S.O.S. Mathematics CyberBoard

[brightdim]

You write the population values {1, 3, 5, 7} on slips of paper and put them in a box. Then you randomly choose two slips of paper, with replacement. List all possible samples of size n = 2 and calculate the mean of each. These means form the sampling distribution of the sample means. Find the mean, variance, and standard deviation of the sample means.


I started out by making a table for the possible combinations of samples and their sample means and used that to make a probability distribution for the mean (x) of the sample means, their frequencies, and probability. I tried to use that to find the mean, variance, and standard deviation but I'm not getting the same answers as my book. Can anyone tell me how to solve this?

[Bilbo]

[brightdim]

The book listed the variance as 2.236 and the standard deviation as 1.495. My work and answers are below. I hope it's legible enough.

Sample --> Sample Mean
1,1 --> 1
1,3 --> 2
1,5 --> 3
1,7 --> 4
3,1 --> 2
3,3 --> 3
3,5 --> 4
3,7 --> 5
5,1 --> 3
5,3 --> 4
5,5 --> 5
5,7 --> 6
7,1 --> 4
7,3 --> 5
7,5 --> 6
7,7 --> 7

mean = 64/16 = 4

x --> P(x) --> (x-mean)^2, P(x)(x-mean)^2
1 --> .0625 --> 9 --> .5625
2 --> .1250 --> 4 --> .5000
3 --> .1875 --> 1 --> .1875
4 --> .2500 --> 0 --> 0
5 --> .1875 --> 1 --> .1875
6 --> .1250 --> 4 --> .5000
7 --> .0625 --> 9 --> .5625

variance = EP(x)(x-mean)^2 = 2.5
standard deviation = (2.5)^(1/2) = 1.581

[/list]

[Bilbo]

[brightdim]

how did you know the variance was five just from the sample?

[Bilbo]

[brightdim]

just to clarify...to find the variance from the population, you can use the population variance equation: (E(x-mean)^2) / N,. And from there, I can divide it by the square root of the number in the sampe (here it would be 2) and that would also give me the standard deviation?

I'm sorry for so many questions, but I want to understand how the problem's worked rather than just have someone solve it for me.

[Bilbo]