Question

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. $0.20$inch

b. $0.31$inch

c. $0.65$inch

d. $1.21$inches

e. $1.56$inches

Short answer

The standard deviation of the heights of all children of this age is (e) $1.56$inches.

Step-by-step solution

Given Information

We have been given that the height of a child is $40$inches which places her at the $90th$percentile of all children of similar age. And the mean of the Normal distribution is $38$inches.

Simplification

Let, $z=(40-38)/s$(where, s = standard deviation)

$P(z<(40-38)/s)=0.90$(given)

So, we get

$P(z<1.2816)=0.90$

So,

$(40-38)/s=1.2816$

$2/1.2816=s$

$s=1.56$