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Statistics · Statistical sampling

Chapter 1 · 3

The idea

Estimation and confidence intervals

The distribution of the sample mean and the Central Limit Theorem, unbiased estimates of a population mean and variance from sample data, and confidence intervals for a mean and for a proportion — building them, interpreting them, and choosing the sample size for a required width.

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Statistics · Statistical sampling

Estimation and confidence intervals

The distribution of the sample mean and the Central Limit Theorem, unbiased estimates of a population mean and variance from sample data, and confidence intervals for a mean and for a proportion — building them, interpreting them, and choosing the sample size for a required width.

Why it works

The sample mean is itself random

Take a sample of nn values from a population with mean μ\mu and variance σ2\sigma^2 and average them: a different sample would give a different average. As a random variable, the sample mean Xˉ\bar{X} satisfies E(Xˉ)=μ,Var(Xˉ)=σ2n,\mathrm{E}(\bar{X}) = \mu, \qquad \mathrm{Var}(\bar{X}) = \frac{\sigma^2}{n}, by the linear-combination rules. Averaging doesn't move the centre, but it shrinks the spread — the standard deviation of Xˉ\bar{X} is σ/n\sigma/\sqrt{n}, so quadrupling the sample size halves it.

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