Answer:
a) The range of lengths from 28.3 cm to 43.3 cm covers almost all (99.7%) of this distribution.
b) 16% of women over 20 have upper arm lengths less than 33.3 cm.
Explanation:
The Empirical Rule(68-95-99.7 Rule) states that, for a normally distributed random variable:
68% of the measures are within 1 standard deviation of the mean.
95% of the measures are within 2 standard deviation of the mean.
99.7% of the measures are within 3 standard deviations of the mean.
In this problem, we have that:
Mean = 35.8 cm
Standard deviation = 2.5 cm
(a) What range of lengths covers almost all (99.7%) of this distribution?
This range is from 3 standard deviations below the mean to three standard deviations above the mean.
So from 35.8 - 3*2.5 = 28.3 cm to 35.8 + 3*2.5 = 43.3 cm
The range of lengths from 28.3 cm to 43.3 cm covers almost all (99.7%) of this distribution.
(b) What percent of women over 20 have upper arm lengths less than 33.3 cm?
68% of the women over 20 have upper arm length between 33.3 cm and 38.3 cm. The other 32% have upper arm length lower than 33.3 cm or higher than 38.3. The distribution is symmetric, so 16% of the have upper arm length lower than 33.3 cm and 16% have upper arm length higher than 38.3 cm
So 16% of women over 20 have upper arm lengths less than 33.3 cm.