Chapter 11.5, Problem 25E

To determine

a.

To State:

Null hypotheses and alternative hypotheses.

Answer to Problem 25E

Solution:

Null Hypotheses –

$H_0:{\sigma }_1^2={\sigma }_2^2$

Alternative Hypotheses –

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

Explanation of Solution

Given:

One study claims that the variance in the resting heart rates of smokers is different than the variance in the resting heart rates of nonsmokers.

A medical student decides to test this claim.

The sample variance of resting heart rates, measured in beats per minute, for a random sample of 5 smokers is 545.1.

The sample variance for a random sample of 5 nonsmokers is 103.7.

Test the study’s claim using a 0.01 level of significance.

Calculation:

Let population variances in the resting heart rates of smokers be represented by ${\sigma }_1^2$ and population variances in the resting heart rates of non-smokers be represented by ${\sigma }_2^2$. One study claims that the variance in the resting heart rates of smokers is different than the variance in the resting heart rates of nonsmokers that is ${\sigma }_1^2\ne {\sigma }_2^2$.

The mathematical opposite of this claim is ${\text{σ}}_{\text{1}}^{\text{2}}{\text{ = σ}}_{\text{2}}^{\text{2}}$.

The null hypothesis and alternative hypothesis are stated as follows-

$H_0:{\sigma }_1^2={\sigma }_2^2$

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

To determine

b.

The type of distribution to use for the test statistics and state the level of significance.

Answer to Problem 25E

Solution:

The F- test statistic is appropriate and the level of significance for this test is $\alpha =0.01$.

Explanation of Solution

Given:

One study claims that the variance in the resting heart rates of smokers is different than the variance in the resting heart rates of nonsmokers.

A medical student decides to test this claim.

The sample variance of resting heart rates, measured in beats per minute, for a random sample of 5 smokers is 545.1.

The sample variance for a random sample of 5 nonsmokers is 103.7.

Test the study’s claim using a 0.01 level of significance.

Calculation:

For comparing the variances of two normally distributed populations using independent, simple random samples, F- test statistic is used.

The level of significance for this test is $\alpha =0.01$.

Therefore, the F-test statistic is appropriate and the level of significance for this test is $\alpha =0.01$

To determine

c.

To Calculate:

The test statistic.

Answer to Problem 25E

Solution:

The test statistic 5.2565.

Explanation of Solution

Given:

One study claims that the variance in the resting heart rates of smokers is different than the variance in the resting heart rates of nonsmokers.

A medical student decides to test this claim.

The sample variance of resting heart rates, measured in beats per minute, for a random sample of 5 smokers is 545.1.

The sample variance for a random sample of 5 nonsmokers is 103.7.

Test the study’s claim using a 0.01 level of significance.

Formula used:

When the samples are given to be independent, the given population distribution are approximately normal, then the test statistics for the hypothesis test for two population variances is given by,

$F=\frac{s_1^2}{s_2^2}$

Where $s_1^2$ and $s_2^2$ are the sample variances.

$n_1$ is the total number of data of population 1.

$n_2$ is the total number of data of population 2.

The degree of freedom for the numerator is $d{f}_1=n_1-1$

The degree of freedom for the denominator is $d{f}_2=n_2-1$

Given $n_1=5,s_1^2=545.1,n_2=5,s_2^2=103.7,\alpha =0.01$

Null Hypotheses –

$H_0:{\sigma }_1^2={\sigma }_2^2$

Alternative Hypotheses –

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

The test statistic value is given by,

$F=\frac{s_1^2}{s_2^2}$

$\begin{array}{rll}F& =& \frac{545.1}{103.7}\\ & =& 5.2565\end{array}$

Therefore, the test statistic is 5.2565.

To determine

d.

To Draw:

The conclusion and interpret the decision.

Answer to Problem 25E

Solution:

The null hypothesis is accepted and it is concluded that there is no sufficient evidence at the 0.01 level of significance to support the claim that variances in the resting heart rates of smokers is different than the variance in the resting heart rates of nonsmokers.

Explanation of Solution

Given:

One study claims that the variance in the resting heart rates of smokers is different than the variance in the resting heart rates of nonsmokers.

A medical student decides to test this claim.

The sample variance of resting heart rates, measured in beats per minute, for a random sample of 5 smokers is 545.1.

The sample variance for a random sample of 5 nonsmokers is 103.7.

Test the study’s claim using a 0.01 level of significance.

Approach:

The null hypothesis is rejected if,

$F\le F_{\left(1-\alpha \right)}$ for the left tailed test.

$F\ge F_{\alpha }$ for the right tailed test.

$F\le F_{\left(1-\frac{\alpha }{2}\right)}$ or $F\ge F_{\alpha /2}$ for the two tail test.

Calculation:

The level of significance, $\alpha =0.01$.

The degree of freedom for numerator is,

$d{f}_1=n_1-1$

Substitute 4 for $n_1$ in the above equation.

$\begin{array}{rll}d{f}_1& =& n_1-1\\ & =& 5-1\\ & =& 4\end{array}$

The degree of freedom for denominator is,

$d{f}_2=n_2-1$

Substitute 4 for $n_2$ in the above equation.

$\begin{array}{rll}d{f}_2& =& n_2-1\\ & =& 5-1\\ & =& 4\end{array}$

From the $F$-table, the critical values for the required degrees is,

$\begin{array}{rll}{F}_{\alpha /2}& =& F_{0.005}\\ & =& 23.1545\end{array}$

Another critical value for the same degrees of freedom is,

$\begin{array}{rll}{F}_{1-\frac{\alpha }{2}}& =& F_{0.995}\\ & =& 0.0432\end{array}$

By comparing the test statistic value and the critical value, the F value is less than the critical value, so by the two tailed test the null hypothesis is rejected.

That is,

$\text{Reject}{\text{H}}_{\text{0}}\text{if}\text{F}\le \text{0}\text{.0432}\text{or}\text{F}\ge \text{23}\text{.1545}$

The critical value is, $\text{p-value}\approx 0.1369$

Conclusion:

Thus, there is no sufficient evidence at 0.01 level of significance to support the claim that the variances in the resting heart rates of smokers is different than the variance in the resting heart rates of nonsmokers