Question

a set of middle school student heights are normally distributed with a mean of 150 centimeters and a standard deviation of 20 centimeters. uma is a middle school student with a height of 165 centimeters. what proportion of student heights are higher than umas height? you may round your answer to four decimal places.

Explanation:

Step1: Calculate the z - score

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x = 165$ (Uma's height), $\mu = 150$ (mean), and $\sigma=20$ (standard deviation).
$z=\frac{165 - 150}{20}=\frac{15}{20}=0.75$

Step2: Find the proportion of values above the z - score

We know that the total area under the normal - distribution curve is 1. Using a standard normal distribution table (or z - table), the area to the left of $z = 0.75$ is $P(Z<0.75)=0.7734$.
The area to the right of $z = 0.75$ (proportion of heights higher than Uma's) is $P(Z>0.75)=1 - P(Z<0.75)$.
$P(Z>0.75)=1 - 0.7734 = 0.2266$

Answer:

$0.2266$