🎲 Critical Value

Critical Value#

The cutoff from a reference distribution that a test statistic must pass to reject the null.

Important

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What it is#

A critical value is the cutoff on a distribution that marks the rejection boundary for a hypothesis test. It separates the acceptance region (fail to reject \(H_0\)) from the rejection region (reject \(H_0\)), and depends on three things: the significance level \(\alpha\), whether the test is one- or two-tailed, and the distribution used (Z, t, \(\chi^2\), F).

The decision rule#

If the test statistic exceeds the critical value in magnitude, reject \(H_0\); if it falls inside, fail to reject.

Values by distribution#

Z (large \(n\), known \(\sigma\)) — two-tailed \(\alpha = 0.05\) gives \(\pm 1.96\); one-tailed, \(1.645\).
t (small \(n\), unknown \(\sigma\)) — two-tailed, \(\alpha = 0.05\), \(df = 10\) gives \(\pm 2.228\); as \(n\) grows, \(t \to z\).
χ² (goodness-of-fit, independence) — \(\alpha = 0.05, df = 4\) gives \(\approx 9.49\).
F (ANOVA) — \(\alpha = 0.05, df_1 = 3, df_2 = 20\) gives \(\approx 3.10\).

Two roles#

The same critical value drives both confidence intervals — \(\text{estimate} \pm (\text{critical value}) \times \text{SE}\) (a 95% z-interval for a mean of 100 with \(\text{SE} = 2\) is \([96.08, 103.92]\)) — and hypothesis tests.

Worked test#

Test \(H_0 : \mu = 50\) vs \(H_1 : \mu \neq 50\) with sample mean 53, \(\sigma = 10, n = 100\):

\[z = \frac{53 - 50}{10/\sqrt{100}} = \frac{3}{1} = 3.0.\]

At \(\alpha = 0.05\) two-tailed the critical value is \(\pm 1.96\); since \(3.0 > 1.96\), reject \(H_0\).

Theme: Probability & Statistics Foundations · All terminology

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Mind map — connected ideas

Margin of Error (MoE) · Standard Error (SE) · Type I Error · Frequentist · True Mean (Population Mean) · Bootstrap Confidence Intervals (CIs)

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More in Probability & Statistics Foundations

Beta Distribution · Confidence Level · Correlation · Cumulative Distribution Function (CDF) · Frequentist · IID (Independent and Identically Distributed) · Likelihood · Margin of Error (MoE) · Mean · Median · Normal Distribution · Outlier · Population Proportion · Probability