Answer to Question #165921 in Statistics and Probability for Angela

Question #165921

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.)


1
Expert's answer
2021-02-24T06:03:14-0500

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.


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