Chapter 10: Q4BSC (page 468)
What is the relationship between the linear correlation coefficient rand the slope\({b_1}\)of a regression line?
There is a direct relationship between the correlation coefficient and slope of the regression line.
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Global Warming If we find that there is a linear correlation between the concentration of carbon dioxide (\(C{O_2}\)) in our atmosphere and the global mean temperature, does that indicate that changes in (\(C{O_2}\))cause changes in the global mean temperature? Why or why not?
In exercise 10-1 12. Clusters Refer to the following Minitab-generated scatterplot. The four points in the lower left corner are measurements from women, and the four points in the upper right corner are from men.
a. Examine the pattern of the four points in the lower left corner (from women) only, and subjectively determine whether there appears to be a correlation between x and y for women.
b. Examine the pattern of the four points in the upper right corner (from men) only, and subjectively determine whether there appears to be a correlation between x and y for men.
c. Find the linear correlation coefficient using only the four points in the lower left corner (for women). Will the four points in the upper left corner (for men) have the same linear correlation coefficient?
d. Find the value of the linear correlation coefficient using all eight points. What does that value suggest about the relationship between x and y?
e. Based on the preceding results, what do you conclude? Should the data from women and the data from men be considered together, or do they appear to represent two different and distinct populations that should be analyzed separately?
Super Bowl and\({R^2}\)Let x represent years coded as 1, 2, 3, . . . for years starting in 1980, and let y represent the numbers of points scored in each Super Bowl from 1980. Using the data from 1980 to the last Super Bowl at the time of this writing, we obtain the following values of\({R^2}\)for the different models: linear: 0.147; quadratic: 0.255; logarithmic: 0.176; exponential: 0.175; power: 0.203. Based on these results, which model is best? Is the best model a good model? What do the results suggest about predicting the number of points scored in a future Super Bowl game?
Finding a Prediction Interval. In Exercises 13–16, use the paired data consisting of registered Florida boats (tens of thousands) and manatee fatalities from boat encounters listed in Data Set 10 “Manatee Deaths” in Appendix B. Let x represent number of registered boats and let y represent the corresponding number of manatee deaths. Use the given number of registered boats and the given confidence level to construct a prediction interval estimate of manatee deaths.
Boats Use x = 85 (for 850,000 registered boats) with a 99% confidence level.
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.
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