Question

Scores of an IQ test have a​ bell-shaped distribution with a mean of 100 and a standard deviation of 20.

Use the empirical rule to determine the following:

​(a) What percentage of people has an IQ score between
60 and 140?

​(b) What percentage of people has an IQ score less than
60 or greater than 140?

​(c) What percentage of people has an IQ score greater than 160?

Answer 1

Let X = Score of an IQ test

Given that, X has bell shaped distribution with a mean of 100 and standard deviation of 20.

Now,

That is, Z is standard normal variate.

a)

First we find probability of an IQ score lies between 60 and 140.

P(IQ score lies between 60 and 140)

=P(60<X<140)

Hence

P(IQ score between 60 and 140) = 0.9544

Now,

Percentage of people has an IQ score between 60 and 140 = 100*P(IQ score between 60 and 140)

Percentage of people has an IQ score between 60 and 140 = 100*0.9544 = 95.44%

Therefore percentage of people has an IQ score between 60 and 140 is 95.44%.

b)

First we find probability of an IQ score less than 60 or greater than 140.

P(IQ score less than 60 or greater than 140)

=P(X<60 or X>140) = 1- P(60<X<140) = 1- 0.9544 = 0.0456

Hence

P(IQ score less than 60 or greater than 140) = 0.0456

Now,

Percentage of people has an IQ score less than 60 or greater than 140 = 100*P(IQ score bless than 60 or greater than 140)

Percentage of people has an IQ score less than 60 or greater than 140 = 100*0.0.0456 = 4.56%

Therefore percentage of people has an IQ score less than 60 or greater than 140 is 4.56%.

c)

First we find probability of an IQ score greater than 160.

P(IQ score greater than 160)

=P(X>160)

Hence

P(IQ score greater than 160) = 0.0013

Now,

Percentage of people has an IQ score greater than 160 = 100*P(IQ score greater than 160)

Percentage of people has an IQ score greater than 160 = 100*0.0013 = 00.13%

Therefore percentage of people has an IQ score greater than 160 is 0.13%.

Reference: Standard Normal Probability Table: