Chapter 13.4, Problem 1AC

Pain Medication

A researcher decides to see how effective a pain medication is. Eight randomly selected subjects were asked to determine the severity of their pain by using a scale of 1 to 10, with 1 being very minor and 10 being very severe. Then each was given the medication, and after 1 hour, they were asked to rate the severity of their pain, using the same scale.

1. What is the purpose of the study?

2. Are the samples independent or dependent?

3. What are the hypotheses?

4. What nonparametric test could be used to test the claim?

5. What significance level would you use?

6. What is your decision?

7. What parametric test could you use?

8. Would the results be the same?

To determine

To explain: The purpose of the study.

Answer to Problem 1AC

The purpose of the study is to determine how effective the pain medication is.

Explanation of Solution

Justification:

The researcher had conducted a study for seeing the effectiveness of the pain medication and selected the subjects randomly and recorded on a scale of 1 to 10 about the severity of the pain. This implies that the purpose of the study is to know about the pain medication effectiveness.

Thus, the purpose of the study is to determine how effective the pain medication is.

To determine

To explain: Whether the samples are independent or dependent.

Answer to Problem 1AC

The samples are dependent samples.

Explanation of Solution

Justification:

In this study the researcher has randomly selected 8 subjects to determine about the effectiveness of the pain medication. First the eight subjects are asked to give the scale about the pain and after one hour the same subjects are asked to give the scale based on the pain medication. This implies that the same subjects are treated twice for the study and the scale is recorded, and the samples are dependent samples.

Thus, the samples are dependent samples.

To determine

To give: the hypotheses for the test.

Explanation of Solution

Justification:

The claim of the study is to determine the effectiveness of the pain medication. The hypotheses for the test are,

Null hypothesis:

$H_0:$ The severity of the pain before and after the medication is the same.

Alternative hypothesis:

$H_1:$ The severity of the pain after medication is less than before the medication.

To determine

What parametric test that can be used to test the claim.

Answer to Problem 1AC

The parametric test that can be used to test the claim is Wilcoxon signed rank test.

Explanation of Solution

Justification:

Wilcoxon sign rank test:

The Wilcoxon sign rank test is a non-parametric test that is used to compare ranks the population means of the paired samples or the matched samples. This implies that the Wilcoxon sign rank test is used to compare the matched pair groups that is, the same sample is measured two times. Wilcoxon signed-rank is used for testing two samples of the dependent groups come from the population having same distribution or not.

In this study the same subjects are treated before and after the medication and the samples are dependent samples.

Thus, the parametric test that can be used to test the claim is Wilcoxon signed rank test.

To determine

What significance level that is used.

Answer to Problem 1AC

The significance level that can be used is 0.05.

Explanation of Solution

Justification:

In general when the confidence level is not mentioned the level that is used for any of the hypotheses testing is 95% and the level of significance is 0.05.

Thus, the significance level that can be used is 0.05.

To determine

The decision of the study.

Answer to Problem 1AC

The Null hypothesis is rejected.

Explanation of Solution

Calculation:

Critical value:

The data represent the value for $\alpha$ is 0.05. The alternative hypothesis is denoted as the one-tailed test.

From Table K, The Wilcoxon Signed-Rank Test, the critical value for $\alpha =0.05$ and $n=8$ is 6.

Hence, the critical value for is 6.

The sum of the signed ranks is obtained below:

Before	After	Difference $\left(D\right)$	Absolute value $\left(\left|D\right|\right)$	Rank	Signed rank
8	6	2	2	6	6
6	5	1	1	2.5	2.5
2	3	–1	1	2.5	–2.5
3	1	2	2	6	6
4	2	2	2	6	6
6	6	0	0	–	–
2	1	1	1	2.5	2.5
7	6	–1	1	2.5	–2.5

The sum of minus ranks is,

$\begin{array}{c}\text{Negative ranks sum}=\left(-2.5\right)+\left(-2.5\right)\\ =-5\end{array}$

The sum of plus ranks is,

$\begin{array}{c}\text{Positive ranks sum}=6+2.5+6+6+2.5\\ =23\end{array}$

The test value is 5 which are taken as the smallest absolute values of the sums.

Thus, the test value is $\left|w_s\right|=5$.

Decision Rule:

If the test value is less than the critical value, then reject the null hypothesis $H_0$.

Conclusion:

It is clear that the critical value is 6 and the test value is 5.

Here, the test value is less than the critical value.

Therefore, by the rule, the null hypothesis $H_0$ is rejected.

There is sufficient evidence to reject the claim that “the severity of the pain after medication is less than before the medication”.

To determine

Which parametric test that can be used.

Answer to Problem 1AC

The parametric test that can be used is paired t test.

Explanation of Solution

In nonparametric tests the Wilcoxon signed rank test is used for testing two dependent samples. Likewise in parametric tests the paired t test is used to test the difference between two populations for dependent samples.

Thus, the parametric test that can be used is paired t test.

To determine

Whether the results are same or not.

Answer to Problem 1AC

Yes, the results would be same.

Explanation of Solution

When the same data is used for testing for parametric paired t test the results would be same as obtained from nonparametric tests. But, the results obtained would be appropriate only if the assumptions for the parametric paired t test are satisfied.

Thus, the results would be the same.