math X2 Calculus final...

Problem (multiple regression).  Following data are taken from textbook (p.614)

Student

Math-proficiency  

X1

SAT-Math X2

Calculus final (college 1st year)  Y

1

72

462

71

2

96

545

92

3

68

585

72

4

86

580

82

5

70

592

74

6

73

516

71

7

91

638

100

8

75

615

87

9

76

596

81

V= 56331/9 = 6259   S=sqrt6259 = 79.1

The least-square equation is Y=-26.6+0.777X1+0.082 X2, that is :

Calculus final=-26.6+0.777 *math-proficiency +0.082* Sat-Math

  1. Find the approximate 95% confidence interval for the mean calculus final for all the students having X1=70 and X2=592. (who scored 70 in Math-proficiency and 592 on Sat-Math).
  2. Compute the TSS and ESS from data and regression equation. Please take a screen shot of you lists in TI-84 and submit the screen shot showing the data, the predicted value, the residuals and the squared residuals.  This is a proof that you did the problem.
  3. Compute the coefficient of multiple determination and interpret its value in the context of the problem.  You must write down TSS, ESS and how r2 is computed from them.
  4. Conduct ANOVA F-test for the entirety of the regression. You must write down the ANOVA table with all the details.  (30 points)

Solutions

Expert Solution

  1. Find the approximate 95% confidence interval for the mean calculus final for all the students having X1=70 and X2=592. (who scored 70 in Math-proficiency and 592 on Sat-Math).
95% Confidence Interval
X1 X2 lower upper
70 592 71.359 81.206
  1. Compute the TSS and ESS from data and regression equation. Please take a screen shot of you lists in TI-84 and submit the screen shot showing the data, the predicted value, the residuals and the squared residuals.  This is a proof that you did the problem.
SS  
Residual 97.3230
Total 848.8889

The regression equation is:

y = -26.6155 + 0.7763*x1 + 0.0820*x2

  1. Compute the coefficient of multiple determination and interpret its value in the context of the problem.  You must write down TSS, ESS and how r2 is computed from them.

R2 = 0.885

88.5% of the variability in the model is explained.

  1. Conduct ANOVA F-test for the entirety of the regression. You must write down the ANOVA table with all the details.  (30 points)
Source SS   df   MS F p-value
Regression 751.5659 2   375.7829 23.17 .0015
Residual 97.3230 6   16.2205
Total 848.8889 8  

The hypothesis being tested is:

H0: β1 = β2 = 0

H1: At least one βi ≠ 0

The p-value is 0.0015.

Since the p-value (0.0015) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that the model is significant.

The calculations are:

0.885
Adjusted R² 0.847
R   0.941
Std. Error   4.027
n   9
k   2
Dep. Var. Y
ANOVA table
Source SS   df   MS F p-value
Regression 751.5659 2   375.7829 23.17 .0015
Residual 97.3230 6   16.2205
Total 848.8889 8  
Regression output confidence interval
variables coefficients std. error    t (df=6) p-value 95% lower 95% upper
Intercept -26.6155
X1 0.7763 0.1465 5.300 .0018 0.4179 1.1347
X2 0.0820 0.0270 3.039 .0228 0.0160 0.1481
Predicted values for: Y
95% Confidence Interval 95% Prediction Interval
X1 X2 lower upper lower upper
70 592 71.359 81.206 65.266 87.299

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