Chapter 12: Q144SE (page 807)

Suppose you fit the regression model $E (y) = β_{0} + β_{1} x_{1} + β_{2} x_{2} + β_{3} {x_{2}}^{2} + β_{4} x_{1} x_{2} + β_{5} x_{1} {x_{2}}^{2} 2$ to n = 35 data points and wish to test the null hypothesis $H_{0} : β_{4} = β_{5} = 0$
State the alternative hypothesis.
Explain in detail how to compute the F-statistic needed to test the null hypothesis.
What are the numerator and denominator degrees of freedom associated with the F-statistic in part b?
Give the rejection region for the test if α = .05.

Expert verified

The alternate hypothesis to test the significance of interaction terms would be Ha: At least one of the parameters β₄or β₅is nonzero.
The F-statistic to check the goodness of fit of the model can be computed by F test statistic = $\frac{S S E}{n - (k + 1)}$ .
In part b, the degrees of freedom for numerator is (n-k) while the degree of freedom for denominator is [n-(k+1)].
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

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.

The F-statistic to check the goodness of fit of the model can be computed by F test statistic = $\frac{S S E .}{n - (k + 1)}$

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

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.}

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