Assume the heights in a female population are normally distributed with mean 65.7 inches and standard deviation 3.2 inches. Then the probability that a typical female from this population is between 5 feet and 5 feet 7 inches tall is (to the nearest three decimals) which of the following? a. 0.620
b. 0.658
c. 0.963

Choice a. approximately .

Step-by-step explanation:

This explanation shows how to solve this problem using a typical -score table. Consider a normal distribution with mean and variance . The -score for a measurement of value would be .

Convert all heights to inches:

Let represent the height (in inches) of a female from this population. By the assumptions in this question: . The question is asking for the probability . Calculate the score for the two boundary values:

Look up the corresponding probabilities on a typical -score table.

For the -score of the upper bound, the corresponding probability is approximately . In other words:

On the other hand, some -score table might not include the probability for negative scores. That missing part can be found using the symmetry of the normal distribution PDF.

The probability corresponding to (that's the opposite of the -score at the lower bound) is approximately . By the symmetry of the normal PDF:

.

Therefore:

.

Calculate the probability of the interval between the two bounds:

.