Question:Suppose you fit the first-order model $\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{3}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{4}}{\mathbf{x}}_{\mathbf{4}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{5}}{\mathbf{x}}_{\mathbf{5}}\mathbf{+}\mathbf{\epsilon }$to n=30 data points and obtain SSE = 0.33 and ${\mathbf{R}}^{\mathbf{2}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{92}$

(A) Do the values of SSE and ${\mathbf{R}}^{\mathbf{2}}$suggest that the model provides a good fit to the data? Explain.

(B) Is the model of any use in predicting Y ? Test the null hypothesis ${\mathbf{H}}_{\mathbf{0}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{1}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{3}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{4}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{5}}\mathbf{=}\mathbf{0}$ against the alternative hypothesis $\left\{H\right\}$at least one of the parameters ${\mathbf{\beta }}_{\mathbf{1}},{\mathbf{\beta }}_{\mathbf{2}},...,{\mathbf{\beta }}_5$ is non zero.Use$\mathbf{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$ .

A low sum of squares of error values like 0.33 indicate that the there is low variability in the set of observations. It means that the data points are close to the model fitted. Also, high value like 0.92 for${\mathbf{R}}^{\mathbf{2}}$indicates that around 92% of the variation in the data is explained by the model which is a good thing.

Hence, we can say that SSE = 0.33 andis a good measure of good fit to the data.

${\mathbf{H}}_0\mathbf{:}{\mathbf{\beta }}_{\mathbf{1}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{3}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{4}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{5}}\mathbf{=}\mathbf{0}$

${\text{H}}_{\text{a}}\text{:}$At least one of the parameters ${\mathbf{\beta }}_{\mathbf{1}},{\mathbf{\beta }}_{\mathbf{2}},...,{\mathbf{\beta }}_5$is non zero

Here, F test statistic $\mathbf{=}\frac{\mathbf{SSE}}{\mathbf{n}\mathbf{-}\mathbf{(}\mathbf{k}\mathbf{+}\mathbf{1}\mathbf{)}}\mathbf{=}\frac{\mathbf{0}\mathbf{.}\mathbf{33}}{\mathbf{30}\mathbf{-}\mathbf{6}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{01375}$

Value of${\mathbf{F}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{24}\mathbf{,}\mathbf{24}}$is 1.984

${\text{H}}_0$is rejected if F statistic > ${\mathbf{F}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{24}\mathbf{,}\mathbf{24}}$For $\mathbf{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$, since F <${\mathbf{F}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{24}\mathbf{,}\mathbf{24}}$

Do not have sufficient evidence to reject ${\text{H}}_0$ at 95% confidence interval.

Therefore,${\mathbf{\beta }}_{\mathbf{1}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{3}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{4}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{5}}\mathbf{=}\mathbf{0}$