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Q.1

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Found in: Page 136

### The Practice of Statistics for AP

Book edition 4th
Author(s) David Moore,Daren Starnes,Dan Yates
Pages 809 pages
ISBN 9781319113339

# Consider the height distribution for $15$-year-old males. (a) Find its mean and standard deviation. Show your method clearly. (b) What height would correspond to a z-score of$2.5?$ Show your work.

(a) Mean $\sigma =7.5\mathrm{cm}$

Standard deviation $\mu =170\mathrm{cm}$

(b) The height correspond to $\mathrm{z}$-score of $2.5$ is $188.75\mathrm{cm}$

See the 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 Height$=170cm$ z-score of$1$corresponds to a height oflocalid="1649750294635" $=177.5cm$

## Part (a) Step 2: Explanation

The mean of the standard normal distribution is $0$.

$0$ is the mean of the standard normal distribution.
Accordingly, the mean of the normal height distribution is zero with a $z$-score of$0$ :

$\mu =170\mathrm{cm}$

$1$ is the standard normal distribution.

A standard deviation for a normal height distribution is then equal to the difference between the $z-$score of $1$ corresponding to the mean and the standard deviation:

$\sigma =|\mu -177.5|=7.5\mathrm{cm}$

## 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 the height$=170cm$

$z-$score of $1$corresponds to the height $=177.5cm$

Taking the $z-$score of $2.5$

## Part (b) Step 2: Explanation

Given:

$z=2.5$

1a Result exercise is:

localid="1649943599620" $\mu =170\mathrm{cm}\phantom{\rule{0ex}{0ex}}\sigma =7.5\mathrm{cm}$

A $z-$score is the mean multiplied by a $z-$score plus a standard deviation:

localid="1650000254221" $x=\mu +z\sigma \phantom{\rule{0ex}{0ex}}=170\mathrm{cm}+2.5\left(7.5\mathrm{cm}\right)\phantom{\rule{0ex}{0ex}}=188.75\mathrm{cm}$