| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2006 |
| Session | November |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | Two-tail z-test |
| Difficulty | Moderate -0.8 This is a straightforward two-tail hypothesis test requiring only standard recall: stating H₀: μ = 46 vs H₁: μ ≠ 46, then comparing z = -1.729 to the critical value ±1.96 at 5% level. No calculations needed, just interpretation of a given test statistic—simpler than average A-level questions which typically require some computation or multi-step reasoning. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(H_0: \mu = 46\) and \(H_1: \mu \neq 46\) | B1 | Both correct |
| (ii) Critical value \(z = \pm 1.96\) | M1 | For \(z = +1.96\) or \(-1.96\) and some comparison seen, OR for \(0.0419\) compared with \(0.025\) o.e. |
| No significant difference in times | A1 | For correct comparison and correct conclusion. (SR: For one tail test in (i) allow M1 for \(z = +/-1.645\) or comparison \(0.0419\) with \(0.05\) o.e.) |
**(i)** $H_0: \mu = 46$ and $H_1: \mu \neq 46$ | B1 | Both correct
**(ii)** Critical value $z = \pm 1.96$ | M1 | For $z = +1.96$ or $-1.96$ and some comparison seen, OR for $0.0419$ compared with $0.025$ o.e.
No significant difference in times | A1 | For correct comparison and correct conclusion. (SR: For one tail test in (i) allow M1 for $z = +/-1.645$ or comparison $0.0419$ with $0.05$ o.e.)
---1 The time taken for Samuel to drive home from work is distributed with mean 46 minutes. Samuel discovers a different route and decides to test at the $5 \%$ level whether the mean time has changed. He tries this route on a large number of different days chosen randomly and calculates the mean time.\\
(i) State the null and alternative hypotheses for this test.\\
(ii) Samuel calculates the value of his test statistic $z$ to be - 1.729 . What conclusion can he draw?
\hfill \mbox{\textit{CAIE S2 2006 Q1 [3]}}