Question

Scores of an IQ test have a​ bell-shaped distribution with a mean of 100 and a standard deviation of 18. Use the empirical rule to determine the following. ​(a) What percentage of people has an IQ score between 64 and 136​? ​(b) What percentage of people has an IQ score less than 64 or greater than 136​? ​(c) What percentage of people has an IQ score greater than 154​?

Answer 1

Given,

= 100 , = 18

a)

z-score = ( x - ) /

For X = 64 ,

z = ( 64 - 100) / 18

= -2

That is 64 is 2 standard deviation below the mean.

For X = 136 ,

z = ( 136 - 100) / 18

= 2

That is 136 is 2 standard deviation above the mean.

From empirical (68-95-99.7) rule,

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

Therefore,

P(64 < X < 136) = 95%

b)

P(X < 64 OR X > 136) = 1 - P( 64 < X < 136)

= 1 - 0.95

= 0.05

= 5%

c)

For X = 154 ,

z = ( 154 - 100) / 18

= 3

That is 154 is 3 standard deviation above the mean.

From empirical (68-95-99.7) rule,

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

That is ( 1 - 0.997) = 0.003 data falls outside 3 standard deviation, and 0.003 / 2 = 0.0015 of the data falls

in either side of curve.

Therefore,

P(X > 154) = 0.0015

= 0.15%