14.9 What number can I be?

14.10 Measuring mice. For a biology project, you measure the tail length (millimeters) and weight (grams) of 10 mice.

14.11 Living on campus. A February 2, 2008, article in the Columbus Dispatch reported a study on the distances students lived from campus and average GPA. Here is a summary of the results:

Based on these data, is the association between the distance a student lives from campus and average GPA positive, negative, or near 0?

14.12 The endangered manatee. Manatees are large, gentle, slow-moving creatures found along the coast of Florida. Many manatees are injured or killed by boats. Figure 14.10 is a scatterplot of the number of manatee deaths by boats versus the number of boats registered in Florida (in thousands) for the years between 1977 and 2014.

14.13 Calories and salt in hot dogs. Figure 14.11 shows the calories and sodium content in 17 brands of meat hot dogs. Describe the overall pattern of these data. In what way is the point marked A unusual?

14.14 The endangered manatee. Is the correlation r for the data in Figure 14.10 near −1, clearly negative but not near −1, near 0, clearly positive but not near 1, or near 1? Explain your answer.

14.15 Calories and salt in hot dogs. Is the correlation r for the data in Figure 14.11 near −1, clearly negative but not near −1, near 0, clearly positive but not near 1, or near 1? Explain your answer.

14.16 Comparing correlations. Which of Figure 14.2, 14.10, and 14.11 has a correlation closer to 0? Explain your answer.

14.17 Outliers and correlation. In Figure 14.11, the point marked A is an outlier. Will removing the outlier increase or decrease r ? Why?

14.18 Global warming. Have average global temperatures been increasing in recent years? Here are annual average global temperatures for the last 21 years in degrees Celsius.

14.19 Death by intent. Homicide and suicide are both intentional means of ending a life. However, the reason for committing a homicide is different than that for suicide, and we might expect homicide and suicide rates to be uncorrelated. On the other hand, both can involve some degree of violence, so perhaps we might expect some level of correlation in the rates. Table 14.1 gives data from 2008–2010 for 26 counties in Ohio. Rates are per 100,000 people. The data also indicate that the homicide rates for some counties should be treated with caution because of low counts (Y = Yes, treat with caution, and N = No, do not treat with caution).

14.20 Marriage. Suppose that men always married women three years younger than themselves. Draw a scatterplot of the ages of six married couples, with the husband’s age as the explanatory variable. What is the correlation r for your data? Why?

14.21 Stretching a scatterplot. Changing the units of measurement can greatly alter the appearance of a scatterplot. Return to the fossil data from Example 3:

These measurements are in centimeters. Suppose a deranged scientist measured the femur in meters and the humerus in millimeters. The data would then be

14.22 Global warming. Exercise 14.18 gives data on the average global temperatures, in degrees Celsius, for the years 1994 to 2014.

14.23 Death by intent. Table 14.1 gives data on on homicide and suicide rates from 2008–2010 for 26 counties in Ohio. The homicide rates for 14 of the counties should be treated with caution because of low counts. You made a scatterplot of these data in Exercise 14.19.

14.24 Strong association but no correlation. The gas mileage of an automobile first increases and then decreases as the speed increases. Suppose that this relationship is very regular, as shown by the following data on speed (miles per hour) and mileage (miles per gallon):

Make a scatterplot of mileage versus speed. Use a calculator to show that the correlation between speed and mileage is r = 0. Explain why the correlation is 0 even though there is a strong relationship between speed and mileage.

14.25 Death by intent. The data in Table 14.1 are given in deaths per 100,000 people. If we changed the data from deaths per 100,000 people to deaths per 1,000 people how would the rates change? How would the correlation between homicide and suicide rates change?

14.26 What are the units? How sensitive to changes in water temperature are coral reefs? To find out, measure the growth of corals in aquariums (where growth is the change in weight, in pounds, of the coral before and after the experiment) when the water temperature (in degrees Fahrenheit) is controlled at different levels. In what units are each of the following descriptive statistics measured?

14.27 Teaching and research. A college newspaper interviews a psychologist about student ratings of the teaching of faculty members. The psychologist says, “The evidence indicates that the correlation between the research productivity and teaching rating of faculty members is close to zero.” The paper reports this as “Professor McDaniel said that good researchers tend to be poor teachers, and vice versa.” Explain why the paper’s report is wrong. Write a statement in plain language (don’t use the word “correlation”) to explain the psychologist’s meaning.

14.28 Sloppy writing about correlation. Each of the following statements contains a blunder. Explain in each case what is wrong.

14.29 Guess the correlation. Measurements in large samples show that the correlation

The answers (in scrambled order) are

r = 0.2  r = 0.5  r = 0.8

Match the answers to the statements and explain your choice.

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

14.31 Investment diversification. A mutual funds company’s newsletter says, “A well-diversified portfolio includes assets with low correlations.” The newsletter includes a table of correlations between the returns on various classes of investments. For example, the correlation between municipal bonds and large-cap stocks is 0.50, and the correlation between municipal bonds and small-cap stocks is 0.21.

14.32 Take me out to the ball game. What is the relationship between the price charged for a hot dog and the price charged, per ounce, for beer in Major League Baseball stadiums? Table 14.2 gives some data. Make a scatterplot appropriate for showing how beer price helps explain hot dog price. Describe the relationship that you see. Are there any outliers?

14.33 When it rains, it pours. Figure 14.12 plots the highest yearly precipitation ever recorded in each state against the highest daily precipitation ever recorded in that state. The points for Alaska (AK), Hawaii (HI), and Texas (TX) are marked on the scatterplot.

14.34 How many corn plants is too many? How much corn per acre should a farmer plant to obtain the highest yield? To find the best planting rate, do an experiment: plant at different rates on several plots of ground and measure the harvest. Here are data from such an experiment:

14.35 Why so small? Make a scatterplot of the following data:

Use your calculator to show that the correlation is about 0.4. What feature of the data is responsible for reducing the correlation to this value despite a strong straight-line association between x and y in most of the observations?

14.36 Ecological correlation. Many studies reveal a positive correlation between income and number of years of education. To investigate this, a researcher makes two plots.

Plot 1: Plot the number of years of education (the explanatory variable) versus the average annual income of all adults having that many years of education (the response variable).

Plot 2: Plot the number of years of education (the explanatory variable) versus the individual annual incomes of all adults (the response variable).

Which plot will display a stronger correlation? (Hint: Which plot will display a greater amount of scatter? In particular, will the variation from individual to individual having the same number of years of education create more or less scatter in Plot 2 compared with plotting the average incomes in Plot 1? What effect will increased scatter have on the strength of the association we observe?)

Note: A correlation based on averages rather than on individuals is called an ecological correlation. Correlations based on averages can be misleading if they are interpreted to be about individuals.

14.37 Ecological correlation again. In Exercise 14.11 (page 330), would the association be stronger, weaker, or the same if the data given listed the GPAs of individual students (rather than averages) and the distance they lived from campus?