American women’s heights are approximately normally distributed with mean 63.8 inches and standard deviation 4.2 inches. American men’s heights are approximately normally distributed with mean 69.4 inches and standard deviation 4.7 inches (Source: National Health Statistics Report).
a. At what doorway height would 95% of men be able to pass through without having to duck? Round to the ones place.
b. What percentage of women would have to duck to pass
through a doorway at the height you found in part (a)? Round to the second decimal place.
c. The height of a doorway is typically 80 inches. What percentage of men can pass through such a doorway without ducking their head? Round to the second decimal place.
d. For each gender, the mean height increased over time. For each of parts (a), (b), and (c), determine whether the correct answer to the question in the future will be larger, smaller, or impossible to know from the given information. Assume that for each gender, the standard deviation of heights will be constant.