Work backwards from a test

When a test conclusion is given, find the unknown

Translate the stated decision into an inequality before solving for the unknown. Keep the alternative, significance units and inclusive rejection rule fixed. Check the resulting boundary against the original test, especially when the unknown is an integer sample size.

The source identifies five ingredients: the hypothesised mean $\mu_0$, observed mean $\bar x$, sample size $n$, known population variance $\sigma^2$ or its unbiased estimate $s^2$, and significance level.

When the significance percentage is unknown, compute p. If a is the numerical percentage, solve $p\le a/100$: 5% means a=5. Equivalently, compare p with $\alpha$ expressed as a probability, so 5% means $\alpha=0.05$.

When $\mu_0$, $\bar x$, $n$ or $s^2$ is unknown, critical-value inequalities often make the algebra clearer.

These are method-selection guides, not exclusions of equivalent CDF inversion.

For a large random sample, no individual normality assumption is needed for the CLT approximation; independence, finite variance and adequate approximation remain necessary. If the population is normal with known variance, the exact sample-mean distribution works at any n.

Respect the domain of the unknown

For a positive integer sample size, solve the inequality before choosing an integer. For a real-valued mean, preserve the unrounded boundary: printing an endpoint to 3 s.f. does not restrict all possible means to a 3 s.f. grid.