Chapter 13.2, Problem 6E

a.

To determine

To identify: The claim and state $H_0$ and $H_1$.

Answer to Problem 6E

The claim is “the median number for the paid attendance of football games is 3,000”.

Null hypothesis: $H_0:\text{Median}=3,000$

Alternative hypothesis: $H_1:\text{Median}\ne 3,000$

Explanation of Solution

Given info:

The data shows the number of paid attendance for 20 randomly selected football games.

Justification:

Here, the claim is that median number for the paid attendance of football games is 3,000.

This can be written as $\text{Median}=3,000$. The complement of the claim is $\text{Median}\ne 3,000$. In the experiment, the null hypothesis indicates the claim.

b.

To determine

To find: The critical value.

Answer to Problem 6E

The critical value is 5.

Explanation of Solution

Determine sample size n:

Subtract the hypothesized median of 3,000 from the data.

Data	$\text{Difference}=\text{Data}-3,000$	Sign	Data	$\text{Difference}=\text{Data}-3,000$	Sign
6210	$6,210–3,000=3,210$	+	2792	$2,792–3,000=–208$	–
3150	$3,150–3,000=150$	+	2800	$2,800–3,000=–200$	–
2700	$2,700–3,000=–300$	–	2500	$2,500–3,000=–500$	–
3012	$3,012–3,000=12$	+	3700	$3,700–3,000=700$	+
4875	$4,875–3,000=1,875$	+	6030	$6,030–3,000=3,030$	+
3540	$3,540–3,000=540$	+	5437	$5,437–3,000=2,437$	+
6127	$6,127–3,000=3,127$	+	2758	$2,758–3,000=–242$	–
2581	$2,581–3,000=–419$	–	3490	$3,490–3,000=490$	+
2642	$2,642–3,000=–358$	–	2851	$2,851–3,000=–149$	–
2573	$2,573–3,000=–427$	–	2720	$2,720–3,000=–280$	–

The sample size is,

$\begin{array}{c}n=\text{sum of }(+)\text{ and }(-)\text{ signs}\\ =10+10\\ =20\end{array}$

Critical value:

From table” critical values for sign test” for $n=20$ at $\alpha =0.05$.

The critical value is 5.

c.

To determine

To find: the test statistic $x$

Answer to Problem 6E

The test statistic is 10.

Explanation of Solution

Condition:

If $n\le 25,$ the test statistic for sign test is $x$.

If $n>25$, the test statistic for sign test is $z=\frac{\left(x+0.5\right)-0.5n}{\frac{\sqrt{n}}{2}}$

Where $x$ is the smaller number of $(+)$& $(-)$ and $n$ is the sample size.

Here, the number of $(+)$ sign in the table is 10 and number of $(-)$ sign in the table is 10.

Thus, the test statistic is 10.

d.

To determine

To decide: Whether to reject or fail to reject the null hypothesis at level of significance $\alpha =0.05$.

Answer to Problem 6E

The decision is “fail to reject the null hypothesis”.

Explanation of Solution

Decision rule:

• If the test statistic is less than or equal to the critical value, then reject the null hypothesis $H_0$.
• If the test statistic is greater than the critical value, then fail to reject the null hypothesis $H_0$.

Conclusion:

The level of significance is, $\alpha =0.05$ and the critical value is 5.

Here, the test statistic value is greater than the critical value.

That is, $\text{Test statistic}\left(=10\right)>\text{criticalvalue}\left(=5\right)$

Thus, the decision is “fail to reject the null hypotheses”.

e.

To determine

To interpret: The decision in the context of the original claim.

Answer to Problem 6E

The conclusion is that there is no enough evidence to reject the claim that the median number of paid attendance for local football game is 3,000.

Explanation of Solution

From the part (d), the null hypothesis is not rejected. Thus, there is no enough evidence to reject the claim that median number of paid attendance for local football game is 3,000 at $5\%$ level of significance.