# Solved (Free): A small pilot study is run to compare a new drug for chronic pain to one that is currently available. Participants are

#### ByDr. Raju Chaudhari

Aug 23, 2020

A small pilot study is run to compare a new drug for chronic pain to one that is currently available. Participants are randomly assigned to receive either the new drug or the currently available drug and report improvement in pain on a 5-point ordinal scale: 1 = Pain is much worse, 2 = Pain is slightly worse, 3 = No change, 4 = Pain improved slightly, 5 = Pain much improved. Is there a significant difference in self-reported improvement in pain? Use the Mann-Whitney U test with a 5% level of significance.

New Drug 4 5 3 3 4 2
Standard Drug 2 3 4 1 2 3

### Solution

Let $X$ denote improvement in chronic pain due to new drug and $Y$ denote the improvement in chronic pain due to standard drug.

Hypothesis testing problem is

$H_0:$ There is no significant difference in self-reported improvement in pain.

$H_1:$ There is a significant difference in self-reported improvement in pain

First Combine the two groups and arrange them in order of magnitude. Assign ranks to each.

X Y rx ry

4 2 10.0 3.0
5 3 12.0 6.5
3 4 6.5 10.0
3 1 6.5 1.0
4 2 10.0 3.0
2 3 3.0 6.5

The Ranks of New Drugs and standard drugs after combining the two samples are 10, 12, 6.5, 6.5, 10, 3 and 3, 6.5, 10, 1, 3, 6.5

Sum of ranks of new drugs $R_1 = 48$, sum of ranks of standard drug $R_2 = 30$.

Define
\begin{aligned} U_1 & = n_1n_2 + \frac{n_1(n_1+1)}{2}-R_1\\ &= 6* 6 + \frac{6*(6 +1)}{2}- 48\\ & = 9\\ U_2 & = n_1n_2 + \frac{n_2(n_2+1)}{2}-R_2\\ &= 6* 6 + \frac{6*(6 +1)}{2}- 30\\ & = 27\\ \end{aligned}
The test statistics is $U = min$ { $U_1, U_2$} $= 9$.

The critical value of $U$ at 0.05 level of significance is $U_{\alpha, n1,n2} = 5$.

As the observed value of $U= 9$ is not less than $U_{crit}=5$, we do not reject the null hypothesis at $0.05$ level of significance, i.e. we fail to reject the null hypothesis.

We conclude that there is a significant difference in self-reported improvement in pain.
The text passage describes a hypothesis test using the Mann-Whitney U test to compare the effectiveness of a new drug for chronic pain to a standard drug. It includes the data, the hypothesis, the calculation of the test statistic, and the conclusion.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
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- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Solved (Free): A small pilot study is run to compare a new drug for chronic pain to one that is currently available. Participants are

#### ByDr. Raju Chaudhari

Aug 23, 2020

A small pilot study is run to compare a new drug for chronic pain to one that is currently available. Participants are randomly assigned to receive either the new drug or the currently available drug and report improvement in pain on a 5-point ordinal scale: 1 = Pain is much worse, 2 = Pain is slightly worse, 3 = No change, 4 = Pain improved slightly, 5 = Pain much improved. Is there a significant difference in self-reported improvement in pain? Use the Mann-Whitney U test with a 5% level of significance.

New Drug 4 5 3 3 4 2
Standard Drug 2 3 4 1 2 3

### Solution

Let $X$ denote improvement in chronic pain due to new drug and $Y$ denote the improvement in chronic pain due to standard drug.

Hypothesis testing problem is

$H_0:$ There is no significant difference in self-reported improvement in pain.

$H_1:$ There is a significant difference in self-reported improvement in pain

First Combine the two groups and arrange them in order of magnitude. Assign ranks to each.

X Y rx ry

4 2 10.0 3.0
5 3 12.0 6.5
3 4 6.5 10.0
3 1 6.5 1.0
4 2 10.0 3.0
2 3 3.0 6.5

The Ranks of New Drugs and standard drugs after combining the two samples are 10, 12, 6.5, 6.5, 10, 3 and 3, 6.5, 10, 1, 3, 6.5

Sum of ranks of new drugs $R_1 = 48$, sum of ranks of standard drug $R_2 = 30$.

Define
\begin{aligned} U_1 & = n_1n_2 + \frac{n_1(n_1+1)}{2}-R_1\\ &= 6* 6 + \frac{6*(6 +1)}{2}- 48\\ & = 9\\ U_2 & = n_1n_2 + \frac{n_2(n_2+1)}{2}-R_2\\ &= 6* 6 + \frac{6*(6 +1)}{2}- 30\\ & = 27\\ \end{aligned}
The test statistics is $U = min$ { $U_1, U_2$} $= 9$.

The critical value of $U$ at 0.05 level of significance is $U_{\alpha, n1,n2} = 5$.

As the observed value of $U= 9$ is not less than $U_{crit}=5$, we do not reject the null hypothesis at $0.05$ level of significance, i.e. we fail to reject the null hypothesis.

We conclude that there is a significant difference in self-reported improvement in pain.
user
The text passage describes a hypothesis test using the Mann-Whitney U test to compare the effectiveness of a new drug for chronic pain to a standard drug. It includes the data, the hypothesis, the calculation of the test statistic, and the conclusion.

TYPE: SINGLE_QUESTION_WITH_ANSWER