Question

Assume that human body temperatures are normally distributed with a mean of 98.18°F and a standard deviation of 0.61°F.

a. A hospital uses 100.6°F as the lowest temperature 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 100.6°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​ 5.0% 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.)

Answer 1

Mean temprature, $\mu=98.18^{\circ}F$

Standard deviation, $\sigma=0.61^{\circ}F$

(a) The cut off temprature P=$100.6^{\circ}F$

The z-value is given by-

$z=\dfrac{p-\mu}{\sigma}$

$=\dfrac{100.6^{\circ}-98.18^{\circ}}{0.61^{\circ}}=\dfrac{2.12}{0.61}=3.967$

Found the Standard normal (z) curve a z of 3.9 is basically 1.0000 or 100.00%.

This means there are no false positives and probably a lot of false negatives. The cutoff is too high.

With 5.0% false positives allowed, to find the new cutoff first have to determine the z value for 100.0% - 5.0% i.e 95%

(B) From the Standard normal (z) curve we find that 95.0% is $1.645.$

$1.645 = \dfrac{\text{(new cutoff} - 98.18)}{0.61}$ .

$1.645 \times 0.62 = \text{new cutoff} - 98.18 \\\Rightarrow1.02 = \text{new cutoff }- 98.18$

$\text{New cutoff} = 1.02 + 98.18$ .

New cutoff equals 99.2$^{\circ}$ Fahrenheit.