Question

Is Paul tall? According to the National Center for Health Statistics, the distribution of heights for $15-$year-old males has a mean of $170$ centimeters (cm) and a standard deviation of $7.5$ cm. 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

Part (a) $z=1.20$

Part (b) The researcher is taller than $85\%$ of the population of $15-$year-old boys or shorter than $15\%$ of the population of $15-$year-old males.

Step-by-step solution

Part (a) Step 1: Given information

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

Part (a) Step 2: Concept

The Formula used: $z=\frac{x-\mu }{\sigma }$

Part (a) Step 3: Calculation

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ z=179-170/7.5 \\ z=1.20 \end{gathered}$$

It means that the researcher's average height is $170$cm, with a standard deviation of $1.20$The researcher's height would be between $168.8$and $171.2$cm.

Part (b) Step 1: Explanation

The $85th$ percentile is a data point in which $85$ percent of the data is less than or equal to it. It also signifies that $15\%$ of the data is larger than or equal to the data point, implying that the researcher is taller than $85\%$ of the population of $15-$year-old boys or shorter than $15\%$ of the population of $15-$year-old males.