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Sampling distribution: population 2,4,6,8

Short answer

Using N=4 population 2,4,6,8, all six samples of size two without replacement have means 3,4,5,5,6,7. Population mean is 5 and mean of sample means is 30/6=5, so the equality is verified.

A population consists of five numbers 2, 4, 6 and 8 a) Enumerate all possible sample of size two without replacement b) Show that the mean of the sampling distribution of sample mean is equal to population mean

The population has four numbers: 2, 4, 6, 8 (N = 4; the question says "five" but lists four, so N = 4 is used). n = 2, without replacement, so there are ⁴C₂ = 6 samples.

a) All possible samples of size two

No.SampleSample mean
1(2, 4)3
2(2, 6)4
3(2, 8)5
4(4, 6)5
5(4, 8)6
6(6, 8)7

b) Mean of sample means equals population mean

Population mean: μ = (2 + 4 + 6 + 8)/4 = 20/4 = 5

Sum of sample means = 3 + 4 + 5 + 5 + 6 + 7 = 30

Mean of sample means = 30/6 = 5

Since the mean of the sampling distribution of x̄ = 5 = μ = 5, the statement is shown to be true.

✓ Answer checked by Claude Sonnet 5.5

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