Question

Suppose you fit the second-order model $\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}\mathbf{x}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{\epsilon }$to n = 25 data points. Your estimate of${\mathbf{\beta }}_{\mathbf{2}}$is${\hat{\mathbf{\beta }}}_{\mathbf{2}}$= 0.47, and the estimated standard error of the estimate is 0.15.

1. Test${\mathbf{H}}_{\mathbf{0}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0}$against${\mathbf{H}}_{\mathbf{a}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{2}}\ne \mathbf{0}$. Use$\mathbf{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$.
2. Suppose you want to determine only whether the quadratic curve opens upward; that is, as x increases, the slope of the curve increases. Give the test statistic and the rejection region for the test for$\mathbf{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$. Do the data support the theory that the slope of the curve increases as x increases? Explain.

Short answer

1. At 95% significance level, it can be concluded that ${\mathbf{\beta }}_{\mathbf{2}}\ne \mathbf{0}$.
2. At 95% significance level, ${\mathbf{\beta }}_{\mathbf{2}}\mathbf{>}\mathbf{0}$. There is sufficient evidence to conclude that the slope of the curve increases as x increases.

Step-by-step solution

Significance of β2

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

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

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$\mathbf{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$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,${\mathbf{\beta }}_{\mathbf{2}}\ne \mathbf{0}$

Signification of β2

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

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

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 $\mathbf{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$, 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" ${\mathbf{\beta }}_{\mathbf{2}}\mathbf{>}\mathbf{0}$. There is sufficient evidence to conclude that the slope of the curve increases as x increases.