Question

in an all boys school, the heights of the student body are normally distributed with a mean of 68 inches and a standard deviation of 2.5 inches. using the empirical rule, what percentage of the boys are between 65.5 and 70.5 inches tall?

Explanation:

Step1: Calculate the number of standard deviations from the mean

The formula to find the number of standard deviations \(z=\frac{x - \mu}{\sigma}\), where \(\mu = 68\) (mean), \(\sigma=2.5\) (standard deviation).
For \(x = 65.5\): \(z_1=\frac{65.5 - 68}{2.5}=\frac{- 2.5}{2.5}=-1\)
For \(x = 70.5\): \(z_2=\frac{70.5 - 68}{2.5}=\frac{2.5}{2.5}=1\)

Step2: Apply the empirical rule

The empirical rule (68 - 95 - 99.7 rule) for a normal distribution states that approximately \(P(\mu-\sigma<X<\mu + \sigma)\) is the proportion of data within \(1\) standard deviation of the mean.

Answer:

68%