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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 98.1998.19degrees°F and a standard deviation of 0.560.56degrees°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?

Respuesta :

Answer:

at least 99.908% of the body temperatures of healthy adults are between 116.67 °F ( maximum possible temperature for 33 standard deviations) and 79.71 °F ( minimum possible temperature for 33 standard deviations)

Step-by-step explanation:

from Chebyshev's theorem:

P( |X-μ| ≤ k*σ ) ≥ 1- 1/k²

where

X = random variable = body temperatures of healthy adults

μ = expected value of X ( mean)

σ = standard deviation of X

k = parameter

P( |X-μ| ≤ k*σ ) = probability that X is within k-standard deviations from the mean

for our case k=33 , then

P( |X-μ| ≤ 33*σ ) ≥ 1- 1/33² = 0.99908= 99.908%

therefore at least 99.908% of the body temperatures of healthy adults are between

X max =μ + 33*σ =  98.19°F + 33* 0.56 °F = 116.67 °F ( maximum possible temperature)

and

X min =μ + 33*σ =  98.19°F - 33* 0.56 °F = 79.71 °F ( minimum possible temperature)

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