List all possible samples of size n=3, with replacement, from the population (1,3,5). Calculate the mean of each sample. Construct a probability distribution of the sample means and compute the mean, variance, and standard deviation of the sample means and compare to the mean, variance, and standard deviation of the population.
for population:
x | P(x) | xP(x) | x^2P(x) |
1 | 1/3 | 0.33 | 0.33 |
3 | 1/3 | 1.00 | 3.00 |
5 | 1/3 | 1.67 | 8.33 |
total | 3.00 | 11.67 |
mean =E(X)=3.00
Variance =E(X2)-(E(X))2 =11.67-32 =2.67
and standard deviation =sqrt(2.67)=1.633
for probability distribution of the sample means :
x1 | x2 | x3 | P(x1,x2) | xbar | xbar*P(xbar) | xbar^2*P(xbar) | |
1 | 1 | 1 | 1/27 | 1.000 | 0.037 | 0.037 | |
1 | 1 | 3 | 1/27 | 1.667 | 0.062 | 0.103 | |
1 | 1 | 5 | 1/27 | 2.333 | 0.086 | 0.202 | |
1 | 3 | 1 | 1/27 | 1.667 | 0.062 | 0.103 | |
1 | 3 | 3 | 1/27 | 2.333 | 0.086 | 0.202 | |
1 | 3 | 5 | 1/27 | 3.000 | 0.111 | 0.333 | |
1 | 5 | 1 | 1/27 | 2.333 | 0.086 | 0.202 | |
1 | 5 | 3 | 1/27 | 3.000 | 0.111 | 0.333 | |
1 | 5 | 5 | 1/27 | 3.667 | 0.136 | 0.498 | |
3 | 1 | 1 | 1/27 | 1.667 | 0.062 | 0.103 | |
3 | 1 | 3 | 1/27 | 2.333 | 0.086 | 0.202 | |
3 | 1 | 5 | 1/27 | 3.000 | 0.111 | 0.333 | |
3 | 3 | 1 | 1/27 | 2.333 | 0.086 | 0.202 | |
3 | 3 | 3 | 1/27 | 3.000 | 0.111 | 0.333 | |
3 | 3 | 5 | 1/27 | 3.667 | 0.136 | 0.498 | |
3 | 5 | 1 | 1/27 | 3.000 | 0.111 | 0.333 | |
3 | 5 | 3 | 1/27 | 3.667 | 0.136 | 0.498 | |
3 | 5 | 5 | 1/27 | 4.333 | 0.160 | 0.695 | |
5 | 1 | 1 | 1/27 | 2.333 | 0.086 | 0.202 | |
5 | 1 | 3 | 1/27 | 3.000 | 0.111 | 0.333 | |
5 | 1 | 5 | 1/27 | 3.667 | 0.136 | 0.498 | |
5 | 3 | 1 | 1/27 | 3.000 | 0.111 | 0.333 | |
5 | 3 | 3 | 1/27 | 3.667 | 0.136 | 0.498 | |
5 | 3 | 5 | 1/27 | 4.333 | 0.160 | 0.695 | |
5 | 5 | 1 | 1/27 | 3.667 | 0.136 | 0.498 | |
5 | 5 | 3 | 1/27 | 4.333 | 0.160 | 0.695 | |
5 | 5 | 5 | 1/27 | 5.000 | 0.185 | 0.926 | |
total | 3.000 | 9.889 |
mean of sampling distribution of sample means =E(Xbar)=3.00
and variance of sampling distribution of sample means =9.889-32 =0.889
and standrd deviation of sampling distribution of sample means =sqrt(0.889)=0.943
from above population mean =sampling distribution of sample means
standrd deviation of sample means =population standard deviation/sqrt(3)
variance of sample means =population variance/3
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