Chapter 10: Q. 88 (page 602)

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

Short Answer

Expert verified

The correct answer is option(c).

There is insufficient evidence to conclude that the $W$ teams score fewer goals, on average, than the $E$ 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 :
$S E = \sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}$

$= \sqrt{1.62778}$

$= 1.2758$

Determine the test statistic as follows:

$t = \frac{μ_{1} - μ_{2}}{S E}$

$= - 1.43675$

Determine the number of degrees of freedom as follows:

$d f = \frac{S E^{4}}{\frac{s_{1}^{4}}{n_{1}^{2} (n_{1} - 1)} + \frac{s_{2}^{2}}{n_{2}^{2} (n_{2} - 1)}}$

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

$= 9.979$

Explanation

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

As a result, $α = 0.05$ and $p > α$ , 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).