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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

Pain Intensity After Duragesic Treatment

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.

Interpreting the Coefficient of Determination. In Exercises 5鈥�8, use the value of the linear correlation coefficient r to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between the two variables.

Crickets and Temperature r = 0.874 (x = number of cricket chirps in 1 minute, y = temperature in 掳F)

In Exercises 5鈥�8, we want to consider the correlation between heights of fathers and mothers and the heights of their sons. Refer to theStatCrunch display and answer the given questions or identify the indicated items.

The display is based on Data Set 5 鈥淔amily Heights鈥� in Appendix B.

Identify the multiple regression equation that expresses the height of a son in terms of the height of his father and mother.

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?

let the predictor variable x be the first variable given. Use the given data to find the regression equation and the best predicted value of the response variable. Be sure to follow the prediction procedure summarized in Figure 10-5 on page 493. Use a 0.05 significance level.

For 50 randomly selected speed dates, attractiveness ratings by males of their

female date partners (x) are recorded along with the attractiveness ratings by females of their male date partners (y); the ratings are from Data Set 18 鈥淪peed Dating鈥� in Appendix B. The 50 paired ratings yield\(\bar x = 6.5\),\(\bar y = 5.9\), r= -0.277, P-value = 0.051, and\(\hat y = 8.18 - 0.345x\). Find the best predicted value of\(\hat y\)(attractiveness rating by female of male) for a date in which the attractiveness rating by the male of the female is x= 8.

The following exercises are based on the following sample data consisting of numbers of enrolled students (in thousands) and numbers of burglaries for randomly selected large colleges in a recent year (based on data from the New York Times).

The sample data result in a linear correlation coefficient of r= 0.499 and the regression equation\(\hat y = 3.83 + 2.39x\). What is the best predicted number of burglaries, given an enrollment of 50 (thousand), and how was it found?

Stocks and Sunspots. Listed below are annual high values of the Dow Jones Industrial Average (DJIA) and annual mean sunspot numbers for eight recent years. Use the data for Exercises 1鈥�5. A sunspot number is a measure of sunspots or groups of sunspots on the surface of the sun. The DJIA is a commonly used index that is a weighted mean calculated from different stock values.

Confidence Interval Construct a 95% confidence interval estimate of the mean sunspot number. Write a brief statement interpreting the confidence interval.

Interpreting the Coefficient of Determination. In Exercises 5鈥�8, use the value of the linear correlation coefficient r to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between the two variables.

Pizza and Subways r = 0.992 (x = cost of a slice of pizza, y = subway fare in New York City

let the predictor variable x be the first variable given. Use the given data to find the regression equation and the best predicted value of the response variable. Be sure to follow the prediction procedure summarized in Figure 10-5 on page 493. Use a 0.05 significance level.

Head widths (in.) and weights (lb) were measured for 20 randomly selected bears (from Data Set 9 鈥淏ear Measurements鈥� in Appendix B). The 20 pairs of measurements yield\(\bar x = 6.9\)in.,\(\bar y = 214.3\)lb, r= 0.879, P-value = 0.000, and\(\hat y = - 212 + 61.9x\). Find the best predicted value of\(\hat y\)(weight) given a bear with a head width of 6.5 in.

In Exercises 5鈥�8, we want to consider the correlation between heights of fathers and mothers and the heights of their sons. Refer to the

StatCrunch display and answer the given questions or identify the indicated items.

The display is based on Data Set 5 鈥淔amily Heights鈥� in Appendix B.

Identify the following:

a. The P-value corresponding to the overall significance of the multiple regression equation

b. The value of the multiple coefficient of determination\({R^2}\).

c. The adjusted value of \({R^2}\)