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I'm learning stratified random sampling and having hard time to solve this problem. consider the following population of six units:

x1=1, x2=2, x3=2, x4=4 x5=4 x6=5

suppose units 2 3 4 and 5 are in one stratum and units 1 and 6 are in a second stratum. Take a simple random sample of 2 units from the first stratum and a simple random sample of 1 unit from the second stratum. what is the exact distribution of the stratified estimator for the population average.

So, possible mean of sample from first stratum is 2, 3 or 4 and from second stratum is 1 or 5. overall sample mean is

2/3 * 2 + 1/3 * 1 = 5/3
2/2 * 2 + 1/3 * 1 = 7/3
2/3 * 4 + 1/3 * 1 = 3
2/3 * 2 + 1/3 * 5 = 3
2/3 * 3 + 1/3 * 5 = 11/3
2/3 * 4 + 1/3 * 5 = 13/3

and each probability is 1/12, 4/12, 1/12, 1/12, 4/12, 1/12. Now what should I do for finding exact distribution? Thank you

user3818991
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1 Answers1

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You're pretty much done. Count the ways you can get the various combinations leading to specific averages. I did it independently of your work above and get

sample      avg
1,2,2       5/3
1,2,4       7/3
1,2,4       7/3
1,2,4       7/3
1,2,4       7/3
1,4,4       9/3
5,2,2       9/3
5,2,4       11/3
5,2,4       11/3
5,2,4       11/3
5,2,4       11/3
5,4,4       13/3

In other words,the distribution of the average is that you get 5/3 with probability 1/12, 7/3 with probability 4/12 (= 1/3), 9/3 (= 3) with probability 2/12 (= 1/6), 11/3 with probability 4/12 (= 1/3), and 13/3 with probability 1/12.

pjs
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