Sampling Distribution of p̂

Set a population proportion p and sample size n, then draw samples. Each draw gives one sample proportion p̂. The histogram is the sampling distribution of p̂ — centered at p with spread √(p(1−p)/n).

Population proportion (p)0.50

Sample size (n)30

Samples

Mean of p̂

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SD of p̂

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Theory SE √(p(1−p)/n)

Try: p = 0.5, n = 30, Draw 1000 — the SD of p̂ lands near 0.091. Now slide n to 150 and watch the histogram narrow while staying centered at 0.50. Set p = 0.1, n = 40 (np = 4) and the shape is right-skewed; Large Counts fails.