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?
Answer:
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
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