Question

Scores of an IQ test have a​ bell-shaped distribution with a mean of

100100 and a standard deviation of1919.

Use the empirical rule to determine the following.

​(a) What percentage of people has an IQ score between 6262 and138138​?

​(b) What percentage of people has an IQ score less than4343or greater than157157​?

​(c) What percentage of people has an IQ score greater than 138138​?

Answer 1

Given:

Mean, = 100

Standard deviation, =19

Empirical value ( 68%, 95%, 99.7%) rule:

68% data values lies within One standard deviation of the mean

95% data values lies within Two standard deviation of the mean

99.7% data values lies within Three standard deviation of the mean

z- score formula : z = X- /

a) P(62 < X < 138):
Case 1:

Z = (62 - 100)/19 = -2

Case 2:

Z = (138 - 100)/19 =2

By Empirical Rule, 95 % of data falls within two standard deviations from the mean.

So P(62< X <138) = 95%

b) P(X < 43 OR X > 157):

Case 1:
For X < 43:
Z = (43 - 100)/19 =-3

Case 2:

For X > 157:

Z = (157 - 100)/19 = 3

By Emprical Rlule, 99.7 % of data falls within the three standard deviation from the mean.

Therefore, 100% - 99.7% = 0.3% of people has an IQ score less than 43 or greater than 157

c)

P(X>138):

z = (138 - 100)/20 = 2

By empirical rule, 95 % fall within two standard deviations.

Percentage of people having greater than 160 :

95/2 = 47.5%

P(X>138) = 50% - 47.5% =2.5%

Therefore 2.5% of people has an IQ score greater than 138.