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: Recall the empirical rule

The empirical rule for a normal - distribution states that about 68% of the data lies within 1 standard deviation of the mean, about 95% lies within 2 standard deviations of the mean, and about 99.7% lies within 3 standard deviations of the mean. The formula for the range within \(k\) standard deviations of the mean is \(\mu\pm k\sigma\), where \(\mu\) is the mean and \(\sigma\) is the standard deviation.

Step2: Calculate the number of standard - deviations

Given \(\mu = 68\) inches and \(\sigma=2.5\) inches. For the lower value \(x_1 = 65.5\) inches, we calculate \(z_1=\frac{x_1-\mu}{\sigma}=\frac{65.5 - 68}{2.5}=\frac{- 2.5}{2.5}=-1\). For the upper value \(x_2 = 70.5\) inches, we calculate \(z_2=\frac{x_2-\mu}{\sigma}=\frac{70.5 - 68}{2.5}=\frac{2.5}{2.5}=1\).

Step3: Apply the empirical rule

Since the values 65.5 and 70.5 are 1 standard deviation below and 1 standard deviation above the mean respectively, by the empirical rule, the percentage of data within 1 standard deviation of the mean is 68%.

Answer:

68%