# Regression

1. In the following regression, X = weekly pay, Y = income tax withheld, and n = 35 McDonald's employees. (a) Write the fitted regression equation. (b) State the degrees of freedom for a two- tailed test for zero slope, and use Appendix D to find the critical value at á = .05. (c) What is your conclusion about the slope? (d) Interpret the 95 percent confidence limits for the slope. (e) Verify that F = t2 for the slope. (f) In your own words, describe the fit of this regression.

R2 0.202

Std. Error 6.816

n 35

ANOVA table

Source SS df MS F p-value

Regression 387.6959 1 387.6959 8.35 .0068

Residual 1,533.0614 33 46.4564

Total 1,920.7573 34

Regression output confidence interval

variables coefficients std. error t (df =33) p-value 95% lower 95% upper

Intercept 30.7963 6.4078 4.806 .0000 17.7595 43.8331

Slope 0.0343 0.0119 2.889 .0068 0.0101 0.0584

2. In the following regression, X = total assets ($ billions), Y = total revenue ($ billions), and n = 64 large banks. (a) Write the fitted regression equation. (b) State the degrees of freedom for a two- tailed test for zero slope, and use Appendix D to find the critical value at á = .05. (c) What is your conclusion about the slope? (d) Interpret the 95 percent confidence limits for the slope. (e) Verify that F = t2 for the slope. (f) In your own words, describe the fit of this regression.

R2 0.519

Std. Error 6.977

n 64

ANOVA table

Source SS df MS F p-value

Regression 3,260.0981 1 3,260.0981 66.97 1.90E-11

Residual 3,018.3339 62 48.6828

Total 6,278.4320 63

Regression output confidence interval

variables coefficients std.error(df =62) p-value 95% lower 95% upper

Intercept 6.5763 1.9254 3.416 .0011 2.7275 10.4252

X1 0.0452 0.0055 8.183 1.90E-11 0.0342 0.0563

#### Solution Preview

Solution 1

a. Let y denote the income tax withheld and x denote the weekly pay. Then the fitted regression equation is

y = 30.7963 + 0.0343 x

b. The degrees of freedom(d.f) for a two- tailed test for zero slope is

=(n-p) =(35-2)=33, where n is the no of observations and p is the no of parameters estimated in the regression model. Assuming the level of significance á = .05, from Student's t table with (n-p) =33 d.f we get the critical value as 2.034. [This value can be obtained from MS excel by entering the formula =TINV(0.05,33)]

c. Here the P-value for a two- tailed test for zero slope is 0.0068. Since the P-value < á = .05, we may conclude that the slope is significant in the regression model. [Also since the value of the t-statistic = 2.889 >2.034, the critical value, we may reject the null hypothesis H0: the slop of the regression model is zero]

d. Here the 95% ...

#### Solution Summary

The explanation of the output of a regression analysis is discussed in the solution.