Normal population: exact results Let X_1,,X_n be independent observations from N(mu,sigma2). Linear combinations of independent normal variables are normal. T_n N(nmu,nsigma2), X N(mu,(sigma2)/(n)) Non-normal population: Central Limit Theorem If X_1,,X_n are independent identically distributed observations with finite mean mu and finite variance sigma2, then as n becomes large, the standardised total and mean approach a standard normal distribution. T_n N(nmu,nsigma2), X N(mu,(sigma2)/(n)) - The approximation usually improves as n increases. - n>30 is a rough classroom guideline, not a universal threshold; strong skewness or heavy tails may require a larger sample. - Sampling from a non-normal population does not make each observation normal. It is the distribution of the total or mean that becomes approximately normal. - If the population is already normal, the displayed sampling distributions are exact for every positive integer n.

Let \(X_1,\ldots,X_n\) be independent observations from \(N(\mu,\sigma^2)\). Linear combinations of independent normal variables are normal.

\[T_n\sim N(n\mu,n\sigma^2),\qquad \overline X\sim N\left(\mu,\frac{\sigma^2}{n}\right)\]

Exact sampling distributions

Non-normal population: Central Limit Theorem

If \(X_1,\ldots,X_n\) are independent identically distributed observations with finite mean \(\mu\) and finite variance \(\sigma^2\), then as \(n\) becomes large, the standardised total and mean approach a standard normal distribution.

\[T_n\mathrel{\dot\sim}N(n\mu,n\sigma^2),\qquad \overline X\mathrel{\dot\sim}N\left(\mu,\frac{\sigma^2}{n}\right)\]

CLT approximations

The approximation usually improves as \(n\) increases.

\(n>30\) is a rough classroom guideline, not a universal threshold; strong skewness or heavy tails may require a larger sample.

Sampling from a non-normal population does not make each observation normal. It is the distribution of the total or mean that becomes approximately normal.

If the population is already normal, the displayed sampling distributions are exact for every positive integer \(n\).

Sampling Distributions and the Central Limit Theorem

Sampling Distributions and the Central Limit Theorem