Use a simple regression model to test the null hypothesis against the alternative

Ho: Beta1 = 0

H1: Beta1 Does NOT = 0

with alpha = 0.05 , given the following regression statistics:

a. The sample size is 35, SST = 100,000, and the correlationbetween X and Y is 0.46.

b. The sample size is 61, SST = 123,000, and the correlation between X and Y is 0.65.

c. The sample size is 25, SST = 128,000, and the correlation between X and Y is 0.69.

Please see the attachment for solution.

Here we need to test the null hypothesis H0: B1 = 0 against the alternative hypothesis .H1: B1 does not equal 0.

The test statistic is

F = MSR/MSE = SSR/s^2

The decision rule is: Reject H0 if F >= F(1,n-2,infinity)

We have the results

R^2 = r^2, R^2 = SSR/SST = 1 – SSE/SST, and s^2 = SSE/(n-2) .

a) Given that,

n =35, SST = 100,000, r = 0.46

Now,

R^2 = (0.46)^2 = 0.2116.

Substituting this result in R^2 = SSR/SST we get

0.2116 = SSR/100,000

That is, …

The solution illustrates the application of correlation coefficient in the test for significance of the regression coefficient.