1. In each of the following situations, is it more reasonable simply to explore the relationship between the two variables or to view one of the variables as an explanatory variable and the other as a response variable? In the latter case, which is the explanatory variable?

2.Figure 6.22 shows the calories and salt content (in milligrams of sodium) in 17 brands of beef hot dogs. Describe the overall pattern (form, direction, and strength) of these data. In what way is the point marked unusual?

3. Figure 6.23 is a scatterplot of data from the World Bank. The individuals are all the world’s nations for which data are available. The explanatory variable is a measure of how rich a country is, which is the gross domestic product (GDP) per person. GDP is the total value of the goods and services produced in a country, converted into dollars. The response variable is life expectancy at birth. Three African nations are outliers, with lower life expectancy than usual for their GDP. A full study would ask what special circumstances explain these outliers.

4. Global warming is due to increased concentrations of greenhouse gases such as carbon dioxide (). Here are data from the National Oceanic and Atmospheric Administration website (www.noaa.gov), where atmospheric is measured in parts per million per unit of volume. The data below were measured at the Mauna Loa Observatory:

(You will revisit these data in Exercise 20.)

5. Table 5.7 (page 196) gives the city and highway gas mileages for 13 midsized cars. Omit the hybrid car (Toyota Prius) and make a scatterplot, taking city mileage as the explanatory variable. Describe in words the form, direction, and strength of the relationship between highway mileage and city mileage. (You will revisit these data in Exercise 19.)

6. How fast do icicles grow? Here are data on two variables, Time measured in minutes and Length measured in centimeters, for one set of conditions: no wind, temperature −11∘C, and water flowing over the icicle at 12 milligrams per second.

279

Which is the explanatory variable? Make a scatterplot. Describe in words the direction, form, and strength of the relationship.

7. How does the fuel consumption of a car change as its speed increases? Here are data for a British Ford Escort. Fuel consumption is measured in miles per gallon of gasoline used and speed is measured in miles per hour.

8. Give an example of two variables from everyday life that have a positive association. Give an example of two variables that have a negative association.

9. The following table shows excerpted and rounded data collected on crickets by Harvard physics professor George W. Pierce in his 1948 book The Song of Insects:

10. On the NASA space shuttle, six primary O-rings were used to seal the sections of the two solid-fuel rocket motors and keep hot gases from escaping and catastrophically igniting the liquid hydrogen fuel tank. The number of O-ring erosion problems and the launch temperature (in degrees Fahrenheit) are given for 23 successful flights between April 12, 1981, and January 12, 1986:

11. Use spreadsheet software or a graphing calculator for this exercise. (Spotlight 6.1 on page 249 provides instruction for TI-83/84 graphing calculators and Excel.) The presence of mercury in fish is a health hazard, particularly for women who may become pregnant and children. Table 6.8 contains data on mercury concentration in tissue samples from 20 largemouth bass taken from Lake Natoma (California). Only fish of legal/edible size were used in this study. (Save your data and work from this exercise for use in Exercise 31.)

280

12. Use spreadsheet software or a graphing calculator for this exercise. (Spotlight 6.1 on page 249 provides instruction for TI-83l84 graphing calculators and Excel.) Satellites are one of the many tools used for predicting flash floods, heavy rainfall, and large amounts of snow. Geostationary (GEOS) satellites collect data on cloud top brightness temperatures (measured in degrees Kelvin). It turns out that colder cloud temperatures are associated with higher and thicker clouds, which in turn are associated with heavier precipitation. Data consisting of temperature and rainfall rate measured by ground radar appear in Table 6.9. Because ground radar can be limited by location and obstructions, having an alternative for predicting the rainfall rates can be useful. (Save your data and work from this exercise for use in Exercise 32.)

13.Figure 6.22 (page 278) shows the salt content (in milligrams of sodium) and calories in 17 brands of beef hot dogs. If we ignore the outlying point marked , a regression line for predicting sodium from calories passes close to these two observations:

Use this fact to estimate the slope of this regression line (round your answer to two decimal places).

14. Exercise 4 (page 278) gives data on concentration (in parts per million) over time. A regression line for predicting for a given year is

281

15. Researchers studying acid rain measured the acidity of precipitation in a Colorado wilderness area for 150 consecutive weeks. Acidity is measured by pH, and lower pH values mean higher acidity. The acid rain researchers observed a straight-line pattern in acidity levels over time. They reported that the regression line

fit the data well. [Data from W. M. Lewis and M. C. Grant, Acid precipitation in the western United States, Science, 207 (1980): 176–177.]

16. A study at the University of Massachusetts, Amherst, published in the May 2007 Journal of Marriage and Family found that married women do about one fewer hour of housework a week for every $7500 they earn as full-time workers outside the home, regardless of their husband’s income.

