Are the variances for incomes on the East Coast and the West Coast the same? Suppose that Table $13.34$shows the results of a study. Income is shown in thousands of dollars. Assume that both distributions are normal. Use a level of significance of $0.05$.

East	West
$38$	$71$
$47$	$126$
$30$	$42$
$82$	$51$
$75$	$44$
$52$	$90$
$115$	$88$
$67$

Given in the question that, Suppose that Table 13.34 shows the results of a study. Income is shown in thousands of dollars. Assume that both distributions are normal. Use a level of significance of $0.05.$

To find these results on the calculator:

Press STAT. Press 1 EDIT. Put the data into the lists $L_1$and $L_2$

Press STAT, and arrow over to TESTS, and arrow down to $2$Samp FTest. Select Data and press then selectl and press var then select$2$in list 1 and list 2. Select the hypothesis as${\sigma }_1\ne {\sigma }_2$.

The calculator displays :

$F=0.811733$

$p=0.78246$

$S_{x1}=27.5564$

$S_{x2}=30.5855$

${\overline{x}}_1=63.25$

${\overline{x}}_2=73.143$

$n_1=8$

$n_2=7$

We use a solution sheet to conduct the hypothesis tests, and we have:

a) The null hypothesis that three mean commuting mileages are the same is:

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

b) The alternate hypothesis is that at least any two of the means are different:

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

c) The degree of freedom in the numerator - $df($num $)$is $7$,

and the degree of freedom in the denominator - $df($denom) is $6$

d) We use the $F_{(7.6)}$distribution for the test.

e) The value of the test statistic (F-value) is$8.117$

f) The $P$-value for the test is $0.7825$

g) The graph of the distribution is

h) i. Level of significance $\alpha$is $0.05$

ii. Decision: We do not reject the null hypothesis

iii. Reason for decision: $P$-value is $0.7825$which is greater than the $0.05$level of significance.

iv. Conclusion: There is not sufficient evidence to conclude that the variances are different.