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CHAPTER 10 SOLUTIONS AND MINI-PROJECT NOTES CHAPTER 10 RELATIONSHIPS BETWEEN MEASUREMENT VARIABLES EXERCISE SOLUTIONS 10.1 Yes, about five of them. The definition of statistical significance is such that when there really is no relationship we will erroneously detect one about 5% of the time. 10.2 It would still be 0.36. Correlation does not change when the units of measurement are changed. 10.3 a. y = 0.96(x − 0.5) = −0.48 + 0.96x (in pints) b. The intercept is −0.48 and it doesn't really have a physical meaning. It would be the volume of water corresponding to a weight of zero, but zero is not a possible weight since the container itself weights 0.5 pounds. The slope is 0.96, and it represents the additional volume of water (in pints) when the weight is increased by 1 pound. c. This is a deterministic relationship, not a statistical relationship, so the correlation is 1. d. See the figure below. 10.4 a. b. c. d. 10.5 The sample of size 10,000 would be more likely. To quote the textbook, "Even a minor relationship will achieve ‘statistical significance’ if the sample is very large" (p. 183). 10.6 No, you would conclude that the size of the sample was too small to be able to detect the relationship. 10.7 A correlation of −0.6 implies a stronger relationship. The sign determines whether one measurement increases or decreases as the other one increases, but the number determines the strength of the relationship, and 0.6 is higher than 0.4. 10.8 The key is that for a positive correlation they increase together, while for a negative correlation one decreases as the other increases. An example of a positive correlation is height and weight. An example of a negative correlation is amount of exercise per week and resting pulse rate. 10.9 Correlation measures only how closely the points fall to a straight line. A perfect curved relationship where A positive correlation. In summer both will be high and in winter both will be low. A negative correlation. Heavier cars (higher weight) have lower gas mileage. A negative correlation. Students who watch more TV tend to have lower GPAs. A positive correlation. People with more education tend to have higher salaries. Page 1 of 3 CHAPTER 10 SOLUTIONS AND MINI-PROJECT NOTES the best line through the points is flat would have a zero correlation. For example, draw x versus x2 for x = 3, −2, −1, 0, 1, 2, 3. 10.10 a. Predicted GPA = 0.539 + (.00362)(500) = 2.349. b. For each increase of 1 in verbal SAT score we would expect to see an increase of 0.00362 in GPA. c. No. It would be the GPA for someone with an SAT score of 0, but that's not a possible score for the SAT. 10.11 a. Man: 157.5 pounds, woman: 133.9 pounds. Men who weigh 150 pounds would ideally like to weigh slightly more, whereas women who weigh 150 pounds would like to weigh much less. b. No, it would be the ideal weight for a woman whose actual weight was zero, which is obviously meaningless. c. Yes, the slope of 0.6 indicates that on average, for two women whose actual weights differ by one pound, their ideal weights differ by 0.6 pounds. d. Someone whose weight is in the normal range, but who wants to weigh much less or much more than they do, so much so that they don't fit the pattern for their sex. For example, there is a woman whose actual weight is 105 pounds but who would like to weigh 125 pounds. In the other direction is a woman who weights 140 pounds but would like to weigh 100 pounds, much more of a discrepancy than the majority of women. 10.12 a. b. c. d. 10.13 We should not extrapolate that far beyond the range of the data. The decreasing trend in winning times cannot continue at the same rate forever, and it may not still apply by the year 2010. 10.14 a. Success rate = 76.5 − (3.95)(6.5) = 50.825%. b. For each additional foot of distance, success drops by about 3.95%. 10.15 a. At 2 feet, success rate = 76.5 − (3.95)(2) = 68.6%. At 20 feet, success rate = 76.5 − (3.95)(20) = −2.5%. These predictions are clearly considerably different from the actual success rates of 93.3% and 15.8%. b. The relationship that held for the distances of 5 to 15 feet obviously does not extend to distances as short as 2 feet or as long as 20 feet. It is not appropriate to assume that relationships continue to hold beyond the range of the data observed. c. The relationship is a curve that looks like an elongated, tilted letter C. d. Regression lines work only for straight line relationships. Clearly, the relationship between success and distance over the entire range from 2 feet to 20 feet is curved, not linear. 10.16 Yes. The strength of the relationship is the same as for the golfers and the sample is much larger. 10.17 a. Frequency of drinking in the past year and score on the Hangover Symptoms Scale, a measure of severity of 13 hangover symptoms. b. There is a moderately positive relationship on a straight line, between frequency of drinking and severity of hangover symptoms. People with higher drinking frequency are likely to have higher values on the hangover severity scale. c. The quote refers to statistical significance. This means that the researchers were convinced that the relationship observed in their sample reflected a real relationship in the population. d. Frequency of drinking would be the explanatory variable, and hangover symptoms would be the response. It is unlikely that hangover symptoms would lead to drinking with higher frequency, but it is likely that higher drinking frequency would lead to more severe hangover symptoms. The correlation would be negative because the slope is negative. Predicted time is 35.87, 1.45 seconds too high. Winning times decrease by 0.1142 seconds per year. Winning times should decrease by about 0.46 second per 4-year Olympics. Page 2 of 3 CHAPTER 10 SOLUTIONS AND MINI-PROJECT NOTES NOTES ABOUT THE MINI-PROJECTS FOR CHAPTER 10 Mini-Project 10.1 Make sure weight is on the vertical axis and height is on the horizontal axis, or the prediction would not make sense. (The variable y is weight and x is height.) The regression equation can be drawn by plugging in two values of x and finding the corresponding values of y, plotting those two points on the scatter plot, and connecting them with a line. The picture should be well labeled. Mini-Project 10.2 The explanation should include an explanation of the two variables, a description about whether the relationship is positive or negative and a discussion about the strength of the relationship. If a regression equation is given, the explanation can also discuss the meaning of the intercept and slope in the context of the example. Page 3 of 3