Question

in an all boys school, the heights of the student body are normally distributed with a mean of 69 inches and a standard deviation of 3 inches. using the empirical rule, determine the interval of heights that represents the middle 99.7% of male heights from this school.
answer attempt 1 out of 2

Explanation:

Step1: Recall the empirical rule for normal distribution

The empirical rule (also known as the 68 - 95 - 99.7 rule) states that for a normal distribution, approximately 99.7% of the data lies within \( \mu - 3\sigma \) and \( \mu+ 3\sigma \), where \( \mu \) is the mean and \( \sigma \) is the standard deviation.

Given that \( \mu = 69 \) inches and \( \sigma=3 \) inches.

Step2: Calculate the lower bound

The lower bound of the interval is \( \mu - 3\sigma \). Substitute \( \mu = 69 \) and \( \sigma = 3 \) into the formula:
\( \mu-3\sigma=69 - 3\times3=69 - 9 = 60 \)

Step3: Calculate the upper bound

The upper bound of the interval is \( \mu + 3\sigma \). Substitute \( \mu = 69 \) and \( \sigma=3 \) into the formula:
\( \mu + 3\sigma=69+3\times3=69 + 9=78 \)

Answer:

The interval of heights that represents the middle 99.7% of male heights is from 60 inches to 78 inches, so the interval is \( (60, 78) \) (or in the form of \( 60 \) to \( 78 \) inches).