S3 exam technique

Hypothesis tests

• Hypotheses in terms of the parameter: μ, μx - μy, ρ, ρ_s; never x̄ or r_s. Define any subscripts.
• For χ², name the model or the 'no association' statement; 'the claim is true' alone is not accepted.
• Conclusions: reject or do not reject H0, then a sentence in context with the direction, and never contradict yourself.

Accuracy and tables

• z values to 4 d.p. (1.6449, 1.9600, 2.3263, 2.5758); examiners deduct marks for 1.645.
• χ² critical values to 3 d.p. from the table; Spearman's and PMCC values to 4 d.p.
• Final answers to 3 s.f. (awrt) unless told otherwise; keep full accuracy in working.
• When a question says 'use tables', a calculator-only answer loses the mark.

Common lost marks

• Var(X1 + X2 + X3) = 3σ² but Var(3X) = 9σ²: write out each variance.
• Standard errors must include sqrt(n).
• Degrees of freedom: subtract estimated parameters; (r - 1)(c - 1) for contingency tables; count cells after combining.
• Spearman's: subtract from 1.
• Unbiased estimators: show E(T) = μ with expectation notation and write a conclusion.
• Sampling: describe the method fully (frame, numbering, random start or random numbers), in context.

Justifying approximations

'n is large, so by the Central Limit Theorem X̄ is approximately normal' is the standard sentence. For large-sample tests also say s² is used as an estimate of σ².

Worked example

A student writes: 'H0: x̄ = 50, H1: x̄ ≠ 50. z = 1.85 > 1.645, so the mean has changed.' List the errors.

1. Hypotheses must use μ, not x̄.
2. For a two-tailed 5% test the critical value is 1.96, not 1.645 (and z values should be given as 1.6449 or 1.9600).
3. 1.85 < 1.96, so H0 is not rejected.
4. The conclusion should be in context and not claim the mean has changed.

Answer: Correct conclusion: there is insufficient evidence that the mean has changed from 50.

Check yourself

1. Which pair of hypotheses earns the mark for a test of whether two population means differ?
   • H0: μx = μy, H1: μx ≠ μy
   • H0: x̄ = ȳ, H1: x̄ ≠ ȳ
   • H0: μx ≠ μy, H1: μx = μy
   • H0: the means are the same, H1: they are not, with no symbols defined
   Show answer

   H0: μx = μy, H1: μx ≠ μy. Population parameters with defined subscripts.

2. Which z value should be quoted for a two-tailed test at the 1% level? Give your answer to 4 decimal places.
   Show answer

   2.5758. P(Z > z) = 0.005 gives 2.5758.

3. X ~ N(10, 2²). Find Var(X1 + X2 + X3), where X1, X2, X3 are independent observations of X.
   Show answer

   12. 3 × 4 = 12 (not 9 × 4, which is Var(3X)).

4. A χ² goodness of fit test for a normal model uses 6 classes after combining, with μ and σ estimated. Find ν.
   Show answer

   3. 6 - 1 - 2 = 3.

5. The S3 paper has 75 marks in 90 minutes. About how many minutes should an 8-mark question take?
   Show answer

   9.6. 1.2 minutes per mark × 8 = 9.6 minutes.

6. Which justification for using a normal distribution for X̄ earns the mark when the population distribution is unknown?
   • The data are independent
   • X̄ is always normal
   • The variance is known
   • n is large, so by the Central Limit Theorem X̄ is approximately normal
   Show answer

   n is large, so by the Central Limit Theorem X̄ is approximately normal. Examiners require large n and the Central Limit Theorem.

7. A χ² test gives X² = 3.2 with critical value 7.815. Which conclusion is best?
   • Reject H0: the model is unsuitable
   • Do not reject H0: there is insufficient evidence that the model is unsuitable
   • Accept H0: the model is proved correct
   • Do not reject H1
   Show answer

   Do not reject H0: there is insufficient evidence that the model is unsuitable. 3.2 < 7.815, so not significant; a test never proves H0.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S3
• Pearson Edexcel IAL Statistics S3 (WST03) examiners' report, June 2025
• Pearson Edexcel IAL Statistics S3 (WST03) mark scheme, June 2024

Last updated 8 November 2026. Spotted a mistake? Tell us.