Question

Scores of an IQ test have a​ bell-shaped distribution with a mean of 100 and a standard deviation of 16. Use the empirical rule to determine the following. ​(a) What percentage of people has an IQ score between 84 and 116​? ​(b) What percentage of people has an IQ score less than 68 or greater than 132​? ​(c) What percentage of people has an IQ score greater than 148​?

Answer 1

Given ,

$\mu$ = 100 , $\sigma$ = 16

a)  

z-score = (X - $\mu$ ) / $\sigma$

For X = 84 ,

z = ( 84 - 100) / 16

= -1

That is 84 is 1 standard deviation below the mean.

For X = 116 ,

z = ( 116 - 100) / 16

= 1

That is 116 is 1 standard deviation above the mean.

Using empirical (68 - 95 - 99.7) rule,

Approximately, 68% data falls between 1 standard deviation of the mean.

So,

P(84 < X < 116) = 68%

b)

We have to calculate P(X < 68 OR X > 132) = ?

P(X < 68 OR X > 132) = 1 - P( 68 < X < 132)

For X = 68 ,

z = ( 68 - 100) / 16

= -2

That is 68 is 2 standard deviation below the mean.

For X = 132 ,

z = ( 132 - 100) / 16

= 2

That is 132 is 2 standard deviation above the mean.

Using empirical (68 - 95 - 99.7) rule,

Approximately, 95% data falls between 2 standard deviation of the mean.

So,

P(X < 68 OR X > 132) = 1 - 0.95

= 0.05

= 5%

c)

P(X > 148) = ?

For X = 148 ,

z = ( 148 - 100) / 16

= 3

That is 148 is 3 standard deviation above the mean.

Using empirical (68 - 95 - 99.7) rule,

Approximately, 99.7% data falls between 3 standard deviation of the mean.

That is 1 - 0.997 = 0.003 of the data falls outside the 3 standard deviation of the mean.

And 0.003 / 2 = 0.0015 of the data falls of one side outside of the 3 standard deviation of the mean.

Therefore ,

P(X > 148) = 0.0015

= 0.15%