Question

14. A sample of size 144 is taken from a population with an unknown distribution. It is known that the population distribution has mean 32 and standard deviation 15.

(A)What is the distribution of the sample means x̄ ? Justify your reasoning and be sure to completely specific the distribution by stating values of the appropriate parameters.

(B) Compute P( x̄ ≥ 34). You may only use the z-score approach and the probabilities provided in Table A.

(C) How large does the sample size need to be in order to reduce the standard deviation of x̄ to 0.75?

Answer 1

Solution:

Given:

Sample size = n = 144

Mean =

Standard Deviation =

Part a) What is the distribution of the sample means x̄ ?

Since sample size n = 144 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) Compute  

Find z score:

Thus we get:

Look in z table for z = 1.6 and 0.00 and find corresponding area.

P( Z< 1.60) = 0.9452

Thus

Part c) How large does the sample size need to be in order to reduce the standard deviation of x̄ to 0.75?

thus