According to a random sample taken at 12 A. M., body temperatures of healthy adults have a bell-shaped distribution with a mean of 98.1398.13degrees°F and a standard deviation of 0.620.62degrees°F. Using Chebyshev's theorem, what do we know about the percentage of healthy adults with body temperatures that are within 33 standard deviations of the mean? What are the minimum and maximum possible body temperatures that are within 33 standard deviations of the mean?

Complete 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 98.13degrees°F and a standard deviation of 0.62degrees°F. Using Chebyshev's theorem, what do we know about the percentage of healthy adults with body temperatures that are within 3 standard deviations of the mean?

What are the minimum and maximum possible body temperatures that are within 3 standard deviations of the mean?

a) what do we know about the percentage of healthy adults with body temperatures that are within 3 standard deviations of the mean?

This means that at least 88.89% of healthy adults with body temperatures that are within 3 standard deviations of the mean.

b).What are the minimum and maximum possible body temperatures that are within 3 standard deviations of the mean?

The minimum body temperature within 3 standard deviations of the mean = 96.27°F

The maximum body temperature within 3 standard deviations of the mean = 99.99°F

Step-by-step explanation:

We solve this question using Chebyshev's theorem.

Chebyshev's theorem states that:

1) At least 3/4 of the data lies within two standard deviations of the mean. This means, the interval endpoints :

Mean ± 2Standard deviation

2) At least 8/9 of the data lies within three standard deviations of the mean. This means, the interval endpoints :

Mean ± 3Standard deviation

From the above question,

Mean = 98.13°F

Standard deviation = 0.62°F

The question specified 3 Standard deviation from the mean. Hence,

a) At least 8/9 of the data lies within three standard deviations of the mean. This means, the interval endpoints :

Mean ± 3Standard deviation

Converting this to percentage

8/9 × 100 = 88.89%

Therefore, this means that at least 88.89% of healthy adults with body temperatures that are within 3 standard deviations of the mean.

b) For question B,to find the minimum, maximum value

At least 8/9 of the data lies within three standard deviations of the mean. This means, the interval endpoints :

Mean ± 3Standard deviation

Mean = 98.13°F

Standard deviation = 0.62°F

98.13 - 3 × 0.62

= 96.27°F

98.13 + 3 × 0.62

= 99.99°F

Therefore, the minimum body temperature within 3 standard deviations of the mean = 96.27°F

The maximum body temperature within 3 standard deviations of the mean = 99.99°F

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