The National Health Statistics Reports dated Oct. 22, 2008 stated that for a sample size of 277 18- year-old American males, the sample mean waist circumference was 86.3 cm. A somewhat compli- cated method was used to estimate various popula- tion percentiles, resulting in the following values:
5th 10th 25th 50th 75th 90th 95th
69.6 70.9 75.2 81.3 95.4 107.1 116.4
a. Is it plausible that the waist size distribution is at least approximately normal? Explain your reasoning. If your answer is no, conjecture the shape of the population distribution.
b. Suppose that the population mean waist size is 85 cmand that the population standard deviation is 15 cm. How likely is it that a random sample of 277 individuals will result in a sample mean waist size of at least 86.3 cm?
c. Referring back to (b), suppose now that the population mean waist size in 82 cm. Now what is the (approximate) probability that the sample mean will be at least 86.3 cm? In light of this calculation, do you think that 82 cm is a reasonable value for μ?