Question

Chebyshev’sTheorem 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 data from 12 AM on day 2). Using Chebyshev’s theorem, what do we know about the percentage of healthy adults with body temperatures that are within 2 standard deviations of the mean? What are the minimum and maximum body temperatures that are within 2 standard deviations of the mean??

Short answer

75% of adults have a body temperature within two standard deviations of the mean.

The maximum body temperature that falls within two standard deviations of the mean is equal to 99.44 ${}^{\text{o}}\text{F}$, and the minimum body temperature that falls within two standard deviations of the mean is equal to 96.96 ${}^{\text{o}}\text{F}$.

Step-by-step solution

Given information

The body temperatures of adults are provided in degrees Fahrenheit.

The mean body temperature is equal to 98.20${}^{\text{o}}\text{F}$, and the standard deviation of the body temperature is equal to 0.62 ${}^{\text{o}}\text{F}$.

Chebyshev’s theorem

For a dataset that approximately follows the bell-shaped probability distribution, Chebyshev’s theorem can be applied. The theorem states that at least $1-\frac{1}{K^2}$values lie between K standard deviations of the mean.

Here, K=2

$$\begin{gathered} 1-\frac{1}{K^2}=1-\frac{1}{2^2} \\ =1-\frac{1}{4} \\ =\frac{3}{4} \end{gathered}$$

Hence, $\frac{3}{4}$or approximately 75%of healthy adults have a body temperature within two standard deviations of the mean.

According to the given value of K, the limits come out to be equal to:

$$\begin{gathered} \mu -2\sigma =98.20-2\left(0.62\right) \\ =96.96 \\ \mu +2\sigma =98.20+2\left(0.62\right) \\ =99.44 \end{gathered}$$

Therefore, the maximum value of body temperature between these limits is equal to 99.44 ${}^{\text{o}}\text{F}$, and the minimum valueof body temperature between these limits is equal to 96.96 ${}^{\text{o}}\text{F}$.