Information to record **Quantity** | **Notation** Sample size | n Sample mean | x Known population variance | sigma2 Unbiased sample variance when sigma2 is unknown | s2= 1n-1 (x_i- x)2 Choosing the mean-test statistic -> Test H_0:mu= _0 using the sample mean. -> Normal population: the sampling distribution of X is normal for every n. -> Non-normal population: use the CLT approximation only for a sufficiently large iid sample with finite variance. -> Normal, known sigma: exact Z= X- _0sigma/ n. -> Normal, unknown sigma: H2 plug-in z convention; exact t theory is outside this unit. -> Large iid sample, known sigma: CLT z approximation. -> Large iid sample, unknown sigma: H2 plug-in z approximation. -> known -> unknown -> known -> unknown z_ obs=(x- _0)/(sigma/ n), under the H2 large-sample convention, z_ obs=(x- _0)/(s/ n) n>30 is only a rough classroom guide. The validity of a normal approximation also depends on independence, identical distribution, finite variance and the shape of the population.

Test \(H_0:\mu=\mu_0\) using the sample mean.

Normal population: the sampling distribution of \(\overline X\) is normal for every \(n\).

Non-normal population: use the CLT approximation only for a sufficiently large iid sample with finite variance.

Normal, known \(\sigma\): exact \(Z=\dfrac{\overline X-\mu_0}{\sigma/\sqrt n}\).

Normal, unknown \(\sigma\): H2 plug-in z convention; exact t theory is outside this unit.

Large iid sample, known \(\sigma\): CLT z approximation.

Large iid sample, unknown \(\sigma\): H2 plug-in z approximation.

Choosing the mean-test statistic

\[z_{\rm obs}=\frac{\overline x-\mu_0}{\sigma/\sqrt n}\quad\text{or, under the H2 large-sample convention,}\quad z_{\rm obs}=\frac{\overline x-\mu_0}{s/\sqrt n}\]

Observed statistic

\(n>30\) is only a rough classroom guide. The validity of a normal approximation also depends on independence, identical distribution, finite variance and the shape of the population.

Quantity	Notation
Sample size	\(n\)
Sample mean	\(\overline x\)
Known population variance	\(\sigma^2\)
Unbiased sample variance when \(\sigma^2\) is unknown	\(s^2=\dfrac{1}{n-1}\sum(x_i-\overline x)^2\)

Test Statistics and Validity Conditions

Test Statistics and Validity Conditions