Question

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

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

68

and

132​?

​(b) What percentage of people has an IQ score less than

52

or greater than

148?

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

132?

(a)

= 100

= 16

To find P(68 <X < 132):

Case 1:For X from 68 to mid value:

Z = (68 - 100)/16

= - 2

Case 2: For X from mid value to 132:
Z = (132 - 100)/16

=2

By Empirical Rule, 95% of data points lie between 2 Standard deviations from mean.

So,

95%

(b)

To find P(X<52 OR X>148):

Case 1: For X < 52

Z =(52 - 100)/16

= - 3

Case 2: For X > 148

Z = (148 - 100)/16

= 3

By Empirical Rule, 99.7% of data points lie within 3 Standard Deviations from mean.

So,

Percentage outside 3 Standard Deviations from mean.= 100 - 99.7 = 0.3%

So,

0.3%

(c)

To find P(X >132):

Z= (132 - 100)/16

= 2

By Empirical Rue, 95% of data points lie between 2 Standard deviations from mean..

So,

Percentage from mid value to 2 SD on RHS = 95%/2 = 47.5%

So,

P(X>132)= 50% - 47.5% = 2.5%

So,

2.5%

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