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NSWMaths Extension 1Quick questions

Statistical Analysis (ME-S1)

Quick questions on The sample proportion p-hat as a random variable: mean p, variance pq/n, and its normal approximation

3short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the normal approximation p^∼N(p, pq/n)\hat{p} \sim N(p,\, pq/n)?
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For large nn the binomial is well approximated by a normal distribution (the central limit theorem). Dividing through by nn carries that approximation over to p^\hat{p}: it becomes approximately normal with the same mean and variance we just derived,
What is the effect of sample size nn?
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The mean of p^\hat{p} is always pp, no matter the sample size, so a bigger sample does not move the centre, it sharpens it. Since
What is linear thinking about sample size?
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Halving the margin of error needs four times the sample, not twice, because the SD carries n\sqrt{n}, not nn.

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