For Exercises 2.23 and 2.24, see page 75; and for 2.25 and 2.26, see pages 77–78.
2.27 Companies of the world
Refer to Exercise 1.118 (page 61), where we examined data collected by the World Bank on the numbers of companies that are incorporated and are listed on their country's stock exchange at the end of the year. In Exercise 2.10 (page 71), you examined the relationship between these numbers for 2012 and 2002.
inccom
2.27
(a) r=0.9589. (b) Yes, there is a very strong linear relationship between the 2002 and 2012 data.
2.28 Companies of the world
Refer to the previous exercise and to Exercise 2.11 (page 72). Answer parts (a) and (b) for 2012 and 1992. Compare the correlation you found in the previous exercise with the one you found in this exercise. Why do they differ in this way?
inccom
2.29 A product for lab experiments
In Exercise 2.17 (page 73), you described the relationship between time and count for an experiment examining the decay of barium.
decay
2.29
(a) Yes, the data points form a nice curve. (b) r=−0.964. (c) No, the data shows a curve, not a line; a transformation is needed to get a better relationship.
2.30 Use a log for the radioactive decay
Refer to the previous exercise and to Exercise 2.18 (page 73), where you transformed the counts with a logarithm.
decay
2.31 Brand-to-brand variation in a product
In Exercise 2.12 (page 73), you examined the relationship between percent alcohol and calories per 12 ounces for 175 domestic brands of beer.
beer
2.31
(a) r=0.9048. (b) Yes, the relationship between percent alcohol and calories is quite linear, so the correlation gives a good numerical summary of the relationship.
2.32 Alcohol and carbohydrates in beer revisited
Refer to the previous exercise. Delete any outliers that you identified in Exercise 2.12.
beer
2.33 Marketing in Canada
In Exercise 2.14 (page 73), you examined the relationship between the percent of the population over 65 and the percent under 15 for the 13 Canadian provinces and territories.
canadap
2.33
(b) r=−0.851. (c) No, although the relationship is mostly linear, there is an outlier, Nunavut, with a high percent of under 15 and a very low percent of over 65.
2.34 Nunavut
Refer to the previous exercise.
canadap
2.35 Education spending and population with logs
In Example 2.3 (page 66), we examined the relationship between spending on education and population, and in Exercise 2.23 (page 75), you found the correlation between these two variables. In Example 2.6 (page 69), we examined the relationship between the variables transformed by logs.
edspend
2.35
(a) r=0.9808. (b) The correlation went up from 0.9798 before taking the logs to 0.9808 after. Although the correlation went up a little bit, the log didn't help much with the explanation of the data.
2.36 Are they outliers?
Refer to the previous exercise. Delete the four states with high values.
edspend
2.37 Fuel efficiency and CO2 emissions
In Example 2.7 (pages 70–71), we examined the relationship between highway MPG and CO2 emissions for 1067 vehicles for the model year 2014. Let's examine the relationship between the two measures of fuel efficiency in the data set, highway MPG and city MPG.
canfuel
2.37
(b) The relationship is somewhat linear but may also be slightly curved. Hwy MPG and City MPG increase together. (c) r=0.9255. (d) The correlation is a decent numerical summary because the data are somewhat linear, but a curve may provide a better description of the relationship.
2.38 Consider the fuel type
Refer to the previous exercise and to Figure 2.6 (page 71), where different colors are used to distinguish four different types of fuels used by these vehicles.
canfuel
2.39 Match the correlation
The Correlation and Regression applet at the text website allows you to create a scatterplot by clicking and dragging with the mouse. The applet calculates and displays the correlation as you change the plot. You will use this applet to make scatterplots with 10 points that have correlation close to 0.7. The lesson is that many patterns can have the same correlation. Always plot your data before you trust a correlation.
2.40 Stretching a scatterplot
Changing the units of measurement can greatly alter the appearance of a scatterplot. Consider the following data:
stretch
x | −4 | −4 | −3 | 3 | 4 | 4 |
y | 0.5 | −0.6 | −0.5 | 0.5 | 0.5 | −0.6 |
2.41 CEO compensation and stock market performance
An academic study concludes, “The evidence indicates that the correlation between the compensation of corporate CEOs and the performance of their company's stock is close to zero.” A business magazine reports this as “A new study shows that companies that pay their CEOs highly tend to perform poorly in the stock market, and vice versa.” Explain why the magazine's report is wrong. Write a statement in plain language (don't use the word “correlation”) to explain the study's conclusion.
2.41
The magazine report is wrong because they are interpreting a correlation close to 0 as a negative association rather than no association.
2.42 Investment reports and correlations
Investment reports often include correlations. Following a table of correlations among mutual funds, a report adds, “Two funds can have perfect correlation, yet different levels of risk. For example, Fund A and Fund B may be perfectly correlated, yet Fund A moves 20% whenever Fund B moves 10%.” Write a brief explanation, for someone who does not know statistics, of how this can happen. Include a sketch to illustrate your explanation.
2.43 Sloppy writing about correlation
Each of the following statements contains a blunder. Explain in each case what is wrong.
2.43
(a) The correlation is not dependent on order and remains the same between two variables regardless of order. (b) A correlation is reserved for quantitative data; because color is categorical, it cannot have any correlation. (c) A correlation can never exceed 1, which indicates a perfect linear relationship.