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According to a random sample taken at 12 A.M., body temperatures of healthy adults have a bell-shaped distribution with a mean of 98.07°F and a standard deviation of0.56°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 deviationsthe mean?At least % of healthy adults have body temperatures within 3 standard deviations of 98.07°F.(Round to the nearest percent as needed.)

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 98.07°F and a standard deviation of
0.56°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
the mean?
At least % of healthy adults have body temperatures within 3 standard deviations of 98.07°F.
(Round to the nearest percent as needed.)
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According to a random sample taken at 12 A.M., body temperatures of healthy adults have a bell-shaped distribution with a mean of 98.07°F and a standard deviation of 0.56°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 the mean? At least % of healthy adults have body temperatures within 3 standard deviations of 98.07°F. (Round to the nearest percent as needed.)

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Step 1

Chebyshev’s inequality:

Chebyshev’s rule is appropriate for any distribution. That is, Chebyshev’s inequality applies to all distributions, regardless of shape. Moreover, it provides the minimum percentage of the observation that lies within k standard deviations of the mean.

  • It is possible that very few measurements will fall within 1 standard deviation of the mean.
  • If k = 2, at least 3/4 of the measurements lie within 2 standard deviations to either side of the mean.
  • If k = 3, at least 8/9 of the measurements lie within 3 standard deviations to either side of the mean.
  • Generally, for any number k greater than 1, at least (1-1/k2) of the measurements will fall within k standard deviations of the mean.
Step 2

It was found that the body temperatures of healthy adults have a bell shaped distribution with a mean of 98.070F and a standard deviation of 0.560F.

Using Chebyshev’s theorem, at least 8/9 of the measurements lie within 3 standard deviations to either side of the mean.

Thus, 88.9% of healthy adults have body temperatures are within 3 standard deviations of the mean.

That is, according to Chebyshev’s rule, at least 88.9% of the observations fall within the interval x-bar±3s.

Substitute 98.07 and s 0.56. The interval 3s is calculated as shown below:
3s 98.07±3(0.56)
98.07-3(0.56).98.07 3(0.56)
=[98.07-1.68,98.07+1.68]
= (96.39,99.75)
_
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Substitute 98.07 and s 0.56. The interval 3s is calculated as shown below: 3s 98.07±3(0.56) 98.07-3(0.56).98.07 3(0.56) =[98.07-1.68,98.07+1.68] = (96.39,99.75) _

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Step 3

Thus, at least 88.9% (approximately 89%) of the healthy adults have body temperatures between 96.39 and 99.75.

Hence, the minimum an...

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