How to Use a Critical Value to Reject H0

You have a test statistic and a critical value from a table or calculator. The decision is simple: if the test statistic falls in the rejection region, beyond the critical value, you reject H₀. If it does not, you fail to reject H₀. This critical value decision rule is the backbone of hypothesis testing. The confusion starts with which critical value to use and which direction to compare.

Match the Tail to the Alternative Hypothesis

Your rejection region depends on the alternative hypothesis. For a right-tailed test (H₁: parameter > value), the critical value is positive. Reject H₀ if your test statistic is greater than that critical value. For a left-tailed test (H₁: parameter < value), the critical value is negative. Reject H₀ if your test statistic is less than that critical value.

For a two-tailed test (H₁: parameter ≠ value), there are two critical values: one negative and one positive. Reject H₀ if the absolute value of your test statistic exceeds the positive critical value. This is the same as saying the test statistic lies in either tail beyond the critical boundaries.

Failure case: Using a one-tailed critical value when the test is two-tailed inflates the rejection region and increases your Type I error. Check the alternative hypothesis before you look up the number.

Sketch the sampling distribution of the test statistic under H₀. Mark the critical value(s) on the horizontal axis. Shade the area beyond the critical value(s), that is your rejection region. For a right-tailed test at α = 0.05 with Z, draw the standard normal curve, mark 1.645, and shade everything to the right. For a two-tailed test at α = 0.05 with Z, mark both -1.96 and +1.96, and shade both tails beyond those points. The shaded area totals α. Your job is to see whether the test statistic lands in the shaded zone.

Z-Test (Right-Tailed, α = 0.05)

You know the population standard deviation. Critical value = 1.645. Your test statistic is greater than 1.645, so it falls in the rejection region. Reject H₀. There is significant evidence for the alternative.

t-Test (Two-Tailed, α = 0.05, df = 20)

You estimated sigma from the sample. Critical values are ±2.086. Your test statistic is 1.95. The absolute value 1.95 is less than 2.086. Fail to reject H₀. The result is not statistically significant. (OpenIntro Statistics, 4th ed., table D.2 gives 2.086 for df=20.)

Chi-Square Test (Right-Tailed, α = 0.05, df = 5)

Used for a variance test or goodness-of-fit. Critical value from a chi-square table or software is 11.070. Your test statistic is greater than 11.070, so reject H₀. The observed variance differs from the null. (NIST/SEMATECH e-Handbook section 1.3.6.7 gives the 0.05 quantile for χ² df=5 as 11.070.)

F-Test (ANOVA, Right-Tailed, α = 0.05, df₁ = 3, df₂ = 20)

Critical value from an F-table is 3.098. Your F-statistic is greater than 3.098. Reject H₀. At least one group mean differs. (Moore, McCabe & Craig, Introduction to the Practice of Statistics, 9th ed., table E gives 3.098 for df₁=3, df₂=20.)

The critical value you use in a hypothesis test is the same as the critical value you use for a confidence interval at the same confidence level. For a 95% confidence interval, the critical value is 1.96 for a Z-interval (σ known) and the t-critical value for a t-interval (σ estimated). If the null hypothesis value falls outside that interval, you would reject H₀ in a two-tailed test at α = 0.05. This duality is central to statistical inference. The numbers are identical; the only difference is what you compare them to: a test statistic in the decision rule, or a parameter value in the interval.

Reject H₀: "The test statistic of [value] exceeds the critical value of [value] at the [α] significance level. We reject the null hypothesis. There is sufficient evidence to support the alternative hypothesis."

Fail to reject H₀: "The test statistic of [value] does not exceed the critical value of [value] at the [α] significance level. We fail to reject the null hypothesis. The result is not statistically significant."

Never say "accept H₀." You either reject or fail to reject. This is standard convention in every textbook (e.g., OpenIntro Statistics, 4th ed.). The ASA 2016 statement reinforces that statistical significance does not equal practical importance. If you reject, also report an effect size or the magnitude of the difference.

The test statistic is computed from your sample data. The critical value is the threshold from the sampling distribution. They are different numbers and serve different roles. The decision rule is a direct comparison: if the test statistic lies in the rejection region (beyond the critical value), the null is unlikely given the data. If it does not, the data do not provide enough evidence to reject. This is the critical value decision rule in action. Common failure: comparing the test statistic to the wrong critical value (e.g., using a Z critical when sigma is unknown). Use t when sigma is estimated from the sample, regardless of sample size.