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Chapter 4: Q AP1.2. (page 276)

A child is 40 inches tall, which places her at the 90th percentile of all children of similar age. The heights for children of this age form an approximately Normal distribution with a mean of 38 inches. Based on this information,

what is the standard deviation of the heights of all children of this age?

(a)0.20inches(c)0.65inches(e)1.56inches(b)0.31inches(d)1.21inches

Short Answer

Expert verified

The correct option is (e) 1.56inches

Step by step solution

01

Given information

Height of a child= 40inches

Percentile in which the child’s height is =90%

Mean of heights of children of similar age =38inches

The heights are normally distributed.

02

Concept

Z score

z=x−μσ

where

μ is the mean

σ is the standard deviation

03

Calculation

A height in the 90th percentile means that 90% of people are shorter than or equal to 40inches

P(X≤40)=0.90P(z<40−38σ)=0.90

Let Xbe the heights of children.

X~N(38,σ2)

Reading from z tables,

P(z<1.2816)=0.90

Comparing the values

1.2816=40−38σσ=40−381.2816σ=1.56

Therefore, standard deviation of heights of all children of similar age is

is=40inches

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