Question

The Empirical Rule Based on Data Set 3 “Body Temperatures” in Appendix B, body temperatures of healthy adults have a bell-shaped distribution with a mean of 98.20°F and a standard deviation of 0.62°F. Using the empirical rule, what is the approximate percentage of healthy adults with body temperatures

a. within 1 standard deviation of the mean, or between 97.58°F and 98.82°F?

b. between 96.34°F and 100.06°F?

Short answer

a. 68% of adults have body temperatures within one standard deviation of the mean.

b. 99.7% of adults have body temperatures between 93.34 ${}^{\text{o}}\text{F}$and 110.06${}^{\text{o}}\text{F}$.

Step-by-step solution

Given information

A dataset consisting of body temperatures of adults is utilized.

The mean temperature is given as 98.2 ${}^{\text{o}}\text{F}$.

The standard deviation of the temperatures is given as 0.62${}^{\text{o}}\text{F}$.

Empirical rule

For a set of values that have a bell-shaped distribution, the empirical rule can be used.

The rule consists of three main properties:

• 68% of observations lie between or between$\left(\mu -\sigma ,\mu +\sigma \right)$ one standard deviation of the mean.
• 95% of observations lie between $\left(\mu -2\sigma ,\mu +2\sigma \right)$or between two standard deviations of the mean.
• 99.7% of observations lie between $\left(\mu -3\sigma ,\mu +3\sigma \right)$or between three standard deviations of the mean.

a.

The given temperatures equal to (97.58${}^{\text{o}}\text{F}$,98.82${}^{\text{o}}\text{F}$) are above and below one standard deviation of the mean.

Therefore, it can be concluded that 68% of adults have a body temperature between (97.58${}^{\text{o}}\text{F}$,98.82${}^{\text{o}}\text{F}$).

b.

The temperatures are given to be 96.34${}^{\text{o}}\text{F}$and 100.06${}^{\text{o}}\text{F}$.

$$\begin{gathered} \mu -3\sigma =98.2-3\left(0.62\right) \\ =96.34 \\ \mu +3\sigma =98.2+3\left(0.62\right) \\ =100.06 \end{gathered}$$

The given values are above and below three standard deviations of the mean.

Therefore, it can be concluded that 99.7% of adults have a body temperature between (96.34${}^{\text{o}}\text{F}$,100.06${}^{\text{o}}\text{F}$).