Question

A grassroots group opposed to a proposed increase in the gas tax claimed that the increase would hurt working-class people the most, since they commute the farthest to work. Suppose that the group randomly surveyed $24$individuals and asked them their daily one-way commuting mileage. The results are as follows.

working-class	professional (middle incomes)	professional (wealthy)
17.8	$16.5$	$8.5$
$26.7$	$17.4$	$6.3$
$49.4$	$22.0$	$4.6$
$9.4$	$7.4$	$12.6$
$65.4$	$9.4$	$11.0$
$47.1$	$2.1$	$28.6$
19.5	15.4	15.4
51.2	13.9	9.3

Determine whether or not the variance in mileage driven is statistically the same among the working class and professional (middle income) groups. Use a 5% significance level.

Short answer

There is sufficient evidence that variance in mileage driven is statistically different among the working classes and middle income professionals.

Step-by-step solution

Given information

Given in the question that the table,

working-class	professional (middle incomes)	professional (wealthy)
$17.8$	$16.5$	$8.5$
$26.7$	$17.4$	$6.3$
$49.4$	$22.0$	$4.6$
$9.4$	$7.4$	$12.6$
$65.4$	$9.4$	$11.0$
47.1	$2.1$	$28.6$
$19.5$	$6.4$	$15.4$
$51.2$	$13.9$	$9.3$

We have to determine whether or not the variance in mileage driven is statistically the same among the working class and professional (middle income) groups.

Explanation

Consider the data supplied in the table $13.32$to decide whether or not the variance in mileage driven is statistically the same among the working class and middle-income professionals.

Consider the null and alternative hypotheses:

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

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

To test the above hypothesis, use Ti-83 calculator,

click on STAT $\to$ENTER, and then put the data into the list $L1$, and $L2$. The screenshot is given as below:

Now again press STAT $\to$arrow over the TESTS $\to$arrow down to 2 sample F-test $\to$Select data and press then select1 and press var then select 2 in list 1 and list 2 .

Select the hypothesis as ${\sigma }_1\ne {\sigma }_2$. The screen shot is given as below:

Screenshot of output

Press calculate to get the output. The screenshot of the obtained output is given as below:

Use the above results to solve the below given subparts.

a.

The null hypothesis is given as below:

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

b.

The alternate hypothesis is given as below:

$H_1:{\sigma }_1^2\ne {\sigma }_2^2$

Subparts

C.

The degree of freedom in the numerator is $7$, and the degree of freedom in the denominator is $7.$

The $F_{(7,7)}$distribution is used to test the two variances.

e.

The value of the test statistic is $\mathbf{9}\mathbf{.}\mathbf{011}$.

f.

The $P$-value for the test is $\mathbf{0}\mathbf{.}\mathbf{0096}$.

g.

The graph of the distribution can be plotted by using Ti-83 calculator by following the same steps as given above and finally pressing 'Draw' instead of 'Calculate'. It is given as below:

h.

a.

Level of significance $\alpha$is $0.05.$

b. Decision: Null hypothesis will be rejected.

C.

Reason for decision: P-value is$0.0096$which is less than the $0.05$level of significance.

There is sufficient evidence that variance in mileage driven is statistically different among the working classes and middle income professionals.