Given a normal distribution with μ = 100 and ơ= 20 , if you select a sample of n = 25, find:

Question

Given a  normal distribution with μ = 100 and ơ= 20 , if you select a sample of n = 25,  find:

 

 What is the probability that Xbar < than 97

Answer 1

Since this problem involves the xbar distribution, the sample size, n, plays a role in the calculation of z . The size of the random samples affects the standard deviation of the xbar distribution. The standard error of the mean (also known as the standard deviation of the xbar distribution) is used instead of σ.

 

The standard error of the mean, σ_xbar = σ/√n 

 

The formula for z is given as :

 

z = (xbar - µ)     σ/√n = 20/√25 

         σ/√n

 

z = 97 - 100      20/√25 =20/5 = 4

        20/√25

 

z = -3/4 = -0.75

 

Using the normal distribution table, z_-0.75 = 0.2266

P(z<-0.75) = 0.2266

Therefore the probability that xbar < 97 is 0.2266 or 22.66%

Answer 2

z = (x-bar - U)/S

 

   = (97-100)/20 = -3/20 = -0.15

 

Prob ( z < -0.15) = .4404 per the normal distribution table.

 

                          = 44.04%