Question

A typical adult has an average IQ score of $105$with a standard deviation of $20$. If $20$randomly selected adults are given an IQ test, what are the probability that the sample mean scores will be between $85$ and $125$ points?

Short answer

The probability that the sample mean scores will be between $85$ and $125$points is$1$.

Step-by-step solution

Given Information

what is the probability that the sample mean scores will be between $85$and$125$ points?

Explanation

$\text{Letvariable}X\text{represent}IQ\text{score of a person. The distribution of}X\text{:}$

$X~N(105,20)$

The distribution of the average score of 20 people

$\overline{X}~N(105,\frac{20}{\sqrt{20}})~N(20,\sqrt{20})$

Finding the wanted probability:

localid="1650524594045" $P(85<X<125)=P(\frac{85-105}{\sqrt{20}}<\frac{\overline{X}-105}{\sqrt{20}}<\frac{125-105}{\sqrt{20}})$

$=\varphi \left(\frac{125-105}{\sqrt{20}}\right)-\varphi \left(\frac{85-105}{\sqrt{20}}\right)$

$=\varphi \left(4.47\right)-\varphi (-4.47)$

$=2\varphi \left(4.47\right)-1$

=$2-1$

$=1$