Check proportion interval conditions and extreme outcomes
Recognise sparse success counts, impossible normal-interval endpoints and unsupported certainty at zero observed successes.
Exam scope and assessment notes
Probability & Statistics 2 (Paper 6, A Level) · Mathematics 9709, 2026–2027, section 6.4. Paper 6 builds on Paper 5 and assumes Paper 3 calculus. Distinguish population parameters, sample statistics and repeated-sample random variables. Random selection needs a relevant frame and does not automatically remove undercoverage or non-response. Particular named sampling methods are not required. Sample-mean formulae here assume independent observations from one population with finite variance: E(X̄) = μ, Var(X̄) = σ²/n and SE = σ/√n. A mean of independent normal observations is normal; the CLT gives an approximate normal mean for an adequate large independent sample, not normal individual data. No universal sample-size cutoff guarantees that approximation. Unbiased sample variance uses n − 1, with n > 1. Mean intervals use the normal model with known variance or a large-sample approximation; a small sample with unknown variance does not justify the z method taught here. Proportion intervals use the large-sample normal approximation with adequate success and failure counts; extreme or sparse counts may invalidate it. Quoted two-sided critical values are 1.645 for 90%, 1.960 for 95% and 2.576 for 99%; retain other working unrounded. Confidence describes repeated-sampling coverage, not posterior probability for a fixed realised interval, individual observations or a guarantee. A narrower interval does not repair a biased sample. Hypothesis testing belongs to section 6.5. Written practice is self-checked; cumulative assessment and human subject-teacher review remain pending.
Before you begin
- Construct a proportion interval.
- Distinguish observed counts from population probabilities.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which counts matter when assessing a normal proportion interval?
What you will learn
- Assess the normal proportion approximation.
- Flag extreme counts without inventing a corrected interval.
A large n can still contain too few rare events
One success among 1000 observations has a large total sample but only one success. The normal approximation can remain poor in a very sparse tail. Consider both success and failure counts along with independence and the population model; n alone is not enough.
Zero successes among 100 observations gives p̂ = 0 and the plug-in formula degenerates to [0,0]. This is not evidence that the population probability is exactly zero. A zero observation count does not remove uncertainty about rare outcomes.
Impossible endpoints are a warning
One success among ten observations gives the nominal formula 0.1 ± 1.960√(0.09/10), approximately [−0.0859,0.2859]. The negative endpoint cannot represent a population probability and the small-count approximation is not justified.
Silently clipping the interval to [0,1] does not certify its nominal coverage. Appropriate alternative inference needs its own justification; this lesson flags the limit of the large-sample method rather than presenting an unverified replacement as a syllabus requirement.
Pause and explain
What does zero observed success imply about population p?
Worked example
Assess the normal interval for one success from ten.
Show the worked solution
- p̂ = 0.1.
- Nominal margin ≈ 0.1859, producing a negative lower endpoint.
- Small success counts undermine the normal approximation.
Answer The large-sample normal interval is unsuitable here.
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.
Assess a claimed exact [0,0] interval after zero successes from 100 observations.
Give me a hint
Distinguish a degenerate plug-in result from justified uncertainty.
Compare my reasoning
- p̂ = 0 gives zero plug-in SE.
- Finite absence does not prove population p = 0.
- The extreme-count normal approximation does not justify exact certainty.
Unsupported certainty; zero observed successes do not prove p = 0.
Look for these in your work
- State the approximation failure.
- Avoid a false exact interval claim.
Calculate the nominal 95% normal interval for one success from ten and assess it.
Give me a hint
Use the formula to expose its impossible lower endpoint.
Compare my reasoning
- p̂ = 0.1.
- Margin ≈ 0.1859419 gives [−0.0859,0.2859].
- Small counts and an impossible probability endpoint make this approximation unsuitable.
Nominal [−0.0859,0.2859]; unsuitable normal approximation.
Look for these in your work
- Check endpoint meaning.
- Do not claim clipping restores coverage.
Does one success from 1000 automatically justify the approximation because n is large?
Give me a hint
Inspect the two binary outcome counts.
Compare my reasoning
- Success count is only one.
- Failure count is 999.
- Sparse successes can invalidate a normal proportion approximation despite large n.
No; the success count remains too sparse for an automatic normal claim.
Look for these in your work
- Check both counts.
- Do not rely on total size alone.
Common mistakes
- Check both outcome counts.
- Zero observed successes do not prove zero population probability.
- Clipping endpoints does not establish correct coverage.
What can you explain now?
Does n = 1000 guarantee a good approximation if only one success occurs?
Compare with the explanation
No.
The success count is still sparse.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
Large n alone does not validate a proportion interval; inspect both counts and the model.
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.