Question

a set of middle school student heights are normally distributed with a mean of 150 centimeters and a standard deviation of 20 centimeters. darnell is a middle school student with a height of 161.4 centimeters. what proportion of student heights are lower than darnells 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 $x$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation. Given $\mu = 150$, $\sigma=20$, and $x = 161.4$. Then $z=\frac{161.4 - 150}{20}=\frac{11.4}{20}=0.57$.

Step2: Find the proportion

We use the standard normal distribution table (or z - table) to find the proportion of values less than the z - score. Looking up $z = 0.57$ in the standard - normal table, the corresponding proportion is $0.7157$.

Answer:

$0.7157$