Approximate a Poisson test when the null mean is large
Use equal null mean and variance, continuity correction and explicit approximation limits.
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 Poisson moments.
- Standardise a normal variable.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What SD applies to the normal approximation of Po(20)?
What you will learn
- Approximate a large-mean Poisson tail.
- Retain exact calculation for an unsuitable small mean.
Approximate under the null mean
For a fixed exposure with H₀: λ = 20 and H₁: λ > 20, use X ~ Po(20) and approximate Y ~ N(20,20). The mean is sufficiently large for the syllabus normal approximation guide. An observed twenty-nine uses boundary 28.5, giving z = 8.5/√20 ≈ 1.90066.
The approximate upper tail is about 0.02867, so reject at 5% with evidence of an increased expected count. For observed twenty-seven, boundary 26.5 gives z ≈ 1.45344 and tail about 0.07305, so do not reject.
A method has conditions
For null mean four, use direct Poisson probabilities rather than automatically invoking a normal approximation. An observed high count does not make the null mean large: adequacy is assessed under the reference model.
A normal count approximation does not preserve all exact masses. If the approximate tail is close to the level, check the original Poisson probability before making an exact-model decision.
Pause and explain
How should a test under Po(4) be evaluated here?
Worked example
Approximate an increase test from λ = 20 using observed twenty-nine.
Show the worked solution
- Y ~ N(20,20) under the null approximation.
- Correct ≥ 29 to > 28.5 and standardise.
- Tail ≈ 0.02867; reject at 5%.
Answer Approximately 0.02867; reject.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
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.
Approximate the Po(20) upper test for X = 29 at 5%.
Give me a hint
Use boundary 28.5 and SD √20.
Compare my reasoning
- Approximate by N(20,20).
- z = 8.5/√20 ≈ 1.90066.
- Tail ≈ 0.02867; reject at 5%.
Approximately 0.02867; reject.
Look for these in your work
- Use equal mean and variance.
- Correct the integer event.
Approximate the same test for X = 27.
Give me a hint
Use the lower edge of the cell centred on twenty-seven.
Compare my reasoning
- Boundary is 26.5.
- z = 6.5/√20; tail ≈ 0.07305.
- Do not reject at 5%.
Approximately 0.07305; do not reject.
Look for these in your work
- Use the correct cell edge.
- Keep the same level.
Choose the method for an upper test under Po(4) with observed nine.
Give me a hint
Judge approximation by the null parameter rather than the observation.
Compare my reasoning
- The reference mean is four.
- That is not a suitable large-mean normal case here.
- Evaluate the inclusive Poisson upper tail directly.
Use exact Poisson probabilities under λ = 4.
Look for these in your work
- Check null-model adequacy.
- Do not change the parameter.
Common mistakes
- Use null mean for approximation suitability.
- Poisson SD is √λ.
- Correct the count boundary.
What can you explain now?
Does observed nine make a Po(4) null suitable for a large-mean approximation?
Compare with the explanation
No.
The approximation is judged under mean four, not the observed count.
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Make it stick
A large-mean Poisson test approximation uses N(λ₀,λ₀) with corrected count boundaries.
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