Question

Assume that human body temperatures are normally distributed with a mean of

98.19°F

and a standard deviation of

0.63°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.)

a. The percentage of normal and healthy persons considered to have a fever is

nothing​%.

​(Round to two decimal places as​ needed.)

Does this percentage suggest that a cutoff of

100.6°F

is​ appropriate?

A.

​No, because there is a large probability that a normal and healthy person would be considered to have a fever.

B.

​No, because there is a small probability that a normal and healthy person would be considered to have a fever.

C.

​Yes, because there is a large probability that a normal and healthy person would be considered to have a fever.

D.

​Yes, because there is a small probability that a normal and healthy person would be considered to have a fever.

b. The minimum temperature for requiring further medical tests should be

nothingdegrees Upper F°F

if we want only​ 5.0% of healthy people to exceed it.

​(Round to two decimal places as​ needed.)

Answer 1

Let X be the random variable denoting the temperature of human body. X ~ N(98.19, 0.63) i.e. (X - 98.19)/0.63 ~ N(0,1).

a.The probability that the temperature will be greater than 100.6°F = P(X > 100.6) = P[(X - 98.19)/0.63 > (100.6 - 98.19)/0.63] = P[(X - 98.19)/0.63 > 3.8254] = 1- P[(X - 98.19)/0.63 < 3.8254] = 1 - (3.8254) = 1 - 0.9999 = 0.0001

The cutoff is appropriate since there is a small probability that a normal and healthy person would be considered to have a fever.

b. Let t be the cutoff temperature if we want only 5% of the health people to exceed it.

Hence, P(X > t) = 0.05 i.e. P[(X - 98.19)/0.63 > (t - 98.19)/0.63] = 0.05 i.e. 1 - P[(X - 98.19)/0.63 < (t - 98.19)/0.63] = 0.05 i.e. [(t - 98.19)/0.63] = 0.05 i.e. (t - 98.19)/0.63 = (0.05) = 1.64 i.e. t = 98.19 + (0.63 * 1.64) = 99.2232°F

So, the required minimum temperature is 99.2232°F (Ans).