Question

the body temperatures of a group of healthy adults have a bell - shaped distribution with a mean of 98.01°f and a standard deviation of 0.62°f. using the empirical rule, find each approximate percentage below.
a. what is the approximate percentage of healthy adults with body temperatures within 2 standard deviations of the mean, or between 96.77°f and 99.25°f?
b. what is the approximate percentage of healthy adults with body temperatures between 96.15°f and 99.87°f?
a. approximately 95% of healthy adults in this group have body temperatures within 2 standard deviations of the mean, or between 96.77°f and 99.25°f.
(type an integer or a decimal. do not round.)
b. approximately % of healthy adults in this group have body temperatures between 96.15°f and 99.87°f.
(type an integer or a decimal. do not round.)

Explanation:

Step1: Recall the empirical rule

The empirical rule for a normal - distribution states that approximately 68% of the data lies within 1 standard deviation of the mean, approximately 95% lies within 2 standard deviations of the mean, and approximately 99.7% lies within 3 standard deviations of the mean.

Step2: Calculate the number of standard - deviations for part b

The mean $\mu = 98.01^{\circ}F$ and the standard deviation $\sigma=0.62^{\circ}F$.
For the lower bound $x_1 = 96.15^{\circ}F$, the z - score $z_1=\frac{x_1 - \mu}{\sigma}=\frac{96.15 - 98.01}{0.62}=\frac{- 1.86}{0.62}=-3$.
For the upper bound $x_2 = 99.87^{\circ}F$, the z - score $z_2=\frac{x_2 - \mu}{\sigma}=\frac{99.87 - 98.01}{0.62}=\frac{1.86}{0.62}=3$.
Since the data lies within 3 standard deviations of the mean, approximately 99.7% of healthy adults have body temperatures between $96.15^{\circ}F$ and $99.87^{\circ}F$.

Answer:

a. 95
b. 99.7