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Chapter 10: Q4FDD (page 468)

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.

Short Answer

Expert verified

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

Using the methods of 9-3, it can be concluded that there is enough evidence to support the claim that the drug Duragesic is effective in reducing pain.

Step by step solution

01

Given information

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

02

Hypotheses

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.

03

Differences in the values of each matched pair

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

Before treatment

After treatment

Differences

1.2

0.4

0.8

1.3

1.4

-0.1

1.5

1.8

-0.3

1.6

2.9

-1.3

8

6

2

3.4

1.4

2

3.5

0.7

2.8

2.8

3.9

-1.1

2.6

0.9

1.7

2.2

1.8

0.4

3

0.9

2.1

7.1

9.3

-2.2

2.3

8

-5.7

2.1

6.8

-4.7

3.4

2.3

1.1

6.4

0.4

6

5

0.7

4.3

4.2

1.2

3

2.8

4.5

-1.7

3.9

2

1.9

5.2

1.6

3.6

6.9

2

4.9

6.9

2

4.9

5

6.8

-1.8

5.5

6.6

-1.1

6

4.1

1.9

5.5

4.6

0.9

8.6

2.9

5.7

9.4

5.4

4

10

4.8

5.2

7.6

4.1

3.5

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.

04

Compute the test statistic, critical value and the p-value

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.

05

Decision and conclusion of the test

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.

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Most popular questions from this chapter

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










Correlation Use the given data to construct a scatterplot, then use the methods of Section 10-1 to test for a linear correlation between the pain intensity before and after treatment. If there does appear to be a linear correlation, can we conclude that the drug treatment is effective?

Best Multiple Regression Equation For the regression equation given in Exercise 1, the P-value is 0.000 and the adjusted \({R^2}\)value is 0.925. If we were to include an additional predictor variable of neck size (in.), the P-value becomes 0.000 and the adjusted\({R^2}\)becomes 0.933. Given that the adjusted \({R^2}\)value of 0.933 is larger than 0.925, is it better to use the regression equation with the three predictor variables of length, chest size, and neck size? Explain.

In Exercises 9 and 10, use the given data to find the equation of the regression line. Examine the scatterplot and identify a characteristic of the data that is ignored by the regression line.

Testing for a Linear Correlation. In Exercises 13鈥�28, construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r from Table A-6. Use a significance level of A = 0.05. Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section 10-2 exercises.)

Tips Listed below are amounts of bills for dinner and the amounts of the tips that were left. The data were collected by students of the author. Is there sufficient evidence to conclude that there is a linear correlation between the bill amounts and the tip amounts? If everyone were to tip with the same percentage, what should be the value of r?

Bill(dollars)

33.46

50.68

87.92

98.84

63.6

107.34

Tip(dollars)

5.5

5

8.08

17

12

16

Interpreting r. In Exercises 5鈥�8, use a significance level of A = 0.05 and refer to the accompanying displays.

5. Bear Weight and Chest Size Fifty-four wild bears were anesthetized, and then their weights and chest sizes were measured and listed in Data Set 9 鈥淏ear Measurements鈥� in Appendix B; results are shown in the accompanying Statdisk display. Is there sufficient evidence to support the claim that there is a linear correlation between the weights of bears and their chest sizes? When measuring an anesthetized bear, is it easier to measure chest size than weight? If so, does it appear that a measured chest size can be used to predict the weight?
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