Question

For the population 2,3,4,5  take sample size 2 and apply central limit theorem and prove that mean of sample means is equal to population mean and standard deviation of sampling distribution of statistic is equal to population standard deviation divide by the square root of sample size.

Answer 1

for sampling with replacement there are 4² = 16 samples of size 2

samples mean

2,2 2   

2,3 2.5

2,4 3   

2,5 3.5

3,2 2.5

3,3 3

3,4 3.5   

3,5 4

4,2 3   

4,3 3.5   

4,4 4   

4,5 4.5

5,2 3.5   

5,3 4   

5,4 4.5   

5,5 5

Mean of sample means = (2+2.5+.....+5)/16 = 3.5

population mean = (2+3+4+5)/4 = 3.5

theefore mean of the sample means = population mean

Variance of sampling distribution of sample mean = {(2-3.5)² + (2.5-3.5)² +......+(5-3.5)²}/16 = 0.625

Standard deviation of sampling distribution of sample mean = 0.7906

population variance = {(2-3.5)² +.....(5-3.5)²}/4 = 1.25

population standard deviation = 1.118

population sd/ square root of sample size = 1.118/sqrt(2) = 0.7906

thus we can see that ,Standard deviation of sampling distribution of sample mean is equal to popultion standard deviation divided by sqaure root of sample size