Response to the following problem:
Allstate Insurance Company identified the 10 safest and 10 least-safe U.S. cities from among the 500 largest cities in the United States, based on the mean number of years drivers went between automobile accidents. The cities on both lists were the smaller cities on the list of the 500 largest. Using facts about the sampling distribution model of the sample mean, why is this not surprising?
In the smaller cities the distribution of the mean number of years drivers go between accidents is bimodal, which makes it more likely they will be in the safest and least-safe cities. Large cities are not on the safest list because their mean time between accidents is reduced by people who "stage" accidents for purposes of insurance fraud. More people in larger cities drive older cars that are more accident prone. Accident statistics are more accurate in the smaller cities. Smaller cities are safer, but a few accident-prone drivers (outliers) in some small cities decreases the mean number of years between accidents, resulting in these cities being listed on the least-safe list. Larger cities have a higher incidence of hit-and-run accidents. Traffic congestion in larger cities decreases the mean number of years drivers go between accidents but not enough to place larger cities in the least-safe group. Cities in which the mean is based on a smaller number of drivers will have greater variation in their means and are therefore more likely to be both safest and least safe.