Chapter 16, Problem 14E

a.

To determine

State the null hypothesis.

Explanation of Solution

In this context, the sample size n is 14.

The test hypotheses are given below:

Null hypothesis:

$H_0:$ The production is same using the old work environment and new work environment.

b.

To determine

Decide the alternative hypothesis.

Explanation of Solution

The answers will vary. One of the possible answers is given below:

Here, it is assumed that the production has been increased in the new work environment.

Alternative hypothesis:

$H_1:$ The production using the new work environment has increased than old work environment.

c.

To determine

Decide the level of significance.

Explanation of Solution

The answers will vary. One of the possible answers is given below:

In general, the level of significance may decide upon the interest of the researcher. Assume that the level of significance is $\alpha =0.05$.

d.

To determine

Provide the decision rule.

Explanation of Solution

The table represents the computations of the test statistic:

Employee	Difference = $\left\{\begin{array}{l}\text{Old}-\\ \text{New}\end{array}\right\}$	Absolute Difference	Rank	Signed Rank
				R+	R-
1	$-10\left(=23-33\right)$	10	13		$-$
2	$0\left(=26-26\right)$	-		-	-
3	$-6\left(=24-30\right)$	6	9		$-$
4	$-8\left(=17-25\right)$	8	11		$-$
5	$1\left(=20-19\right)$	1	1	+
6	$2\left(=24-22\right)$	2	4	+
7	$1\left(=30-29\right)$	1	1	+
8	$-4\left(=21-25\right)$	4	7		$-$
9	$\text{3 }\left(=25-22\right)$	3	6	+
10	$\text{-2 }\left(=21-23\right)$	2	4		$-$
11	$\text{-1 }\left(=16-17\right)$	1	1		$-$
12	$5\left(=20-15\right)$	5	8	+
13	$8\left(=17-9\right)$	8	11	+
14	$-7\left(=23-30\right)$	7	10		$-$

Decision rule:

• If $T\le T_{\alpha }$, reject the null hypothesis.
• Otherwise, fail to reject the null hypothesis.

In this context, the alternative is two-tailed test and the total number of ranks is 13.

From Appendix B, Table B.8 Wilcoxon T values:

Procedure:

1. 1. In the Wilcoxon T values, locate n as 13.
2. 2. Locate the value of 0.05 in α.
3. 3. Locate the intersection value of the row and column. This gives the critical value.

The critical value for 0.05 the significance level is 21.

Decision rule:

• If $T\le 21$, reject the null hypothesis.
• Otherwise, fail to reject the null hypothesis.

e.

To determine

Make the necessary conclusions regarding the null hypothesis.

Answer to Problem 14E

The null hypothesis is not rejected.

Explanation of Solution

The sum of positive ranks is denoted as R⁺, and the sum of negative ranks is denoted as R^-.

Here,

 $\begin{array}{c}{R}^{+}=1+4+1+6+8+11\\ =31\end{array}$

$\begin{array}{c}{R}^{-}=\left\{\begin{array}{l}13+9+11+7+4+\\ 1+10\end{array}\right\}\\ =55\end{array}$

For the Wilcoxon signed-rank test, the test statistic will be obtained as follows:

$T=\text{Min}\left(R^{+},R^{-}\right)$

In this context, the test statistic for sample size 13 is $\text{Min}\left(31,55\right)$. Therefore, the test statistic is 31.

Conclusion:

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

Therefore, by the decision rule, fail to reject the null hypothesis.

Therefore, there is no enough evidence that the production using the new work environment has increased than the old.

f.

To determine

State the assumption that has been assumed about the distribution of difference.

Explanation of Solution

In this context, none of the assumption is required for the distribution of difference.