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Q.1

Expert-verified

The Practice of Statistics for AP

Found in: Page 136

Book edition 4th

Author(s) David Moore,Daren Starnes,Dan Yates

Pages 809 pages

ISBN 9781319113339

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Short Answer

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 $σ = 7.5 cm$

Standard deviation $μ = 170 cm$

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

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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 $= 170 c m$ z-score of $1$ corresponds to a height oflocalid="1649750294635" $= 177.5 c m$

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 $0$ :

$μ = 170 cm$

$1$ 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:

$σ = | μ - 177.5 | = 7.5 cm$

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 $= 170 c m$

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

Taking the

z -

score of

2.5

Part (b) Step 2: Explanation

Given:

$z = 2.5$

1a Result exercise is:

localid="1649943599620" $μ = 170 cm σ = 7.5 cm$

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

localid="1650000254221" $x = μ + z σ = 170 cm + 2.5 (7.5 cm) = 188.75 cm$

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