Question

In the EAI sampling problem, the population mean is $51,200 and the population standard deviation is $5,000. When the sample size is n=30, there is a 0.4908 probability of obtaining a sample mean within plus or minus $600 of the population mean. Use z-table.

a. What is the probability that the sample mean is within $600 of the population mean if a sample of size 60 is used (to 4 decimals)?

b. What is the probability that the sample mean is within $600 of the population mean if a sample of size 120 is used (to 4 decimals)?

Answer 1

Solution:

Given:

the population mean = and

the population standard deviation =

when sample size = n = 30 , the probability of obtaining a sample mean within plus or minus $600 of the population mean is 0.4908

Part a) We have to find the probability that the sample mean is within $600 of the population mean if a sample of size 60 is used

Find standard error when n = 60:

Now use following steps:

Look in z table for z = 0.9 and 0.03 as well as for z = -0.9 and 0.03 and find area.

P( Z < 0.93) =0.8238

P( Z < -0.93) = 0.1762

Thus

Part b) Find the probability that the sample mean is within $600 of the population mean if a sample of size 120 is used

Find standard error when n = 120:

Thus

Look in z table for z = 1.3 and 0.01 as well as for z = -1.3 and 0.01

and find area.

P( Z < 1.31 ) = 0.9049

P( Z< -1.31) = 0.0951

Thus