Chapter 5.5, Problem 66E

The paper “How Lead Exposure Relates to Temporal Changes in IQ, Violent Crime, and Unwed Pregnancy” (Environmental Research [2000]: 1–22) investigated whether childhood lead exposure is related to criminal behavior in young adults. Using historical data, the author paired y = Assault rate (assaults per 100,000 people) for each year from 1964 to 1997 with a measure of lead exposure (tons of gasoline lead per 1000 people) 23 years earlier. For example, the lead exposure from 1974 was paired with the assault rate from 1997.

The author chose to go back 23 years for lead exposure because the highest number of assaults are committed by people in their early twenties, and 23 years earlier would represent a time when those in this age group were infants.

A least-squares line was used to describe the relationship between assault rate and lead exposure 23 years prior. Summary statistics given in the paper are

Use the information provided to answer the following questions.

1. a. What is the value of the correlation coefficient for x = Lead exposure 23 years prior and y = Assault rate? Interpret this value. Is it reasonable to conclude that increased lead exposure is the cause of increased assault rates? Explain.
2. b. What is the equation of the least-squares line? Use the line to predict assault rate in a year in which gasoline lead exposure 23 years prior was 0.5 tons per 1000 people.
3. c. What proportion of year-to-year variability in assault rates can be explained by the relationship between assault rate and gasoline lead exposure 23 years earlier?
4. d. The graph in the figure on the next page appeared in the paper. Note that this is not a scatterplot of the (x, y) pairs—it is two separate time series plots. The time scale 1941, 1942, ..., 1986 is the time scale used for the lead exposure data and the time scale 1964, 1965, ..., 2009 is used for the assault rate data. Also note that at the time the graph was constructed, assault rate data was only available through 1997. Spend a few minutes thinking about the information contained in this graph and then briefly explain what aspect of this graph accounts for the reported positive correlation between assault rate and lead exposure 23 years prior.