Chapter 6, Problem 6.90SE

Normal Distribution? The chest measurements for 5738 Scottish militiamen in the early 19th century are given below. ${}^9$ Chest sizes are measured in inches, and each observation reports the number of soldiers with that chest size.

Notice the approximate normality of the histogram of the 5738 chest measurements.

a. The mean of this distribution is $\bar{x}=39.83$ and the standard deviation is $s=2.05$ . What is the 95th percentile of this distribution based on a normal curve with $\mu =39.83$ and $\sigma =2.05$ ?

b. Find the empirical estimate of the 95th percentile and compare with your answer in part a. (HINT: The 95th percentile will be in position $95\left(n+1\right)=.95\times 5739=5452.05$ from the left tail of the distribution or in position $5738-5452.05=285.95$ from the right tail of the distribution.)

c. Find the 90th percentile of this distribution based on a normal curve with $\mu =39.83$ and $\sigma =2.05$ . What is the value of the empirical 90th percentile? How does it compare with the value assuming normality?