Question

In the country of United States of Heightlandia, the height measurements of ten-year-old children are approximately normally distributed with a mean of 53.9 inches, and standard deviation of 3 inches.A) What is the probability that a randomly chosen child has a height of less than 47.4 inches?Answer= (Round your answer to 3 decimal places.)B) What is the probability that a randomly chosen child has a height of more than 59.1 inches?Answer= (Round your answer to 3 decimal places.)

Answer 1

Solution:

Given:

$\begin{gathered} \mu=53.9 \\ \sigma=3 \\ x=47.4 \end{gathered}$

Part A:

Find the Z-score

$\begin{gathered} Z=\frac{x-\mu}{\sigma} \\ Z=\frac{47.4-53.9}{3} \\ Z=\frac{-6.5}{3} \\ Z=-2.1667 \end{gathered}$

From the Z-scores table, the probability of the height less than 47.4 inches is;

$\begin{gathered} P(xTherefore, the probability that a randomly chosen child has a height of less than 47.4 inches is 0.015

Part B:

[tex]\begin{gathered} \mu=53.9 \\ \sigma=3 \\ x=59.1 \end{gathered}$

Find the Z-score

$\begin{gathered} Z=\frac{x-\mu}{\sigma} \\ Z=\frac{59.1-53.9}{3} \\ Z=\frac{5.2}{3} \\ Z=1.7333 \end{gathered}$

From the Z-scores table, the probability that the height of more than 59.1 inches is;

$\begin{gathered} P(x>Z)=0.041521 \\ \\ To\text{ 3 decimal places;} \\ P(x>Z)=0.042 \end{gathered}$

Therefore, the probability that a randomly chosen child has a height of more than 59.1 inches is 0.042