Scores of an IQ test have a bellshaped distribution with a m

Scores of an IQ test have a bell-shaped distribution with a mean of 100 and a standard deviation of 13. Use the empirical rule to determine the following. What percentage of people has an IQ score between 61 and 139? What percentage of people has an IQ score less than 87 or greater than 113? What percentage of people has an IQ score greater than 126? % (Type an integer or a decimal.) % (Type an integer or a decimal.) % (Type an integer or a decimal.)

Solution

Normal Distribution
Mean ( u ) =100
Standard Deviation ( sd )=13
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 61) = (61-100)/13
= -39/13 = -3
= P ( Z <-3) From Standard Normal Table
= 0.00135
P(X < 139) = (139-100)/13
= 39/13 = 3
= P ( Z <3) From Standard Normal Table
= 0.99865
P(61 < X < 139) = 0.99865-0.00135 = 0.9973                  

b)
To find P( X > a or X < b ) = P ( X > a ) + P( X < b)
P(X < 87) = (87-100)/13
= -13/13= -1
= P ( Z <-1) From Standard Normal Table
= 0.1587
P(X > 113) = (113-100)/13
= 13/13 = 1
= P ( Z >1) From Standard Normal Table
= 0.1587
P( X < 87 OR X > 113) = 0.1587+0.1587 = 0.317311                  

c)
P(X > 126) = (126-100)/13
= 26/13 = 2
= P ( Z >2) From Standard Normal Table
= 0.0228                  

[ANSWERS]
a. 99.73%
b. 31.73 ~ 32%
c. 2.28% ~ 3%