Question

The upper arm length of females over 20 years old in a country is approximately Normal with mean 35.8 centimeters (cm) and standard deviation 2.5 cm. Use the 68-95-99.7 rule to answer the following questions. (Enter your answers to one decimal place.) (a) What range of lengths covers almost all (99.7%) of this distribution? From 33.3 Incorrect: Your answer is incorrect. cm to 38.3 Incorrect: Your answer is incorrect. cm (b) What percent of women over 20 have upper arm lengths less than 33.3 cm? 2.5 Incorrect: Your answer is incorrect. %

Answer 1

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.