Question

a set of elementary school student heights are normally distributed with a mean of 105 centimeters and a standard deviation of 10 centimeters. faisal is an elementary school student with a height of 103.1 centimeters. what proportion of student heights are lower than faisals height? you may round your answer to four decimal places.

Explanation:

Step1: Calculate the z - score

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value from the data set, $\mu$ is the mean, and $\sigma$ is the standard deviation. Given $\mu = 105$, $\sigma=10$, and $x = 103.1$.
$z=\frac{103.1 - 105}{10}=\frac{- 1.9}{10}=-0.19$

Step2: Find the proportion using the standard normal table

We want to find $P(X < 103.1)$, which is equivalent to $P(Z < - 0.19)$ in the standard - normal distribution. Looking up the value of $-0.19$ in the standard normal table (the cumulative distribution function of the standard normal distribution $\varPhi(z)$), we get $P(Z < - 0.19)=0.4247$

Answer:

$0.4247$