Question

What is the smallest your IQ score can be so that you are among the smartest 1% of the population with a mean of 100 and a standard deviation of 15?

A random sample of 25IQ scores is selected. Find the probability that the mean of the sample is between 90 and 110?

A random sample of 100 IQ scores is selected. Find the probability that the sample mean is less than 85%. Find k.

Answer 1

a)

Given,  

= 100, = 15

We convert this to standard normal as

P( X < x) = P( Z < x - / )

We have to calculate x such that

P( X > x) = 0.01

P( X < x) = 0.99

That is

P( Z < x - / ) = 0.99

From the Z table , z-score for the probability of 0.01 is 2.3263

  x - / = 2.3263

x - 100 / 15 = 2.3263

Solve for x

x = 134.8945

b)

Using central limit theorem,

P( < x) = P( Z < x - / / sqrt(n) )

So,

P(90 < < 110) = P( < 110) - P( < 90)

= P( Z < 110 - 100 / 15 / sqrt(25) ) - P( Z < 90 - 100 / 15 / sqrt(25) )

= P( Z < 3.3333) - P( Z < -3.3333)

= 0.9991