Body Temperatures Based on the sample results in Data Set 3 “Body Temperatures” in Appendix B, assume that human body temperatures are normally distributed with a mean of 98.20°F and a standard deviation of 0.62°F.

a. According to emedicinehealth.com, a body temperature of 100.4°F or above is considered to be a fever. What percentage of normal and healthy persons would be considered to have a fever? Does this percentage suggest that a cutoff of 100.4°F is appropriate?

b. Physicians want to select a minimum temperature for requiring further medical tests. What should that temperature be, if we want only 2.0% of healthy people to exceed it? (Such a result is a false positive, meaning that the test result is positive, but the subject is not really sick.)

Body temperature follows a normal distribution with a mean of $98.20°\text{F}$and a standard deviation of $0.62°\text{F}$

Let X be the human body temperature in degree Fahrenheit that follows normal distribution.

Thus,

$$\begin{gathered} X~N\left(\mu ,{\sigma }^2\right) \\ ~ N\left(\text{98.20}, {\text{0.62}}^2\right) \end{gathered}$$

a.

A body temperature of or above is considered to be a fever, that is, $X⩾100.4$.

The z score associated with a human body temperature of 100.4:

$$\begin{gathered} z= \frac{x-\mu }{\sigma } \\ = \frac{100.4-98.20}{0.62} \\ = 3.5484 \\ =3.55 \end{gathered}$$

The probability that a body temperature is$100.4°\text{F}$or above is expressed as:

$$\begin{gathered} P\left(X⩾100.4\right)= P\left(Z⩾3.55\right) \\ = 1- P(Z⩽3.55) ...1) \end{gathered}$$

By referring to the standard normal table,the cumulative probability of 3.55 is obtained from the intersection of the row with label 3.5 and up and column label 0.0,which is obtained as 0.9999.

Substitute the value in equation 1):

$$\begin{gathered} P\left(X⩾100.4\right) = 1-P\left(Z⩽3.55\right) \\ = 1- 0.9999 \\ =0.0001 \end{gathered}$$

The probability is that a body temperature of$100.4°\text{F}$or above is 0.0001. The percentage of people considered to have a fever is 0.01%.

The result suggests that the percentage is appropriate as the exact temperature is rarely possible for healthy humans. Thus, the cut-off is appropriate.

b.

The probability that2.0% of healthy people exceed a specific body temperatureximplies that 98% of healthy people have a body temperature lesser than x.

Thus,

$$\begin{gathered} P\left(X<x\right)=P\left(Z<z\right) \\ = 0.98 \end{gathered}$$

And

$z=\frac{x-\mu }{\sigma }$

From the standard normal table, the area of 0.98 is observed corresponding to the row value 2.0 and column value between 0.05 and 0.06, which implies the z score, is 2.05.

$$\begin{gathered} z= \frac{x-\mu }{\sigma } \\ x= \mu +\sigma \times z \\ = 98.20+2.05\times 0.62 \\ =99.471 \end{gathered}$$

Thus, 2.0% of healthy people exceed the temperature of , which can be used for further physical tests.