Question

A random sample of size n=100 is taken from an infinite population with mean=75 and variance =256. What is the probability that x bar will fall between 67 and 83?

Could you please answer this explaining the why of every step, I know the solution but don't know how I'm supposed to get there. Thank you!

Answer 1

Given,

= 75 , = sqrt(256) = 16

Using central limit theorem,

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

So,

P(67 < < 83) = P( < 83) - P( < 67)

= P( Z < 83 - 75 / 16 / sqrt(100) ) - P( Z < 67 - 75 / 16 / sqrt(100) )

= P( Z < 5) - P( Z < -5)

= 1.0000 - 0.0000 (Probability calculated from Z table)

= 1