Test means and variances using critical values from a sampling distribution.

A claims analyst sees an average severity of $4,820 in 36 recent files and asks whether that materially exceeds the $4,500 priced into rates. Critical-value testing turns that question into a yes-or-no answer at a stated error rate.

The five-step recipe. State $H_0$ and $H_1$. Pick $\alpha$. Compute the test statistic from the sample. Look up critical values from the relevant sampling distribution. Reject $H_0$ if the statistic falls in the rejection region; otherwise fail to reject.

KEY: The sampling distribution belongs to the test statistic under $H_0$. Critical values are quantiles of that null distribution, not of the data.

Testing a single mean, variance known. When the population variance $\sigma^2$ is known (rare in practice, common on exams), the standardized sample mean is exactly standard normal under $H_0:\mu=\mu_0$.

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

For a two-sided test at $\alpha = 0.05$, reject if $|Z| > 1.96$. One-sided upper: reject if $Z > z_\alpha = 1.645$. One-sided lower: reject if $Z < -1.645$.

Common mistakes

• Using z when $\sigma$ is estimated. A sample $s$ forces a t-test regardless of $n$. On the exam, defaulting to $z = 1.96$ when the problem gave $s$ loses the point.
• Symmetrizing the chi-square test. Cutoffs at $\chi^2_{n-1,\,\alpha/2}$ and $\chi^2_{n-1,\,1-\alpha/2}$ are different numbers; for $n-1=19$ they are 32.85 and 8.91. Comparing $|W|$ to one of them is wrong.
• Splitting $\alpha$ on a one-sided test. One-sided $\alpha = 0.05$ uses $z = 1.645$, not 1.96. Using 1.96 for a one-sided test makes rejection too conservative.

Bottom line

• Reject $H_0$ when the test statistic falls in the rejection region defined by critical values at significance level $\alpha$; the null sampling distribution fixes those critical values.
• Mean, $\sigma$ known: $Z = (\bar{X} - \mu_0)/(\sigma/\sqrt{n}) \sim N(0,1)$. Two-sided cutoff at $\pm z_{\alpha/2}$.
• Mean, $\sigma$ unknown: $T = (\bar{X} - \mu_0)/(s/\sqrt{n}) \sim t_{n-1}$. Two-sided cutoff at $\pm t_{n-1,\alpha/2}$; an estimated $s$ forces a t-test for any $n$.
• Single variance: $W = (n-1)s^2/\sigma_0^2 \sim \chi^2_{n-1}$. Two-sided cutoffs at $\chi^2_{n-1,\alpha/2}$ (upper) and $\chi^2_{n-1,1-\alpha/2}$ (lower), which are not symmetric.

Exam shortcut

If the problem states a sample standard deviation, immediately write t with $n-1$ df; do not pause to consider z. For variance questions, scan for a single $\sigma_0$ (chi-square) versus two sample variances (F). When the F-statistic is below 1, flip the ratio and swap the degrees-of-freedom order; never look up lower-tail F values you do not have to.

The full lesson (about 2,464 words, 16 min read) adds 8 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• B4

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