Math Statistics

Question

Adults have IQ score that are normally distributed with a mean of 100 and a standard deviation of 15.

What percentage of scores are less than 103?

What percentage of scores are between 60 and 130?

What is the IQ score separating the bottom 25% from the rest?

Answer 1

For a) See the following :

Set x to to IQ score.

we want to show that P(x< 103):

P(x<103)=P(x<(103-100)/15)=P(z<0.2)=0.5793. (By checking the Z score table)

For b)

P(60<x<130)=P(x<130)-P(x<60)=P(z<2)-P(z<-2.67)

=P(z<2)+P(z<2.67)-1=0.9772+0.9967-1=0.9739

For c) :

first we set X_75 to be the top 75%, by

P(x < x_75)=0.75, and by checking z value equal to 0.75, we find that z = 0.67.

so we have (x_75 -100)/15 = 0.67, and x_75 = 110. Since X is normal distribution, we can easily find that the bottom z score to be x_25= 100-(110-100)=90. Therefore, 90 is the the IQ score separating the bottom 25% from the rest