Simple random sampling

Basic ideas.

Simplest random scheme where every subset of n units has the same chance of forming the sample, $p = \left( \binom{N}{n} \right)$ .

Distribution of sample mean.

\[E(\bar{y}) = \mu\] \[Var(\bar{y}) = (1-f)\frac{\sigma^2}{n} \] \[E(s^2) = \sum^n_1 \frac{(y_i - \bar{y})^2}{n-1} \]

With independent observations, the CLT ensures that $\bar{y}$ is approximately normal, which we can use to create CI(confidence interval)s etc.

Estimating proportions

\[E(p) = \pi\] \[Var(p) = (1-f)\frac{\sigma^2}{n} = (1-f)\frac{p(1-p)}{n-1} \]