According to a random sample taken at 12 A.M., body temperatures of healthy adul

Question

According to a random sample taken at 12 A.M., body temperatures of healthy adults have a bell-shaped distribution with a mean of 98.09 degrees°F and a standard deviation of 0.56 degrees°F. 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 possible body temperatures that are within 2 standard deviations of the mean? At least what percent of healthy adults have body temperatures within 2 standard deviations of 98.09 degrees°F?

Explanation / Answer

CHEBYSHEV’S THEOREM:

For any numerical data set,
at least 3/4 of the data lie within two standard deviations of the mean, that is, in the interval with endpoints x¯±2s x¯±2s for samples and with endpoints ±2 for populations;
at least 8/9 of the data lie within three standard deviations of the mean, that is, in the interval with endpoints x¯±3s for samples and with endpoints ±3 for populations;
at least 1±1/k2 of the data lie within kk standard deviations of the mean, that is, in the interval with endpoints x¯±ks for samples and with endpoints ±k±k for populations, where kk is any positive whole number that is greater than 11 .

a) Hence for the present question.
Lets assume there are 100 healthy adults, so 3/4 of that adults will have body temperatures within range 96.97 degrees°F
and 99.21 degrees°F.
Hence we know that 3*100/4 = 75% of healthy adults fall in range of 2 standard deviations of the mean.

b) The maximum and minimun body temperature is 99.21 degrees°F and 96.97 degrees°F respectively

c) At least 75 % of healthy adults have body temperatures within 2 standard deviations of 98.09 degrees°F

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