One-tailed binomial tests

Objective. Calculate probabilities and reach contextual conclusions.

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Learn: the key idea

A one-tailed binomial test checks for evidence of change in one specified direction, such as an increase or decrease in p, by calculating the probability of obtaining a result at least as extreme as observed, assuming H₀ is true. You compare this p-value to the significance level: if the p-value is smaller, you reject H₀ in favour of H₁. For example, testing H₀: p=0.3 vs H₁: p<0.3 with observed X=2 from n=20 trials means calculating P(X≤2) and comparing to 5%. A small p-value means the observed result would be rare if H₀ were true.

Worked example

Under H₀: p=0.5, n=10, a one-tailed test has H₁: p<0.5 and observed X=1. Given P(X≤1)=0.0107, state the conclusion at the 5% level.

1 $Compare p-value to significance level$
2 $0.0107 < 0.05$
3 $Since the p-value is smaller, reject H₀$
4 $Conclude there is significant evidence that p<0.5$

Answer: Reject H₀; significant evidence that p<0.5

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Common mistake. Students calculate P(X=observed value) alone instead of the cumulative tail probability P(X≤observed) or P(X≥observed). Always use the appropriate cumulative probability matching the direction of H₁.

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