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 correlation between 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
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.