Question

88. If the level of significance is $0.05$, the conclusion is:
a. There is sufficient evidence to conclude that the $\mathbf{W}$Division teams score fewer goals, on average, than the $\mathbf{E}$ teams
b. There is insufficient evidence to conclude that the $\mathbf{W}$ Division teams score more goals, on average, than the $\mathbf{E}$ teams.
c. There is insufficient evidence to conclude that the $\mathbf{W}$ teams score fewer goals, on average, than the $\mathbf{E}$ teams score.
d. Unable to determine

Short answer

The correct answer is option(c).

There is insufficient evidence to conclude that the $\mathbf{W}$teams score fewer goals, on average, than the $\mathbf{E}$ teams score. .

Step-by-step solution

Given information

To determine the the conclusion, if the level of significance is $0.05$.

Explanation

The data is provided as follows:

	n	Mean	Standard deviation
Western	6	4.5	2.258318
Eastern	6	6.3333	2.160247

To determine the standard error by using student's $t$ test for testing the hypothesis :
$SE=\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}$

$=\sqrt{1.62778}$

$=1.2758$

Determine the test statistic as follows:

$t=\frac{{\mu }_1-{\mu }_2}{SE}$

$=-1.43675$

Determine the number of degrees of freedom as follows:

$df=\frac{S{E}^4}{\frac{s_1^4}{n_1^2\left(n_1-1\right)}+\frac{s_2^2}{n_2^2\left(n_2-1\right)}}$

$=\frac{S{E}^4}{0.265488}$

$=9.979$

Explanation

The $p$-value is determined by using the graph as follows:

As a result, $\alpha =0.05$and $p>\alpha$, so, cannot reject hypothesis $H_0$. Hence, there is insufficient evidence to conclude that the $W$teams score fewer goals, on average, than the $E$teams score. So, the correct answer is option(c).