Question

The height of a women in the united states are normally distributed with a mean of 64 inches and a standard deviation of 2.75 inches. Find the area under the curve that represents the percent of women whose heights are at least 64 inches.

Answer 1

Answer:

The area under the curve that represents the percent of women whose heights are at least 64 inches is 0.5.

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

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

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X, or the area under the curve representing values that are lower than x. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X, which is the same as the area under the curve representing values that are higher than x.

In this problem, we have that:

$\mu = 64, \sigma = 2.75$

Find the area under the curve that represents the percent of women whose heights are at least 64 inches.

This is 1 subtracted by the pvalue of Z when X = 64.

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

$Z = \frac{64 - 64}{2.75}$

$Z = 0$

$Z = 0$ has a pvalue of 0.5.

1 - 0.5 = 0.5

The area under the curve that represents the percent of women whose heights are at least 64 inches is 0.5.