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7. Sampling distribution, sample by sample
A small population, typed in. Every possible sample of size n, its mean, and the exact distribution of , with or without replacement.
| x̄ | Samples | P(x̄) | P(x̄ or less) |
|---|---|---|---|
| 1 | 1 | 0.0625 | 0.0625 |
| 1.5 | 2 | 0.125 | 0.1875 |
| 2 | 3 | 0.1875 | 0.375 |
| 2.5 | 2 | 0.125 | 0.5 |
| 3 | 1 | 0.0625 | 0.5625 |
| 3.5 | 2 | 0.125 | 0.6875 |
| 4 | 2 | 0.125 | 0.8125 |
| 4.5 | 2 | 0.125 | 0.9375 |
| 6 | 1 | 0.0625 | 1 |
- Population mean μ3Σx / N = 12 / 4
- Population SD σ1.8708Divide by N = 4 because this is the whole population. σ² = 3.5.
- Possible samples16N^n = 4^2. Order counts and repeats are allowed, so every one is equally likely.
- Average of all the sample means3It equals μ: is an unbiased estimator.
- Standard error of 1.3229σ / √n = 1.8708 / √2. Same as the SD of the list of means.
- P( ≤ 2), exact0.3756 of the 16 samples have a mean of 2 or less.
- Normal approximation0.2248z = (2 − 3) / 1.3229 = -0.7559. Rough at a small n; the exact count above is the answer.