Construct an equal-tail two-sided Poisson decision
Allocate a two-sided level explicitly when the discrete distribution is asymmetric.
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
- Evaluate Poisson cumulative probabilities.
- Understand two-sided tail allocation.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
How should the two-sided Poisson tails be found?
What you will learn
- Find two Poisson critical tails.
- Calculate actual size without assuming symmetry.
Use separate tail searches
For Po(4), H₁: λ ≠ 4 and nominal 5%, allocate at most 0.025 to each tail. The lower event X ≤ 0 has probability e⁻⁴ ≈ 0.0183156389, while X ≤ 1 has probability about 0.0915781944. The upper event X ≥ 9 has probability about 0.0213634345, while X ≥ 8 has probability about 0.0511336158.
The stated equal-tail rejection rule is X = 0 or X ≥ 9. Its disjoint probabilities sum to about 0.0396790734. The two tail probabilities are unequal even though their allowances are equal.
State the convention in the conclusion
Under this rule, zero or nine events reject; eight events do not. The conclusion is evidence of a rate change, with the observed direction noted separately. Non-rejection still does not prove the original rate.
This lesson uses equal-tail allocation. Do not silently apply a symmetric-tail shortcut to Poisson masses or substitute a different two-sided convention after seeing the count.
Pause and explain
What is the actual size of the stated rule?
Worked example
Construct the equal-tail 5% Po(4) rejection rule.
Show the worked solution
- Lower budget admits only X = 0.
- Upper budget admits X ≥ 9.
- Actual size is e⁻⁴ + P₀(X ≥ 9) ≈ 0.0396790734.
Answer X = 0 or X ≥ 9; actual size 0.0396790734.
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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.
Find the equal-tail 5% Po(4) rejection rule.
Give me a hint
Allow at most 0.025 in each direction.
Compare my reasoning
- The lower boundary is zero.
- The upper boundary is nine.
- Reject for X = 0 or X ≥ 9.
X = 0 or X ≥ 9.
Look for these in your work
- Check separate tails.
- Use their union.
Calculate its actual Type I probability.
Give me a hint
Add the two disjoint null region probabilities.
Compare my reasoning
- Lower probability is e⁻⁴ ≈ 0.0183156389.
- Upper probability is approximately 0.0213634345.
- Sum gives 0.0396790734.
Actual size approximately 0.0396790734.
Look for these in your work
- Use the null mean.
- Do not impose symmetry.
Classify observed eight under this rule.
Give me a hint
Check membership of the pre-specified union.
Compare my reasoning
- Eight is not zero.
- Eight is below nine.
- It lies in the non-rejection region.
Do not reject at the stated equal-tail 5% level.
Look for these in your work
- Use the specified rule.
- Avoid changing conventions afterwards.
Common mistakes
- Poisson tails are not symmetric.
- Each tail must fit its budget.
- Do not double one actual tail to obtain region size.
What can you explain now?
Are equal tail allowances the same as equal achieved tail probabilities?
Compare with the explanation
No.
Discrete probabilities need not use equal amounts of their allowances.
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Make it stick
Equal-tail budgets for asymmetric count models require two separate boundary searches.
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.