What Sample Summary Must You Calculate Before Choosing a Critical Value?

The most common mistake is treating the rule "n ≥ 30 means use Z" as fact. The choice between Z and t depends on whether the population standard deviation (σ) is known or estimated from the sample, not on sample size alone. What sample summary must you calculate before choosing a critical value? You must calculate the sample size (n), sample mean (x̄), sample standard deviation (s), and determine whether σ is known. These descriptive quantities and sample facts determine whether z, t, chi-square, or F is appropriate.

A concrete example makes the decision clear. Suppose you have a sample of n=25, x̄=52, and s=8, and the population σ is unknown. Because σ is unknown, you must use the t-distribution. The degrees of freedom are n−1 = 24. For a two-tailed test at α=0.05, the critical value from the t-distribution is t_{0.975,24}=2.064. If you had used the Z critical value of 1.96 instead, you would have a threshold that is too small, increasing the chance of a Type I error (rejecting a true null).

Now consider a second sample: n=100, x̄=50, and σ=10 is known. Because σ is known, you use the Z-distribution. For a two-tailed test at α=0.05, the critical value is 1.96. The sample size here is 100, but the deciding factor is that σ is known, not the sample size. Both examples follow the rule: known σ → Z, estimated σ → t.

The calculation of the test statistic itself requires the sample summary. For a single mean test, if σ is known, the test statistic is Z = (x̄ − μ₀)/(σ/√n). If σ is unknown, the test statistic is t = (x̄ − μ₀)/(s/√n) with df=n−1.

For chi-square and F tests, the sample summary must include additional information: the number of categories or groups to compute degrees of freedom. For a chi-square goodness-of-fit test, df = k−1 (where k is the number of categories). For an F-test in a one-way ANOVA, the numerator df is the number of groups minus one, and the denominator df is the total sample size minus the number of groups. Without these values, you cannot correctly identify the critical value from a table or a statistics calculator.

The critical value is the threshold in a test statistic's sampling distribution that defines the rejection region for a null hypothesis. The decision rule is straightforward: if σ is known, use Z; if σ is estimated from the sample by s, use t; for goodness-of-fit or independence tests, use chi-square; for ANOVA or comparing two variances, use F.

The degrees of freedom parameter changes the critical value. For t, more df brings the critical value closer to the Z critical value. For chi-square and F, the critical value increases with df. The significance level (alpha) also matters: a smaller alpha makes the critical value larger, making it harder to reject the null. A two-tailed test splits alpha between both tails, so the one-tailed critical value at α/2 has the same magnitude as the two-tailed value at α. These choices involve a trade-off between Type I and Type II error, not a mathematical rule about sample size.

The most common failure case is using Z when σ is unknown, even with a large sample. For example, take a sample of n=50, s=5, and σ unknown. The correct approach is to use the t-distribution with df=49. The t critical value for α=0.05 two-tailed is t_{0.975,49} ≈ 2.009. If you incorrectly use the Z critical value of 1.96, your rejection region is larger than it should be, increasing the probability of a Type I error. The difference of 0.049 may seem small, but it can change the outcome of a hypothesis test when the test statistic is close to the threshold.

The decision rule “known σ → Z, estimated σ → t, goodness-of-fit → chi-square, ANOVA → F” prevents misapplication, but it does not cover every situation. For chi-square and F tests, the sample summary must include the number of categories or groups to compute degrees of freedom. Without that information, you cannot find the correct critical value. Additionally, the ASA 2016 statement reminds you that statistical significance (critical-value-based decisions) is not equivalent to practical importance. The critical value is a threshold for decision-making, not a measure of effect size.

The single thing that most often goes wrong is ignoring the distinction between known and unknown sigma. Check your sample summary before choosing a critical value. If you have any doubt about whether σ is known, assume it is unknown and use t.