Answer true or false to each of the following statements and explain your answers. a. If the F-test for the utility of a multiple linear regression equation rejects the null hypothesis, then each of the predictor variables in the regression equation is useful in predicting the response variable. b. The t-test for the utility of a particular predictor variable in a multiple linear regression equation can be affected by the other predictor variables in the equation.

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Answer true or false to each of the following statements and explain your answers.

a. If the F-test for the utility of a multiple linear regression equation rejects the null hypothesis, then each of the predictor variables in the regression equation is useful in predicting the response variable.

b. The t-test for the utility of a particular predictor variable in a multiple linear regression equation can be affected by the other predictor variables in the equation.

Expert Solution
Step 1

a.

Hypotheses for testing the utility of multiple regression equation:

 

The null and alternative hypotheses to test the significance of a regression equation is,

 

H0:β1=β2=...=βk=0.

 

H1: at least one of the βi's is non-zero (i=1,2,...,k).

 

If the F-test for the utility of a multiple linear regression rejects the null hypothesis, then automatically, the alternative hypothesis is accepted. Now, the alternative hypothesis can be accepted when at least one of the βi's is non-zero. This condition does not require for all βi's to be non-zero. Having only one non-zero βi is sufficient to reject the null hypothesis. This means that the null hypothesis is rejected, if there is only one useful predictor variable in the regression equation.

 

Thus, the statement “If the F-test for the utility of a multiple linear regression equation rejects the null hypothesis, then each of the predictor variables in the regression equation is useful in predicting the response variable.” is False.

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