Critical Thinking: Is the pain medicine Duragesic effective in reducing pain? Listed below are measures of pain intensity before and after using the drug Duragesic (fentanyl) (based on data from Janssen Pharmaceutical Products, L.P.). The data are listed in order by row, and corresponding measures are from the same subject before and after treatment. For example, the first subject had a measure of 1.2 before treatment and a measure of 0.4 after treatment. Each pair of measurements is from one subject, and the intensity of pain was measured using the standard visual analog score. A higher score corresponds to higher pain intensity.

Pain Intensity Before Duragesic Treatment

1.2	1.3	1.5	1.6	8	3.4	3.5	2.8	2.6	2.2
3	7.1	2.3	2.1	3.4	6.4	5	4.2	2.8	3.9
5.2	6.9	6.9	5	5.5	6	5.5	8.6	9.4	10
7.6

Pain Intensity After Duragesic Treatment

0.4	1.4	1.8	2.9	6	1.4	0.7	3.9	0.9	1.8
0.9	9.3	8	6.8	2.3	0.4	0.7	1.2	4.5	2
1.6	2	2	6.8	6.6	4.1	4.6	2.9	5.4	4.8
4.1

Matched Pairs The methods of Section 9-3 can be used to test a claim about matched data. Identify the specific claim that the treatment is effective, then use the methods of Section 9-3 to test that claim.

The pain intensities of a group of subjects are recorded before and after using the drug Duragesic.

It is claimed that the drug Duragesic is effective in reducing pain.

The following hypotheses are noted:

Null Hypothesis: The mean value of the pain intensity before the treatment is equal to the mean value of the pain intensity after the treatment.

\({H_0}:{\mu _d} = 0\)

Alternative Hypothesis: The mean value of the pain intensity before the treatment is greater than the mean value of the pain intensity after the treatment.

\({H_1}:{\mu _d} > 0\)

Here,\({\mu _d}\)represents the population difference in the pain intensities before and after the treatment.

The test is right-tailed.

The following table shows the differences in the pain intensities before and after the treatment:

The number of pairs is equal to\(n = 31\).

The mean value of the differences is computed below:

\(\begin{aligned} \bar d &= \frac{{0.8 + \left( { - 0.1} \right) + ...... + 3.5}}{{31}}\\ &= 1.38\end{aligned}\)

The standard deviation of the differences is computed below:

\(\begin{aligned} {s_d} &= \sqrt {\frac{{\sum\limits_{i = 1}^n {{{({d_i} - \bar d)}^2}} }}{{n - 1}}} \\ &= \sqrt {\frac{{{{\left( {0.8 - 1.38} \right)}^2} + {{\left( {\left( { - 0.1} \right) - 1.38} \right)}^2} + ....... + {{\left( {3.5 - 1.38} \right)}^2}}}{{31 - 1}}} \\ &= 2.92\end{aligned}\)

The mean value of the differences for the population of matched pairs \(\left( {{\mu _d}} \right)\) is considered to be equal to 0.

The value of the test statistic is computed as shown:

\(\begin{array}{c}t = \frac{{\bar d - {\mu _d}}}{{\frac{{{s_d}}}{{\sqrt n }}}}\\ = \frac{{1.38 - 0}}{{\frac{{2.92}}{{\sqrt {31} }}}}\\ = 2.623\end{array}\)

The degrees of freedom are computed below:

\(\begin{array}{c}df = n - 1\\ = 31 - 1\\ = 30\end{array}\)

The critical value of t at\(\alpha = 0.05\)and degrees of freedom equal to 30 for a right-tailed test is equal to 1.6973.

The corresponding p-value is equal to 0.0068.

Since the value of the test statistic (2.623) is greater than the critical value and the p-value is less than 0.05, the null hypothesis is rejected.

There is enough evidence to conclude that the drug Duragesic is effective in reducing pain.