Chapter 18.2, Problem 4JC

Every 10 years, the United States takes a census. The census tries to count every resident. There have been two forms, known as the “short form,” answered by most people, and the “long form,” slogged through by about one in six or seven households chosen at random. (For the 2010 Census, the long form was replaced by the American Community Survey.) According to the Census Bureau (www.census.gov), “. . . each estimate based on the long form responses has an associated confidence interval.”

The Census Bureau goes on to say, “These confidence intervals are wider . . . for geographic areas with smaller populations and for characteristics that occur less frequently in the area being examined (such as the proportion of people in poverty in a middle-income neighborhood).”

To deal with this problem, the Census Bureau reports long-form data only for “. . . geographic areas from which about two hundred or more long forms were completed—which are large enough to produce good quality estimates. If smaller weighting areas had been used, the confidence intervals around the estimates would have been significantly wider, rendering many estimates less useful. . . .”

4. Suppose the Census Bureau decided to report on areas from which only 50 long forms were completed. What effect would that have on a 95% confidence interval for, say, the mean cost of housing? Specifically, which values used in the formula for the margin of error would change? Which would change a lot and which would change only slightly?