Set up the test

Choose a tail from the claim

Define the population quantity and translate the claim into an alternative hypothesis before using the sample. Next check the sampling assumptions and choose a reference distribution. Only then standardise the observed mean.

Always define the population mean $\mu$ and state $H_0:\mu=\mu_0$. Choose the alternative from the claim:

• A suspected decrease uses $H_1:\mu<\mu_0$.
• An increase uses $H_1:\mu>\mu_0$.
• A change in either direction uses $H_1:\mu\ne\mu_0$. Choose this before looking at the observed sign. The source’s table reverses the words “increase” and “decrease” beside its correct inequalities; follow the inequalities. Self-Practice 1 below applies this distinction.

Match the statistic to its assumptions

A statistic such as $\bar X$ is random before sampling; its observed value $\bar x$ is a number. Under $H_0$, standardise using $(\bar X-\mu_0)/(\sigma/\sqrt n)$ when the population variance is known. Choose the reference model before calculating:

• For independent observations from a normal population with known variance, the sample mean has an exact normal distribution at any sample size.
• For a sufficiently large independent random sample with finite variance, use a justified CLT approximation. If the population variance is unknown, use its unbiased estimate and state that the resulting normal test is approximate.
• A small sample from an unspecified population does not supply a normal reference model merely because a mean and a variance can be calculated.

Define the statistic before substituting its observed value. For a known population standard deviation, write

$$Z=\frac{\bar X-\mu_0}{\sigma/\sqrt n}.$$

Under the exact normal null model, $Z\sim N(0,1)$. Replacing $\sigma$ by $s$ needs the large-sample qualification above.

Reference: the retained variance-case comparison

The source’s four cases need these qualifications:

Normal, known $\sigma^2$, any sample size

Sampling justification: Exact normal sample mean

Reference distribution: Exact standard normal

Non-normal or unspecified, known $\sigma^2$, large random sample

Sampling justification: CLT, with independent observations and finite variance

Reference distribution: Approximately standard normal

Normal, unknown variance estimated by $S^2$

Sampling justification: The source labels this exactly normal for any $n$

Reference distribution: In fact $(\bar X-\mu_0)/(S/\sqrt n)$ has a $t_{n-1}$ distribution; a normal approximation requires large-sample justification

Non-normal or unspecified, unknown variance, large random sample

Sampling justification: CLT and a suitable variance estimate

Reference distribution: Approximately standard normal

The source uses $n\ge30$ as its classroom large-sample guideline; this is not a guarantee for every population. Small-sample tests with unknown variance are not made exact z-tests by assuming normality. Keep this distinction when selecting the method. See NIST’s known-versus-estimated standard deviation distinction.