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Statistics · Hypothesis testing

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Two-tailed tests

Carrying out a two-tailed binomial test (H1 : p ≠ p0), where a change in either direction counts as evidence. Splitting the significance level α into two tails of α/2, finding both critical regions and the actual significance level (the sum of the two tail probabilities), the equivalent p-value method (compare the observed tail with α/2, or 2× the tail with α), and writing a full conclusion in context.

Statistics · Hypothesis testing

Two-tailed tests

Carrying out a two-tailed binomial test (H1 : p ≠ p0), where a change in either direction counts as evidence. Splitting the significance level α into two tails of α/2, finding both critical regions and the actual significance level (the sum of the two tail probabilities), the equivalent p-value method (compare the observed tail with α/2, or 2× the tail with α), and writing a full conclusion in context.

Why it works

A two-tailed test asks whether a proportion has changed, with no direction specified, so H1:p≠p0H_1 : p \ne p_0. Evidence can come from the result being unusually high or low, so the significance level α\alpha is split between the two tails — α2\tfrac{\alpha}{2} in each. A 5%5\% two-tailed test therefore uses 2.5%2.5\% at each end.

Critical-region method. Assume H0:p=p0H_0 : p = p_0, so X∼B(n,p0)X \sim B(n, p_0).
  • Lower critical value c1c_1: the largest cc with P(X≤c)≤α2P(X \le c) \le \tfrac{\alpha}{2}.
  • Upper critical value c2c_2: the smallest cc with P(X≥c)≤α2P(X \ge c) \le \tfrac{\alpha}{2}.
The critical region is X≤c1X \le c_1 or X≥c2X \ge c_2, and you reject H0H_0 if the observed value lies in either piece. The actual significance level is the sum of the two actual tail probabilities, P(X≤c1)+P(X≥c2)P(X \le c_1) + P(X \ge c_2) — usually a little under α\alpha.

p-value method. Find which tail the observation is in — compare it with the expected value np0np_0. If the observed xx is below np0np_0, find P(X≤x)P(X \le x); if above, find P(X≥x)P(X \ge x). Compare that tail probability with α2\tfrac{\alpha}{2} (equivalently, double it to get the p-value and compare with α\alpha). If it is smaller, reject H0H_0.

Two common slips. Compare the single tail with α2\tfrac{\alpha}{2}, not α\alpha (or double the tail before comparing with α\alpha); and choose the tail by which side of np0np_0 the observation lies. As always, finish with a conclusion in context.