Homework #19 Solutions (Chapter 12: exercises 12-20 even)
12. Ho: b1 = b2 = b3 = b4 = b5 = 0 H1: Not all b‘s equal zero.
df1 = 5 df2 = 20 – (5 + 1) = 14, so Ho is rejected if F > 2.96
Source SS df MSE F
Regression 448.28 5 89.656 17.58
Error 71.40 14 5.10
Total 519.68 19
So Ho is rejected. Not all the regression coefficients equal zero.
14. a. The strongest relationship is between sales and income (0.964). A problem could occur if both “cars” and “outlets” are part of the final solution. Also, outlets and income are strongly correlated (0.825). This is called multicollinearity.
b.
c. Ho is rejected. At least one regression coefficient is not zero. The computed value of F is 140.42.
d. Delete “outlets” and “bosses” Critical values are -2.776 and 2.776
e.
There was little change in the
coefficient of determination
f. The normality assumption appears reasonable.
g. There is nothing unusual about the plots.
16. a. The correlation matrix is:
Salary Years Rating
Years 0.868
Rating 0.547 0.187
Master 0.311 0.208 0.458
Years has the strongest correlation with salary. There does not appear to be a problem with multicollinearity.
b. The regression equation is: Y¢ = 9.92 + 0.899X1 + 0.154X2 - 0.67 X3
Y¢ = 23.655 or $23,655
c. Ho is rejected if F > 3.24 Computed F = 301.06/5.71 = 52.72
Ho is rejected. At least one regression coefficient is not zero.
d. A regression coefficient is dropped if computed t is to the left of -2.120 or right of 2.120. Keep “years” and “rating”; drop “masters.”
e. Dropping “masters”, we have:
Salary = 10.1157 + 0.8926(years) + 0.1464(rating)
f. The stem and leaf display and the histogram revealed no problem with the assumption of normality. Again using MINITAB:
Midpoint Count
-4 1 *
-3 1 *
-2 3 * * *
-1 3 * * *
0 4 * * * *
1 4 * * * *
2 0
3 3 * * *
4 0
5 1 *
g. There does not appear to be a pattern to the residuals according to the following MINITAB plot.
18. a. The regression equation is: Y¢ = 1480.7 + 0.7315X1 + 9.991X2 - 2.308X3
b. R2 = 83.5% or 0.835
c. Ho: b1 = b2 = b3 = 0 H1: Not all bI’s = 0 Reject Ho if F > 3.59
Ho is rejected. Some of the net regression coefficients do not equal zero.
d. b1 = 0 b2 = 0 b3 = 0
b1 ¹ 0 b2 ¹ 0 b3 ¹ 0
Reject Ho if t < -2.201 or t > 2.201
Reject Ho for area and spaces, do not reject for income. Delete income.
e. R2 = 0.804, Y¢ = 1342.49 + 0.7727X1 + 11.634X2
20. a. Y¢ = 28.2 + 0.0287X1 + 0.650X2 - 0.049X3 - 0.00040X4 - 0.723X5
b. R2 = 0.750
c & d. Predictor Coef Stdev t-ratio P
Constant 28.242 2.986 9.46 0.000
Value 0.028669 0.004970 5.77 0.000
Years 0.6497 0.2412 2.69 0.014
Age -0.04895 0.3126 -1.57 0.134
Mortgage -0.000405 0.001269 -0.32 0.753
Sex 0.7227 0.2491 2.90 0.009
s = 0.5911 R-sq = 75.0% R-sq(adj) = 68.4%
Analysis of Variance
Source DF SS MS F p
Regression 5 19.8914 3.9783 11.39 0.000
Error 19 6.6390 0.3494
Total 24 26.5304
Consider dropping the variables age and mortgage.
e. The new regression equation is: Y¢ = 29.81 + 0.0253X1 + 0.41X2 + 0.708X5
The value of R2 is 0.7161