17. A 21-year-old college student drinks heavily at a party until his BAC is 0.15 g/dl—almost twice the legal driving limit of 0.08. He now stops drinking, and each hour his BAC falls by 0.015 g/dl.

18. Suppose that the slope of the regression line of weight on height for a group of young men is when we measure height in centimeters and weight in kilograms. That is, when height increases by 1 centimeter, weight increases by 1.1 kilograms. There are 1000 grams in a kilogram. If we measured weight in grams, what would the slope be?

19. Find the correlation between the city and highway gas mileages for the 12 nonhybrid midsized cars (i.e., omit the Toyota Prius) in Table 5.7 (page 196). Explain why the value of supports the scatterplot that you created in Exercise 5.

20. Exercise 4 (page 278) gives data on concentration (in parts per million) over time.

21. Find the correlation between city and highway mileage for all 13 midsized cars in Table 5.7 (page 196), including the Toyota Prius. Compare your with the value you found in Exercise 19. Explain why adding the Prius changes in this direction.

22. In Example 11, Table 6.7 (page 270), the positive association between family income and SAT score is clear from the lockstep pattern, but to calculate a specific numerical value for the correlation, we need to make a simplifying assumption. For each income bracket with fixed endpoints, choose its midpoint. Now, use a calculator (or Excel) to calculate a correlation value (refer to Spotlight 6.2 on page 259).

23. Exercise 7 (page 279) gives data on gas used versus speed for a small car. Make a scatterplot, if you did not do so in Exercise 7. Calculate the correlation. Explain why is close to 0 despite a strong relationship between speed and gas use.

24. Consider the data in Exercise 6 (page 278) on forming icicles.

25. If heterosexual women always dated men who are three years older than themselves, what would the correlation be between the ages of each man and each woman? (Hint: Draw a scatterplot for several ages.)

26. We want to find the correlation between

The answers (in scrambled order) are , , and . Match the values to the variable pairings and explain your choices.

27. For each of the following pairs of variables, would you expect a substantial negative correlation, a substantial positive correlation, or a small correlation?

28. Each of the following statements contains a mistake. Explain what is wrong in each case.

29. Mutual fund reports often give correlations to describe how the prices of different investments are related. You look at the correlations between three Fidelity funds and the Standard & Poor’s 500 Stock Index, which describes stocks of large U.S. companies. The three funds are Dividend Growth (stocks of large U.S. companies), Small Cap Stock (stocks of small U.S. companies), and Emerging Markets (stocks in developing countries). For a recent year, the three correlations are , , and .

30.Archaeopteryx is an extinct animal that had feathers like a bird but teeth and a long bony tail like a reptile. Only six fossil specimens are known. If the specimens belong to the same species and differ in size because some are younger than others, there should be a straight-line relationship between the lengths of a pair of bones from all individuals. An outlier from this relationship would suggest a different species. Here are data on the lengths in centimeters of the femur (a leg bone) and the humerus (a bone in the upper arm) for the five specimens that preserve both bones:

31. Exercise 11, Table 6.8 (page 280) presented data on fish length and mercury concentration in fish tissue.

32. Exercise 12, Table 6.9 (page 280) presented data on cloud top brightness temperatures (measured from a satellite) and radar rain rate.

33. In Exercise 5, you made a scatterplot of city and highway gas mileage for the 12 nonhybrid midsized cars (omitting the Prius) in Table 5.7 (page 196).

34. In Exercise 6 (page 278), you made a scatterplot of the length of an icicle and the number of minutes that water has been flowing over the icicle.

35. Exercise 7 (page 279) gives data on fuel consumption in miles per gallon (mpg) and speed in miles per hour (mph) for a small car. From these data, the least-squares regression line to predict gas performance from speed is

36. Scientists are concerned that rising sea temperatures will have an adverse effect on coral growth. A small study on this issue produced the data in the table below:

37. A random sample of femur bone lengths (in millimeters) and heights (in centimeters) from six males appears in the table below (these data are from the Forensic Data Bank at the University of Tennessee):

284

The equation of the least-squares regression line for predicting height from femur length is

38. The length of the icicle in Exercise 6 (page 278) is measured in centimeters. There are 2.54 centimeters in an inch. If length were measured in inches, how would the slope of the regression line given in Exercise 34 change?

39. The mean height of American women in their early twenties is about 64.5 inches and the standard deviation is about 2.5 inches. The mean height of men the same age is about 68.5 inches, with a standard deviation of about 2.7 inches.

40. These data, from the National Oceanic and Atmospheric Administration website (www.noaa.gov), are the mean annual number of named Atlantic storms (hurricanes, tropical storms, and tropical depressions), during five-year windows ending with the year shown in the table:

41. Use the general equation for the least-squares regression line to show that this line always passes through the point (). That is, set and show that the line predicts that .

