Question

Question: The complete model$\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}{\mathbf{x}}_3\mathbf{+}{\mathit{\beta }}_{\mathbf{4}}{\mathbf{x}}_4\mathbf{+}\mathit{\epsilon }$was fit to n = 20 data points, with SSE = 152.66. The reduced model,$\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}\mathit{\epsilon }$, was also fit, with

SSE = 160.44.

a. How many β parameters are in the complete model? The reduced model?

b. Specify the null and alternative hypotheses you would use to investigate whether the complete model contributes more information for the prediction of y than the reduced model.

c. Conduct the hypothesis test of part b. Use α = .05.

Short answer

Answer

a. The no. of β parameters in complete model are 5 and the no of β parameters in reduced model are 3.

b. The null and alternate hypothesis to test whether the complete model contributes more information for the prediction of y than the reduced model can be written as H₀: β₃ = β₄ = β₅ = 0 while H_a: At least one of β parameters are nonzero

c. At 95% confidence interval there is enough evidence to not reject H₀

Step-by-step solution

No of β parameters

The no. of β parameters in complete model are 4 and the no of β parameters in reduced model are 3.

 Step 2: Hypotheses

The null and alternate hypothesis to test whether the complete model contributes more information for the prediction of y than the reduced model can be written as

H₀: β₃ = β₄ = 0 while H_a: At least one of β parameters are nonzero.

Thesis testing

H₀: β₃ = β₄ = 0 and H_a: At least one of β parameters are nonzero

$Teststatistic=\frac{\left(\right(SS{E}_R-SS{E}_C)/(k-g\left)\right)}{(SS{E}_C/[n-(k-1)\left]\right)}=0.38222$

For α = .05, F-test statistic = 3.009. H₀ is rejected if F-statistic > F

Here, 0.38222 < 3.009.

Therefore, at 95% confidence interval there is enough evidence to not reject H₀.