Question

Paul is$15$years old and $179$cm tall.

(a) Find the z-score corresponding to Paul’s height. Explain what this value means.

(b) Paul’s height puts him at the$85th$ percentile among $15$-year-old males. Explain what this means to someone who knows no statistics.

Short answer

(a)The z-score corresponding to Paul's height is$1.2$standard deviations above the mean.
(b)Paul is taller than $85\%$of the entire population of $15$-year-old males.

Step-by-step solution

Part (a) Step 1: Given information

The distribution of heights for $15$-year-old males is symmetric, single-peaked, and bell-shaped.

z-score of $0$corresponds to a heightlocalid="1649751998639" $=170cm$

z-score of $1$corresponds to a height of localid="1649752004677" $=177.5cm$

Find the z-score corresponding to Paul’s height.

Part (a) Step 2: Explanation

Given:

$$\begin{gathered} x=179\mathrm{cm} \\ \mu =170\mathrm{cm} \\ \sigma =7.5\mathrm{cm} \end{gathered}$$

In other words, it is the mean divided by the standard deviation divided by the value of the $z-$score

$z=\frac{x-\mu }{\sigma }=\frac{179-170}{7.5}=1.20$

localid="1649752085470" $1.2$standard deviations above the mean, Paul's height is higher than the average.

Part (b)Step 1: Given information

The distribution of heights for$15$-year-old males is symmetric, single-peaked, and bell-shaped.

z-score of$0$corresponds to a height of

localid="1649752116086" $z-$score of $1$corresponds to a height oflocalid="1649752017672" $=177.5cm$

Part (b) Step 2: Explanation

Given:

Paul's height$=85th$percentile

As a percentage, the $xth$percentile represents a value that has$x\%$of the other values below it.

A $15-$year-old male who is taller than Paul exceeds $85\%$of the entire population.