Overview
Read Chapter 4, and answer the exercise after chapter4: 4.1, 4.2, 4.3
1. In Chapter 2 we discussed the di↵erences between statements about correlations and other factual statements that do not convey information about a correlation. Now that you have a deeper understanding that correlation requires variation, consider the following statements. Which ones describe a correlation and which ones do not?
(a) Most top performing schools have small student bodies.
(b) Married people are typically happier than unmarried people
(c) Among professionals, taller basketball players tend to have lower free-throw per- centages than shorter players.
(d) The locations in the U.S. with the highest cancer rates are typically small towns.
(e) Older houses are more likely to have lead paint than newer ones.
(f) Most colds caught in Cook County are caught on cold days. (This one also doubles as a tongue twister.
2. At least 20 billionaires dropped out of college before earning their fortunes, including Bill Gates and Mark Zuckerberg.
(a) Does this mean that dropping out of college is correlated with becoming a bil- lionaire? Why or why not?
(b) Draw the two-by-two table that would allow you to assess whether dropping out of college is correlated with becoming a billionaire. Let’s assume that exactly 20 people have dropped out of college and become billionaires, so you know what to put in one of the four cells. Make your best guess for the other cells. At the time of this writing, there are about 7.8 billion people in the world, and about 2,000 billionaires. Do you think there is a positive or negative correlation between dropping out of college and becoming a billionaire?
(c) Given ytour guesses from part (b), what proportion of the non-billionaires would need to be college dropouts in order for the correlation to be negative? What proportion of the non-billionaires would need to be college dropouts in order for the correlation to be positive?
(d) If you’re currently a college student deciding whether you want to drop out in the hopes of becoming a billionaire, you may want to restrict attention to people who actually started college. Do you think the correlation between dropping out of college and becoming a billionaire is more or less likely to be positive if we restrict attention to just people who start college?
(e) About 7 percent of the world’s population has a college degree. And about a third of those who start college complete it. If we assume that everyone who becomes a billionaire started college, you should now have all the information you need to assess the correlation between becoming a billionaire and dropping out of college among those who start college. Is it positive, negative, or zero?
3. Identify one recent case where an analyst made the mistake discussed in this chapter. That is, find a case where someone (at least implicitly) makes a claim about a cor- relation but they don’t have variation in one of their variables. Your example might come from a newspaper article, an academic study, a policy memo, or a statement from a politician or business leader. (a) Summarize the claim being made (perhaps implicitly) and explain why the evi- dence does not necessarily support the claim
(b) Explain what additional data collection and analysis would allow the analyst to assess the correlation of interest.
(c) Draw a two-by-two table that illustrates your argument, and discuss what the unknown numbers in the table would have to be in order for the correlation of interest to be positive, negative, or zero.