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 5 inches. using the empirical rule, determine the interval of heights that represents the middle 95% of male heights from this school.

Explanation:

Step1: Recall the empirical rule

The empirical rule states that for a normal distribution, about 95% of the data lies within \( \mu\pm2\sigma\), where \( \mu\) is the mean and \( \sigma\) is the standard deviation.

Step2: Calculate the lower bound

The lower bound is \( \mu - 2\sigma\). Given \( \mu = 68\) and \( \sigma=5\), we have \( 68-2\times5=68 - 10=58\).

Step3: Calculate the upper bound

The upper bound is \( \mu + 2\sigma\). So, \( 68+2\times5=68 + 10=78\).

Answer:

The interval of heights that represents the middle 95% of male heights is \( (58,78)\) inches.