Question

Assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation 15. Find P3 which is the IQ score separating the bottom 3% from the top 97%?

Answer 1

IQ score due to the problem is normally distributed. Let's denote by $\xi$ random variable that corresponds to IQ score. Mean of this distribution is $\mu = 100$ and standard deviation $\sigma = 15$ . Thus

$$\xi \sim N (100, 15)$$

To find point separating the bottom $3\%$ score from the top $97\%$ let's find quantile of normal distribution corresponding to 0.03:

$$q_{1-0.03} = -1.88$$

Using properties of normal distribution we have that required score equals to

$$\mu + q_{1-0.03} \cdot \sigma = 100 - 1.88 \cdot 15 = 71.8$$

ANSWER: 71.8