Question

the body - temperatures of a group of healthy adults have a bell - shaped distribution with a mean of 98.12°f and a standard deviation of 0.44°f. using the empirical rule, find each approximate percentage below.
a. what is the approximate percentage of healthy adults with body temperatures within 3 standard deviations of the mean, or between 96.80°f and 99.44°f?
b. what is the approximate percentage of healthy adults with body temperatures between 97.24°f and 99.00°f?
a. approximately □% of healthy adults in the group have body temperatures within 3 standard deviations of the mean, or between 96.80°f and 99.44°f.
(type an integer or a decimal. do not round.)

Explanation:

Step1: Recall the empirical rule for normal - distribution

The empirical rule (68 - 95 - 99.7 rule) states that for a bell - shaped (normal) distribution: Approximately 99.7% of the data lies within 3 standard deviations of the mean.

Step2: Calculate the z - scores for part b

First, find the z - scores. The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $\mu = 98.12^{\circ}F$ is the mean and $\sigma = 0.44^{\circ}F$ is the standard deviation.
For $x = 97.24^{\circ}F$, $z_1=\frac{97.24 - 98.12}{0.44}=\frac{- 0.88}{0.44}=-2$.
For $x = 99.00^{\circ}F$, $z_2=\frac{99.00 - 98.12}{0.44}=\frac{0.88}{0.44}=2$.
According to the empirical rule, approximately 95% of the data lies within 2 standard deviations of the mean.

Answer:

a. 99.7
b. 95