| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2002 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Unbiased estimates calculation |
| Difficulty | Moderate -0.3 This is a straightforward one-sample t-test with standard calculations. Students must compute sample mean and variance from summary statistics, then perform a two-tailed hypothesis test at 10% significance. All steps are routine and follow a standard template with no conceptual challenges or novel problem-solving required. The large sample size (n=150) makes the calculation easier. Slightly below average difficulty due to its mechanical nature. |
| Spec | 5.05b Unbiased estimates: of population mean and variance5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\bar{x} = 14.2\), \(s^2 = \frac{1}{49}\left(3746 - \frac{2130^2}{150}\right) = 50.3(4)\) | B1, B1 (2 marks) | For correct mean; For correct variance |
| Answer | Marks | Guidance |
|---|---|---|
| Test statistic: \(z = \frac{14.2 - 12}{\sqrt{\frac{50.34}{150}}} = 3.798\) | B1, M1, A1 (5 marks) | Both hypotheses correct; For standardising attempt with se of form \(\frac{s}{\sqrt{n}}\); For 3.80 or comparing \(\Phi(3.798)\) with 0.95 (or equiv. for one tail test) Signs consistent. |
| Answer | Marks | Guidance |
|---|---|---|
| Reject exam boards claim | A1 | For correct conclusion fi on their z and \(H_1\) |
**(i)** $\bar{x} = 14.2$, $s^2 = \frac{1}{49}\left(3746 - \frac{2130^2}{150}\right) = 50.3(4)$ | B1, B1 (2 marks) | For correct mean; For correct variance
**(ii)** $H_0: \mu = 12$ and $H_1: \mu \neq 12$
Test statistic: $z = \frac{14.2 - 12}{\sqrt{\frac{50.34}{150}}} = 3.798$ | B1, M1, A1 (5 marks) | Both hypotheses correct; For standardising attempt with se of form $\frac{s}{\sqrt{n}}$; For 3.80 or comparing $\Phi(3.798)$ with 0.95 (or equiv. for one tail test) Signs consistent.
Compare with 1.645 or 1.282 for one-tail t
Reject exam boards claim | A1 | For correct conclusion fi on their z and $H_1$
---4 The mean time to mark a certain set of examination papers is estimated by the examination board to be 12 minutes per paper. A random sample of 150 examination papers gave $\Sigma x = 2130$ and $\Sigma x ^ { 2 } = 37746$, where $x$ is the time in minutes to mark an examination paper.\\
(i) Calculate unbiased estimates of the population mean and variance.\\
(ii) Stating the null and alternative hypotheses, use a $10 \%$ significance level to test whether the examination board's estimated time is consistent with the data.
\hfill \mbox{\textit{CAIE S2 2002 Q4 [7]}}