Question

Scores of an IQ test have a​ bell-shaped distribution with a mean of 100 and a standard deviation of 10. Use the empirical rule to determine the following

a) What percentage of people has an IQ score between 70 and 130​?

​(b) What percentage of people has an IQ score less than 90 or greater than 110​?

​(c) What percentage of people has an IQ score greater than 120​?

Answer 1

(a)

= 100

= 10

To find P(70 < X < 130):
Case 1: For X from 70 to mid value:

Z = (70 - 100)/10 = -3

Case 2:

For X from mid value to 130:
Z = (130 - 100)/10 = 3

By Empirical Rule, 99.7 % of data falls within three standard deviations from the mean.

So,

Answer is:

99.7 %

(b)

To find P(X < 90 OR X > 100):

Case 1:
For X < 90:
Z = (90 - 100)/10 = = - 1

Case 2:

For X > 110:

Z = (110 - 100)/10 = 1

By Emprical Rlule, 68.27 % of data falls within the first standard deviation from the mean.

So,

Percentage of dta not within the first standard deviation from the mean = 100 - 68.27 = 31.73 %

So,

Answer is:

31.73 %

(c)

To find P(X>120):

z + (120 - 100)/10 = 2

By empirical rule, 95.44 % fall within two standard deviations.

Since here only the upper side is required, Percentage of people having greater than 120 is got as follows:

95.44/2 = 47.72

So,

P(X>120) = 50 - 47.72 = 2.28

So,

Answer is:

2.28 %