Answer to Question #143899 in Statistics and Probability for Vladimyr Lubin

Question

(3.2.42) The body temperatures of a group of healthy adults have a​ bell-shaped distribution with a mean of 98.11F and a standard deviation of 0.56F. Using the empirical​ rule, find each approximate percentage below. ( for A do not round)
a. What is the approximate percentage of healthy adults with body temperatures within 1 standard deviation of the​ mean, or between 97.55F and 98.67​F?
b. What is the approximate percentage of healthy adults with body temperatures between 96.43F and 99.79F

Answer 1

a

$Z=\frac{X-\Mu}{\sigma}$

$\mu=98.11$

$\sigma=0.56$

$P(97.55<X<98.67)$

$Z_l=\frac{97.55-98.11}{0.56}=-1$

$Z_u=\frac{98.67-98.11}{0.56}=1$

97.55 and 98.67 lie within one standard deviation from the mean. The empirical rule states that 68% of all data lie within one standard deviation from the mean. Thus, 68% of healthy people have body temperature between 97.55F and 98.67F.

b

$P(96.43<X<99.79)$

$Z_l=\frac{96.43-98.11}{0.56}=-3$

$Z_u=\frac{98.67-98.11}{0.56}=3$

96.43 and 99.79 lie within three standard deviations from the mean. The emphirical rule states that 99.7% of the data fall within three standard deviations. Thus, 99.7% of healthy individuals have body temperature between 96.43F and 99.79F.