Question

Question: Suppose you fit the regression model$\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{4}}{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{5}}{{\mathbf{x}}_{\mathbf{2}}}^{\mathbf{2}}$to n = 30 data points and wish to test H₀: β₃ = β₄ = β₅ = 0

a. State the alternative hypothesis H_a.

b. Give the reduced model appropriate for conducting the test.

c. What are the numerator and denominator degrees of freedom associated with the F-statistic?

d. Suppose the SSE’s for the reduced and complete models are SSE_R = 1,250.2 and SSE_C = 1,125.2. Conduct the hypothesis test and interpret the results of your test. Test using α = .05.

Short answer

Answer

a. The alternate hypothesis would be at least one of the β parameters under test is nonzero.

b. The reduced model under consideration here is $\mathit{y}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{\mathit{x}}_{\mathbf{2}}$.

c. In the numerator there’s (k-g) and in the denominator there are [n-(k+1)] degrees of freedom where (k – g) = Number of b parameters specified in H₀ (i.e., number of β parameters tested), k + 1 = Number of β parameters in the complete model (including β₀), and n = Total sample size.

d. At 95% confidence interval there is enough evidence to not reject H₀. Hence, at least one of β parameters are nonzero.

Step-by-step solution

Alternate hypothesis

The alternate hypothesis would be at least one of the β parameters under test is nonzero.

Reduced model

The reduced model under consideration here is $\mathit{y}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{\mathit{x}}_{\mathbf{2}}$.

Degrees of freedom

In the numerator there’s (k-g) and in the denominator there are [n-(k+1)] degrees of freedom