Question

Identify the rejection region for each of the following cases. Assume

${\mathbf{\text{v}}}_{\mathbf{\text{1}}}{\mathbf{\text{=7andv}}}_{\mathbf{\text{2}}}\mathbf{\text{=9}}$

a. ${\mathbf{\text{H}}}_{\mathbf{\text{a}}}{{\mathbf{\text{:σ}}}_{\mathbf{\text{1}}}}^{\mathbf{\text{2}}}{{\mathbf{\text{<σ}}}_{\mathbf{\text{2}}}}^{\mathbf{\text{2}}}$,$\mathbf{\text{α=.05}}$

b. ${\mathbf{\text{H}}}_{\mathbf{\text{a}}}{{\mathbf{\text{:σ}}}_{\mathbf{\text{1}}}}^{\mathbf{\text{2}}}{{\mathbf{\text{>σ}}}_{\mathbf{\text{2}}}}^{\mathbf{\text{2}}}$,$\mathbf{\text{α=.01}}$

c. ${\mathbf{\text{H}}}_{\mathbf{\text{a}}}{{\mathbf{\text{:σ}}}_{\mathbf{\text{1}}}}^{\mathbf{\text{2}}}\mathbf{\ne }{{\mathbf{\text{σ}}}_{\mathbf{\text{2}}}}^{\mathbf{\text{2}}}$,$\mathbf{\text{α=.1}}$with${{\mathbf{\text{s}}}_{\mathbf{\text{1}}}}^{\mathbf{\text{2}}}{{\mathbf{\text{>s}}}_{\mathbf{\text{2}}}}^{\mathbf{\text{2}}}$

d. ${\mathbf{\text{H}}}_{\mathbf{\text{a}}}{{\mathbf{\text{:σ}}}_{\mathbf{\text{1}}}}^{\mathbf{\text{2}}}{{\mathbf{\text{<σ}}}_{\mathbf{\text{2}}}}^{\mathbf{\text{2}}}$,$\mathbf{\text{α=.025}}$

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

Step-by-Step Solution Step 1: (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$v_1=7$

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$\alpha =0.01$

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$5.613$

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

(c) 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$H_a:{{\sigma }_1}^2\ne {{\sigma }_2}^2$

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$2.505$

Therefore, the rejection region is $F>2.505$.

(d) 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$\alpha =0.025$

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 islocalid="1652720854474" $4.197$

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