Question

A set of middle school student heights are normally distributed with a mean of 150150150 centimeters and a standard deviation of 202020 centimeters. Darnell is a middle school student with a height of 161.4161.4161, point, 4 centimeters. What proportion of student heights are lower than Darnell's height?

Answer 1

Answer:

71.57% of student heights are lower than Darnell's height

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. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this question, we have that:

$\mu = 150, \sigma = 20$

Darnell has a height of 161.4 centimeters. What proportion of student heights are lower than Darnell's height?

This is the pvalue of Z when X = 161.4.

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

$Z = \frac{161.4 - 150}{20}$

$Z = 0.57$

$Z = 0.57$ has a pvalue of 0.7157

71.57% of student heights are lower than Darnell's height