Question

The body temperatures of a group of healthy adults have a​ bell-shaped distribution with a mean of

98.14°F and a standard deviation of 0.57°F.

Using the empirical​ rule, find each approximate percentage below.

a.	What is the approximate percentage of healthy adults with body temperatures within 2 standard deviations of the​ mean, or between 97.00°F and 99.28°​F?
b.	What is the approximate percentage of healthy adults with body temperatures between 97.57°F and 98.71°​F?

Answer 1

Question 1

X ~ N ( µ = 98.14 , σ = 0.57 )
P ( 97 < X < 99.28 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 97 - 98.14 ) / 0.57
Z = -2
Z = ( 99.28 - 98.14 ) / 0.57
Z = 2
P ( -2 < Z < 2 )
P ( 97 < X < 99.28 ) = P ( Z < 2 ) - P ( Z < -2 )
P ( 97 < X < 99.28 ) = 0.9772 - 0.0228
P ( 97 < X < 99.28 ) = 0.9545

Percentage = 95.45%

Question 2

X ~ N ( µ = 98.14 , σ = 0.57 )
P ( 97.57 < X < 98.71 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 97.57 - 98.14 ) / 0.57
Z = -1
Z = ( 98.71 - 98.14 ) / 0.57
Z = 1
P ( -1 < Z < 1 )
P ( 97.57 < X < 98.71 ) = P ( Z < 1 ) - P ( Z < -1 )
P ( 97.57 < X < 98.71 ) = 0.8413 - 0.1587
P ( 97.57 < X < 98.71 ) = 0.6827

Percentage = 68.27%