Question

Identify the rejection region for each of the following cases. Assume ${\mathbf{\text{v}}}_{\mathbf{\text{1}}}{\mathbf{\text{= 7 andv}}}_{\mathbf{\text{2}}}\mathbf{\text{= 9}}$

$$\begin{gathered} \mathit{a}\mathbf{.}\mathbf{}{\mathbf{\text{H}}}_{\mathbf{\text{a}}}{\mathbf{\text{:σ}}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{\text{<}}{\mathit{\sigma }}_{\mathbf{2}}^{\mathbf{2}}\mathbf{,}\mathbf{}\mathit{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05} \\ \mathit{b}\mathbf{.}\mathbf{}{\mathbf{\text{H}}}_{\mathbf{\text{a}}}{\mathbf{\text{:σ}}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{\text{>}}{\mathit{\sigma }}_{\mathbf{2}}^{\mathbf{2}}\mathbf{,}\mathbf{}\mathit{\alpha }\mathbf{=}\mathbf{.}\mathbf{01} \\ \mathit{c}\mathbf{.}\mathbf{}{\mathbf{\text{H}}}_{\mathbf{\text{a}}}{\mathbf{\text{:σ}}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{\text{≠}}{\mathit{\sigma }}_{\mathbf{2}}^{\mathbf{2}}\mathbf{,}\mathbf{}\mathit{\alpha }\mathbf{=}\mathbf{.}\mathbf{1}\mathbf{}\mathit{w}\mathit{i}\mathit{t}\mathit{h}\mathbf{}\mathbf{}\mathbf{}{{\mathbf{\text{s}}}_{\mathbf{\text{1}}}}^{\mathbf{\text{2}}}\mathbf{\text{>}}{{\mathbf{\text{s}}}_{\mathbf{\text{2}}}}^{\mathbf{\text{2}}} \\ \mathit{d}\mathbf{.}\mathbf{}{\mathbf{\text{H}}}_{\mathbf{\text{a}}}{\mathbf{\text{:σ}}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{\text{<}}{\mathit{\sigma }}_{\mathbf{2}}^{\mathbf{2}}\mathbf{,}\mathbf{}\mathit{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{025} \end{gathered}$$

Short answer

A hypothesis is a tested assertion concerning the relation among two or more factors or a suggested reason for an observable phenomenon.

Step-by-step solution

(a) Find the rejected region 

The numerator degrees of freedom is$v_1=7$

The denominator degrees of freedom is$v_2=9$

The alternative hypothesis is$H_a:{{\sigma }_1}^2<{{\sigma }_2}^2$

The level of significance is$\alpha =0.05$

As the alternative hypothesis is given to us, so we will set up a null hypothesis, that is$H_0:{{\sigma }_1}^2={{\sigma }_2}^2$

Using percentage points of the F-distribution$\alpha =0.05$, the critical value is 3.293.

Therefore, the rejection region is $F>3.293$.

(b) Find the rejected region 

The numerator degrees of freedom is V₁ =7

The denominator degrees of freedom is V₂ = 9

The alternative hypothesis is$H_a:{{\sigma }_1}^2>{{\sigma }_2}^2$

The level of significance is$\alpha =0.01$

As the alternative hypothesis is given to us, so we will set up a null hypothesis, that is$H_0:{{\sigma }_1}^2={{\sigma }_2}^2$

Using percentage points of the F-distribution$\alpha =0.01$, the critical value is

Therefore, the rejection region is $F>5.613$.

(c) Find the rejected region

The numerator degrees of freedom is V₁ =7

The denominator degrees of freedom is V₂ = 9

The alternative hypothesis is$H_a:{{\sigma }_1}^2\ne {{\sigma }_2}^2$

The level of significance is$\alpha =0.1$

As the alternative hypothesis is given to us, so we will set up a null hypothesis, that is$H_0:{{\sigma }_1}^2={{\sigma }_2}^2$

Using percentage points of the F-distribution$\alpha =0.1$, the critical value is

Therefore, the rejection region is 2,505.

(d) Find the rejected region

The numerator degrees of freedom is V₁ =7

The denominator degrees of freedom is V₂ = 9

The alternative hypothesis is$H_a:{{\sigma }_1}^2<{{\sigma }_2}^2$

The level of significance is$\alpha =0.025$

As the alternative hypothesis is given to us, so we will set up a null hypothesis, that is $H_0:{{\sigma }_1}^2={{\sigma }_2}^2$

Using percentage points of the F-distribution $\alpha =0.025$, the critical value is 4.197

Therefore, the rejection region is $F>4.197$.