Question

Suppose you fit the quadratic model $\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}\mathbf{x}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}^{\mathbf{2}}$to a set of n = 20 data points and found ${\mathbf{R}}^{\mathbf{2}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{91}$, ${\mathbf{SS}}_{\mathbf{yy}}\mathbf{=}\mathbf{29}\mathbf{.}\mathbf{94}$, and SSE = 2.63.

a. Is there sufficient evidence to indicate that the model contributes information for predicting y? Test using a = .05.

b. What null and alternative hypotheses would you test to determine whether upward curvature exists?

c. What null and alternative hypotheses would you test to determine whether downward curvature exists?

Short answer

1. At 95% significance level, it can be concluded that${\mathbf{\beta }}_{\mathbf{1}}\ne {\mathbf{\beta }}_{\mathbf{2}}\ne \mathbf{0}$
2. ${\mathbf{H}}_{\mathbf{0}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0}$while${\mathbf{H}}_{\mathbf{a}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{>}\mathbf{0}$
3. ${\mathbf{H}}_{\mathbf{0}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0}$while${\mathbf{H}}_{\mathbf{a}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{<}\mathbf{0}$

Step-by-step solution

Goodness of fit

${\mathbf{H}}_{\mathbf{0}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{1}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0}$

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

Here, F test statistic$\mathbf{=}\frac{\mathbf{SSE}}{\mathbf{n}\mathbf{-}\left(\mathbf{K}\mathbf{+}\mathbf{1}\right)}\mathbf{=}\frac{\mathbf{2}\mathbf{.}\mathbf{63}}{\mathbf{17}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{1547}$

${\mathbf{H}}_{\mathbf{0}}$is rejected if F – statistics < ${\mathbf{F}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{20}\mathbf{,}\mathbf{20}}$. For $\mathbf{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$, since F – statistic <${\mathbf{F}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{20}\mathbf{,}\mathbf{20}}$

Sufficient evidence to reject ${\mathbf{H}}_{\mathbf{0}}$at 95% confidence interval.

Therefore, ${\mathbf{\beta }}_{\mathbf{1}}\ne {\mathbf{\beta }}_{\mathbf{2}}\ne \mathbf{0}$.

Significance of β2

To check the curvature of the hyperbola, the null hypothesis is whether the model parameter${\mathbf{\beta }}_{\mathbf{2}}$is explaining the model where the beta value is zero and the alternate hypothesis is whether the beta value is greater than zero to check whether the hyperbola slopes upwards.

Mathematically,

$$\begin{gathered} {\mathbf{H}}_{\mathbf{0}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0}\mathbf{ } \\ {\mathbf{H}}_{\mathbf{a}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{>}\mathbf{0} \end{gathered}$$

consequence of β2

To check the curvature of the hyperbola, the null hypothesis is whether the model parameter${\mathbf{\beta }}_{\mathbf{2}}$is explaining the model where the beta value is zero and the alternate hypothesis is whether the beta value is less than zero to check if the hyperbola slopes downwards.

Mathematically,

$$\begin{gathered} {\mathbf{H}}_{\mathbf{0}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0} \\ {\mathbf{H}}_{\mathbf{a}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{<}\mathbf{0} \end{gathered}$$