Given the following data: Y X 1 X 2 X 3 X 4 51.4 0.2 17.8 24.6 18.9 72.0 1.9 29.
ID: 3235129 • Letter: G
Question
Given the following data:
Y
X1
X2
X3
X4
51.4
0.2
17.8
24.6
18.9
72.0
1.9
29.4
20.7
8.0
53.2
0.2
17.0
18.5
22.6
83.2
10.7
30.2
10.6
7.1
57.4
6.8
15.3
8.9
27.3
66.5
10.6
17.6
11.1
20.8
98.3
9.6
35.6
10.6
5.6
74.8
6.3
28.2
8.8
13.1
92.2
10.8
34.7
11.9
5.9
97.9
9.6
35.8
10.8
5.5
88.1
10.5
29.6
11.7
7.8
94.8
20.5
26.3
6.7
10.0
62.8
0.4
22.3
26.5
14.3
81.6
2.3
37.9
20.0
0.5
(a) Fit a full multiple regression model to the data, computing the sample partial regression coefficients and Y intercept.
(b) By analysis of variance, test the hypothesis that there is no significant multiple regression relationship between Y and Xs.
(c) If H0 is rejected in part (b), test hypotheses H0: bi = 0 for each independent variable.
(d) Calculate the standard error of estimate and the coefficient of determination.
(e) What is the predicted Y at X1 = 5.2, X2 = 21.3, X3 = 19.7, X4 = 12.2.
(f) Do all four independent variables have a significant effect on Y in the population samples? If not, use the stepwise regression to analyze the data and report the new regression model.
Y
X1
X2
X3
X4
51.4
0.2
17.8
24.6
18.9
72.0
1.9
29.4
20.7
8.0
53.2
0.2
17.0
18.5
22.6
83.2
10.7
30.2
10.6
7.1
57.4
6.8
15.3
8.9
27.3
66.5
10.6
17.6
11.1
20.8
98.3
9.6
35.6
10.6
5.6
74.8
6.3
28.2
8.8
13.1
92.2
10.8
34.7
11.9
5.9
97.9
9.6
35.8
10.8
5.5
88.1
10.5
29.6
11.7
7.8
94.8
20.5
26.3
6.7
10.0
62.8
0.4
22.3
26.5
14.3
81.6
2.3
37.9
20.0
0.5
Explanation / Answer
(a) Fit a full multiple regression model to the data, computing the sample partial regression coefficients and Y intercept.
Y = -30.1369 + 2.0699 X1+ 2.5816 X2+ 0.6360 X3+ 1.1060X4
(b) H0: regression not significant
Ha : regression is sufficient.
F - value = 90.1768 so it is more than Fcritical and significane - F is 2.954 x 10-7
(c) H0 is rejected in part(b)
for independent variables, Null Hypothesis : H0: bi = 0
Alternative Hypothesis : Ha : bi > 0
T - stat and P - value for each variable is given above.
so for X1and X2are significant in nature. and other variables X3and X4 are insignificant in nature.
(d) standard error of estimate = 3.1065
Coefficient of determination = 0.9757
(e) What is the predicted Y at X1 = 5.2, X2 = 21.3, X3 = 19.7, X4 = 12.2.
Y = -30.1369 + 2.0699 X1+ 2.5816 X2+ 0.6360 X3+ 1.1060X4
Y = -30.1359 + 2.0699 * 5.2 + 2.5816 * 21.3 + 0.6360 * 19.7 + 1.1060 * 12.2 = 61.638
SUMMARY OUTPUT Regression Statistics Multiple R 0.9878 R Square 0.9757 Adjusted R Square 0.9648 Standard Error 3.1065 Observations 14 ANOVA df SS MS F Significance F Regression 4 3480.9944 870.2486 90.1768 2.954E-07 Residual 9 86.8542 9.6505 Total 13 3567.8486 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept -30.1369 37.5282 -0.8030 0.4426 -115.0315 54.7577 X1 2.0699 0.4562 4.5374 0.0014 1.0380 3.1019 X2 2.5816 0.7397 3.4898 0.0068 0.9082 4.2550 X3 0.6360 0.4603 1.3817 0.2004 -0.4053 1.6773 X4 1.1060 0.7648 1.4460 0.1821 -0.6242 2.8362Related Questions
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