Describe hypothesis tests and their decision regions
Use a null model to decide whether an observation gives sufficient evidence for a stated alternative.
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
- Calculate probability under a model.
- Distinguish a population parameter from an observation.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does a test statistic summarise?
What you will learn
- Define a test statistic and significance level.
- Interpret rejection and non-rejection in context.
Make the comparison explicit
A factory claims its defect probability is 0.2. H₀: p = 0.2 is the null model; H₁: p > 0.2 expresses a suspected increase. The count X in twenty independent items is the test statistic. Its null distribution is B(20,0.2).
Set the significance level before observing the count. A 5% level bounds the probability of a false rejection under the specified null model. The rejection or critical region contains outcomes sufficiently extreme in the alternative direction.
Use cautious conclusions
The complementary region is called the acceptance region in the syllabus. Say do not reject H₀: the evidence is insufficient at this level. That does not establish that p equals 0.2.
A p-value measures a specified tail extremeness under the null, not the probability that the null is true. A statistical result alone does not establish a cause, an important effect or a guaranteed future outcome.
Pause and explain
How should p-value 0.08 be interpreted at level 0.05?
Worked example
Interpret a one-sided defect test with observed null tail probability 0.032 at the 5% level.
Show the worked solution
- The relevant upper-tail probability is 0.032.
- Compare 0.032 with 0.05.
- Reject H₀; there is evidence that the defect probability has increased.
Answer Reject H₀ at 5%; evidence of an increase, not certainty.
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.
A defect-increase test has null tail 0.032 and level 0.05. State its decision.
Give me a hint
Compare the tail probability with the stated level.
Compare my reasoning
- Use the increase-direction tail.
- 0.032 < 0.05.
- Reject H₀ and state evidence of increased defect probability.
Reject H₀; evidence of increased defect probability.
Look for these in your work
- Use the stated level.
- Give a contextual conclusion.
The same specified test gives tail 0.08. What is the conclusion?
Give me a hint
Keep the 5% rule rather than changing it after the result.
Compare my reasoning
- 0.08 exceeds 0.05.
- The observation lies outside the rejection rule.
- Do not reject H₀; evidence of an increase is insufficient.
Do not reject H₀ at 5%.
Look for these in your work
- Compare numerical values.
- Avoid proving the null.
Does a reported p-value 0.032 mean a 3.2% chance that the null claim is true?
Give me a hint
Separate probability conditional on a model from probability of the model.
Compare my reasoning
- The calculation assumes H₀.
- It measures the specified extremeness of observations.
- No prior or posterior model for hypothesis truth was supplied.
No; it is a null-model tail probability.
Look for these in your work
- State the conditioning model.
- Avoid an unsupported posterior interpretation.
Common mistakes
- Do not interpret the p-value as P(H₀ true).
- Non-rejection is not proof of equality.
- State the contextual alternative.
What can you explain now?
Does not rejecting H₀ establish it as true?
Compare with the explanation
No.
The chosen test did not find enough evidence against it.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Define the null model, statistic, level and alternative before making a contextual decision.
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.