Question

Suppose you fit the regression model $E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}^2+{\beta }_4{x}_1{x}_2+{\beta }_5{x}_1{x_2}^2$2 to n = 35 data points and wish to test the null hypothesis$H_0:{\beta }_4={\beta }_5=0$

1. State the alternative hypothesis.
2. Explain in detail how to compute the F-statistic needed to test the null hypothesis.
3. What are the numerator and denominator degrees of freedom associated with the F-statistic in part b?
4. Give the rejection region for the test if α = .05.

Short answer

1. The alternate hypothesis to test the significance of interaction terms would be H_a: At least one of the parameters β₄ or β₅ is nonzero.
2. The F-statistic to check the goodness of fit of the model can be computed byF test statistic =.$\frac{\mathrm{SSE}}{\mathrm{n}-\left(\mathrm{k}+1\right)}$.
3. In part b, the degrees of freedom for numerator is (n-k) while the degree of freedom for denominator is [n-(k+1)].
4. When α = 0.05, the rejection region for the significance of interaction terms can be defined when the t-statistic < t_{0.025, n-1}.

Step-by-step solution

Alternate hypothesis

The alternate hypothesis to test the significance of interaction terms would be H_a: At least one of the parameters β₄ or β₅ is nonzero.

F-statistic

The F-statistic to check the goodness of fit of the model can be computed by

F test statistic =$\frac{\mathrm{SSE}}{\mathrm{n}-\left(\mathrm{k}+1\right)}$ .

Degrees of freedom

In part b, the degrees of freedom for numerator is (n-k) while the degree of freedom for denominator is [n-(k+1)].

Rejection region

When α = 0.05, the rejection region for the significance of interaction terms can be defined when the t-statistic < t_{0.025, n-1}.