91Ó°ÊÓ

$$\begin{gathered} {\mathbf{H}}_{\mathbf{0}}\mathbf{:}{\mathbf{β}}_{\mathbf{2}}\mathbf{=}\mathbf{0} \\ {\mathbf{H}}_{\mathbf{a}}\mathbf{:}{\mathbf{β}}_{\mathbf{2}}≠\mathbf{0} \end{gathered}$$

Here, t-test statistic$\mathbf{=}\frac{{\widehat{\mathbf{β}}}_{\mathbf{2}}}{\mathbf{s}{\widehat{\mathbf{β}}}_{\mathbf{2}}}\mathbf{=}\frac{\mathbf{0}\mathbf{.}\mathbf{47}}{\mathbf{0}\mathbf{.}\mathbf{15}}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{133}$

Value of ${\mathbf{t}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{24}}$is 1.711.

${\mathbf{H}}_{\mathbf{0}}$is rejected iftstatistics >${\mathbf{t}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{24}}$. For$\mathbf{Î\pm }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$since t >${\mathbf{t}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{24}}$

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

Therefore,${\mathbf{β}}_{\mathbf{2}}≠\mathbf{0}$

To check whether the quadratic curve opens upwards, that is as x increases, the slope of the curve also increases, the hypotheses would be

${\mathbf{H}}_{\mathbf{0}}\mathbf{:}{\mathbf{β}}_{\mathbf{2}}\mathbf{=}\mathbf{0}$against${\mathbf{H}}_{\mathbf{a}}\mathbf{:}{\mathbf{β}}_{\mathbf{2}}\mathbf{>}\mathbf{0}$

The t-test statistic would be$\frac{{\hat{\mathbf{b}}}_{\mathbf{2}}}{\mathbf{s}{\hat{\mathbf{b}}}_{\mathbf{2}}}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{133}$

${\mathbf{H}}_{\mathbf{0}}$is rejected when t statistic > ${\mathbf{t}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{24}}$. For $\mathbf{Î\pm }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$, the value of ${\mathbf{t}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{24}}$is 1.711

Since t > ${\mathbf{t}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{24}}$, there is sufficient evidence to reject ${\mathbf{H}}_{\mathbf{0}}$at a 95% significance level.

Thus,localid="1649833131364" ${\mathbf{β}}_{\mathbf{2}}\mathbf{>}\mathbf{0}$. There is sufficient evidence to conclude that the slope of the curve increases as x increases.