| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2011 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Calculate CI from summary stats |
| Difficulty | Standard +0.3 Part (a) is a standard confidence interval calculation using normal approximation with given sample statistics—routine application of formulas. Part (b) requires understanding that P(interval entirely below μ) = α/2 = 1/16, so α = 12.5%, which involves conceptual understanding of confidence interval tails but is still a straightforward one-step problem once the concept is grasped. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(41.2 \pm z \times \sqrt{\frac{32.6}{50}}\) | M1 | |
| \(z = 1.96\) | B1 | Allow any brackets or none, or < or "to" etc |
| \([39.6, 42.8]\) (3 sfs) | A1 | |
| [3] | ||
| (b) \(2 \times \frac{2}{16}\) or \(\frac{1}{8}\) or 0.125 or 12.5% | M1 | or 0.875 |
| \(\alpha = 87.5\%\) | A1 | |
| [2] |
**(a)** $41.2 \pm z \times \sqrt{\frac{32.6}{50}}$ | M1 |
$z = 1.96$ | B1 | Allow any brackets or none, or < or "to" etc
$[39.6, 42.8]$ (3 sfs) | A1 |
| [3] |
**(b)** $2 \times \frac{2}{16}$ or $\frac{1}{8}$ or 0.125 or 12.5% | M1 | or 0.875
$\alpha = 87.5\%$ | A1 |
| [2] |2
\begin{enumerate}[label=(\alph*)]
\item The time taken by a worker to complete a task was recorded for a random sample of 50 workers. The sample mean was 41.2 minutes and an unbiased estimate of the population variance was 32.6 minutes ${ } ^ { 2 }$. Find a $95 \%$ confidence interval for the mean time taken to complete the task.
\item The probability that an $\alpha \%$ confidence interval includes only values that are lower than the population mean is $\frac { 1 } { 16 }$. Find the value of $\alpha$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2011 Q2 [5]}}