42. Exercise 6 gives data on the growth of an icicle (page 278).

43. Fidelity Investments, like other large mutual fund companies, offers many “sector funds” that concentrate their investments in narrow segments of the stock market. These funds often rise or fall by much more than the market as a whole. Here are the percent returns for 23 Fidelity “Select Portfolios” funds for the years 2002 (when stocks fell) and 2003 (when stocks went up).

285

Do a careful statistical analysis of these data, using both graphs and whatever numerical measures you think are appropriate. Make a side-by-side comparison of the distributions of returns in 2002 and 2003 and also describe the relationship between the returns of the same funds in these two years. What are your most important findings? (The outlier is Fidelity Gold Fund.)

44. Here are data collected on six individuals:

45.Table 6.10 offers four datasets prepared by statistician Frank Anscombe to show the dangers of calculating without first plotting the data.

46. Children who watch many hours of TV get lower grades in school, on average, than those who watch less TV. Explain clearly why this fact does not show that watching TV causes poor grades. In particular, suggest some other characteristics of households where children watch lots of TV that may contribute to poor grades.

47. People who use artificial sweeteners in place of sugar tend to be heavier than people who use sugar. Does this mean that artificial sweeteners cause weight gain? Give a more plausible explanation for this association.

48. Based on an examination of 22 companies that announced large layoffs during 1994, Downs found a strong correlation between the size of the layoffs and the compensation of the CEOs” (K. Phillips, Wealth and Democracy, Broadway Books, New York, 2002, p. 151). Discuss why this positive correlation is probably explained by a third variable, the size of the company as measured by its number of employees.

49. The positive correlation between health ® and income per capita is one of the best-known relations in international development. This correlation is commonly thought to reflect a causal link running from income to health…. Recently, however, another intriguing possibility has emerged: that the health-income correlation is partly explained by a causal link running the other way—from health to income” [D. E. Bloom and D. Canning, The health and wealth of nations, Science, 287 (2000): 1207–1208]. Explain how higher income in a nation can cause better health. Then explain how better health can cause higher national income. There is no simple way to determine the direction of the link.

50. The effect of an outside variable can be surprising when individuals are divided into groups. In recent years, the mean SAT score of all high school seniors has increased. But the mean SAT score has decreased for students at each level of high school grades (A, B, C, etc.). Explain how grade inflation in high school can account for this pattern. A relationship that holds for each group within a population need not hold (and may even be in the opposite direction!) for the population as a whole.

51. Consider the dataset below:

52. A vehicle’s tire pressure can affect tire wear (in terms of the length of life of the tire). When tire pressure is low, the tire flattens out and more of its surface contacts the road. This causes more friction between the tire and the roadway and increases the amount of wear on the tire. Data collected on tire pressure and tire wear appear in the table below:

53. Major recalls of toys with lead paint refocused people on the dangers of lead exposure. Below are data from research exploring the association with student achievement for blood lead levels below the “danger threshold” of 10 mcg/dl set by the Centers for Disease Control [M. L. Miranda et al., The relationship between early childhood blood lead levels and performance on end-of-grade tests, Environmental Health Perspectives, 115 (2007): 1242–1247].

287

54. A study of reading ability in schoolchildren chose 60 fifth-grade children at random from a school. The researchers obtained the children’s scores on an IQ test and on a test of reading ability. Figure 6.24 plots reading test score (response variable) against IQ score (explanatory variable).

55. A student wonders if tall women tend to date taller people than do short women. She measures herself, her sister, and the women in the adjoining dorm rooms. Then she measures the next person each woman dates and obtains the following data (in inches):

56. In Exercise 55, you found the correlation between the heights in inches of several college women and the heights in inches of the next person each woman dates.

57. The equation of the least-squares regression line for predicting dates’ heights from women’s heights for the data in Exercise 55 is

58. From 2000 to 2005, sales and file sharing (i.e., free downloading) intensity were tracked within seven musical genres (rock, alternative, R&B, rap/hip-hop, country, jazz, classical). The correlation between change in sales and file-sharing intensity was −0.648. Is this evidence that file sharing helps or hurts sales? Explain.

59. In issue 49 of Stats: The Magazine for Students of Statistics, Schuyler Huck presents a dataset of 100 ordered pairs in which 25 of them are (17, 1), 25 are (18, 2), 25 are (19, 3), and 25 are (20, 4).

60. Return to Table 5.13 (page 226), which lists the top 100 baseball players ranked by career batting average. Consider only the top 50 players—Ty Cobb to Chuck Klein. Enter the data on career batting averages and career home runs into an Excel spreadsheet or calculator lists.