Question

A simple random sample of size n=64 is obtained from a population with mu equals 76 and sigma equals 8. ​(a) Describe the sampling distribution of x overbar. ​(b) What is Upper P left parenthesis x overbar greater than 77.9 right parenthesis​? ​(c) What is Upper P left parenthesis x overbar less than or equals 73.55 right parenthesis​? ​(d) What is Upper P left parenthesis 74.85 less than x overbar less than 78.1 right parenthesis​?

Answer 1

Solution:

Given:

Sample size = n = 64

Population Mean =

Population Standard Deviation =

Part a) Describe the sampling distribution of  

Since sample size n = 64 is large , we can use Central limit theorem which states that for large sample size n ,
sampling distribution of sample mean is approximately normal with mean of sample means:

and standard deviation of sample means is:

Part b) find:  

Find z score for   = 77.9

Thus we get:

Look in z table for z = 1.9 and 0.00 and find corresponding area.

P( Z <1.90) =0.9713

thus

Part c) Find:

Find z score for   = 73.55

Thus we get:

Look in z table for z = -2.4 and 0.05 and find corresponding area.

P(Z < -2.45) = 0.0071

Thus

Part d) Find:

Find z score for   = 74.85 and for 78.1

and

thus

Look in z table for z = 2.1 and 0.00 as well as for  z = -1.1 and 0.05  and find corresponding area.

P( Z< -1.15) = 0.1251

and

P( Z < 2.10 ) = 0.9821

thus