Q. 5.8 - A First Course in Probability

Question

A randomly chosen $IQ$ test taker obtains a score that is approximately a normal random variable with mean $100$ and standard deviation $15$. What is the probability that the score of such a person is

(a) more than 125; 

(b) between $90$ and $110$?

Step-by-Step Solution

Answer

(a) Probability that the score of a person more than $125$ is $0.0478$ 

(b) Probability that the score of a person between $90$ and $110$ is $0.4972$

Step:1 Find the Probability that the score of a person more than 125 (part a)

Define $X$ as a random variable that represents a random person's $IQ$. We're told that data-custom-editor="chemistry" $X~\mathcal{N}\left(100,{15}^2\right)$. That is,  $Y=\frac{X-100}{15}$has a conventional normal distribution with a cumulative distribution$\varphi$.

$$\begin{gathered} P(X>125)=1-P(X\le 125) \\ =1-P\left(\frac{X-100}{15}\le \frac{125-100}{15}\right) \end{gathered}$$

$=1-\Phi \left(\frac{5}{3}\right)\approx 0.0478$

Step:2 Find the Probability that the score of a person between 90 and 110 (part b)

$P(X\in (90,110\left)\right)=P(X\le 110)-P(X\le 90)$

$=P\left(\frac{X-100}{15}\le \frac{110-100}{15}\right)-P\left(\frac{X-100}{15}\le \frac{90-100}{15}\right)$

$$\begin{gathered} =\Phi \left(\frac{2}{3}\right)-\Phi \left(-\frac{2}{3}\right) \\ =2\Phi \left(\frac{2}{3}\right)-1 \\ =0.4972 \end{gathered}$$