ZIMSEC A Level · Statistics · Form 6
The method
- State the null hypothesis H₀ and the alternative H₁ (one-tailed or two-tailed).
- Choose the significance level (e.g. 5%).
- Calculate the test statistic (or probability).
- Compare with the critical value or region.
- Conclude in context: “there is (in)sufficient evidence at the 5% level that…”.
Tests
- Binomial test for a proportion p.
- Poisson test for a mean rate λ.
- Normal test for a mean: z = (x̄ − μ)/(σ/√n).
Worked example: A manufacturer claims bulbs last 1 000 hours (σ = 100). A sample of 50 has a mean of 970 hours. Test at the 5% level whether the mean is less than claimed.
H₀: μ = 1 000; H₁: μ < 1 000 (one-tailed)
z = (970 − 1 000)/(100/√50) = −30/14.14 = −2.12
Critical value −1.645; −2.12 < −1.645 → reject H₀: evidence that bulbs last less than claimed.
Errors
Type I error: rejecting H₀ when it is true (probability = significance level). Type II error: accepting H₀ when it is false.
Chi-squared tests (if in syllabus)
Goodness of fit and contingency tables: χ² = Σ(O − E)²/E, with degrees of freedom.
Exam tip: Never say H₀ is “proved” – only that there is or is not enough evidence to reject it.
Check yourself
- What is a Type I error?
- State the critical value for a two-tailed z-test at the 5% level.
- For a test of H₀: p = 0.4 against H₁: p > 0.4, is the test one- or two-tailed?
Show answers
- Rejecting the null hypothesis when it is actually true.
- ±1.96
- One-tailed.
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