Allocate both tails in a two-sided binomial test
Use an explicit equal-tail rule and distinguish two-sided evidence from directional evidence.
Section 6.5 · Hypothesis tests
Exam scope and assessment notes
Probability & Statistics 2 (Paper 6, A Level) · Mathematics 9709, 2026–2027, section 6.5. Paper 6 builds on Paper 5 and assumes Paper 3 calculus. Specify population hypotheses, significance level, test statistic and tail direction before inspecting the result. Calculate tail probabilities under H₀, including the observed discrete value. A p-value is not the probability that H₀ is true. Reject H₀ when the specified tail probability is at most the stated significance threshold; otherwise do not reject. The syllabus term acceptance region means the non-rejection region, not proof of H₀. For two-sided discrete examples here, use an explicitly stated equal-tail allocation; asymmetric discrete models admit different conventions, so do not treat twice the smaller tail as a universal definition. Integer critical regions can have actual Type I probability below the nominal level. Normal approximations need a suitable model and continuity correction for counts; check exact probabilities when approximation could change a close decision. Mean tests use a normal population with known variance or an adequate large independent sample with estimated variance. Small samples with unknown variance do not automatically justify the z method. Type I means rejecting a true null; Type II means not rejecting a false null at a specified alternative. Calculate error probabilities using the fixed decision region under the relevant parameter, not by changing the test after seeing data. Power is one minus Type II probability at that alternative. Quoted normal critical values and table probabilities have rounding limits. Interpret evidence in context without claiming certainty, causation or practical importance. Written practice is self-checked; cumulative assessment and human subject-teacher review remain pending.
Before you begin
- Use symmetric B(n,0.5) probabilities.
- Distinguish one-sided from two-sided questions.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the equal-tail budget for a two-sided 5% test?
What you will learn
- Construct a two-sided binomial critical region.
- State the tail allocation and actual size.
Two directions share the level
Test H₀: p = 0.5 against H₁: p ≠ 0.5 with n = 20 and nominal 5%, allocating 2.5% to each tail. P₀(X ≤ 5) = P₀(X ≥ 15) ≈ 0.0206947327; extending either tail by one count gives 0.0576591492.
The rejection region is X ≤ 5 or X ≥ 15. The two disjoint tails give actual null rejection probability about 0.0413894653, below 0.05 because the counts are discrete.
Interpret the observed tail
For X = 15, the symmetric two-sided p-value is twice P₀(X ≥ 15), about 0.0413894653: reject at 5% with evidence of a change. Observed fourteen does not enter the equal-tail rejection region.
Doubling a selected tail is justified here by the stated symmetric convention. For asymmetric discrete distributions, different two-sided definitions can disagree. State the chosen convention instead of silently treating one formula as universal.
Pause and explain
What is the rejection region for the stated B(20,0.5) rule?
Worked example
Construct the equal-tail 5% rule for B(20,0.5).
Show the worked solution
- Allow at most 0.025 in each tail.
- Choose lower ≤ 5 and upper ≥ 15.
- Sum their probabilities to obtain actual size ≈ 0.0413894653.
Answer Reject for X ≤ 5 or X ≥ 15.
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Construct the equal-tail 5% rule for B(20,0.5).
Give me a hint
Split the level and check neighbouring integer boundaries.
Compare my reasoning
- Allocate 0.025 per tail.
- ≤ 5 and ≥ 15 each have probability 0.0206947327.
- Use their union; actual size ≈ 0.0413894653.
X ≤ 5 or X ≥ 15; actual size 0.0413894653.
Look for these in your work
- Split the level.
- Add disjoint probabilities.
Test the observation X = 15 with this rule.
Give me a hint
Compare the symmetric two-tail probability with 0.05.
Compare my reasoning
- The upper observed tail is 0.0206947327.
- Double it to obtain 0.0413894653 under the stated convention.
- Reject H₀; evidence of a population probability change.
Two-sided probability 0.0413894653; reject.
Look for these in your work
- Use both-tail convention.
- State change rather than certainty.
Can twice the smaller tail be used without stating a convention for every asymmetric discrete model?
Give me a hint
Different definitions of two-sided extremeness need not agree.
Compare my reasoning
- Discrete masses can be asymmetric.
- Equal-tail and probability-ordering procedures can differ.
- Specify the adopted rule and its error bound.
No universal unstated definition; specify the rule.
Look for these in your work
- Recognise asymmetry.
- Keep the decision convention explicit.
Common mistakes
- Do not allocate 5% to each tail.
- Use a union of disjoint regions.
- State the two-sided convention.
What can you explain now?
Would two 5% tails create a 5% test automatically?
Compare with the explanation
No.
Their combined null rejection probability can approach 10%.
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Make it stick
A two-sided test divides its level across directions and states its tail convention.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.