| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2023 |
| Session | March |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Confidence interval for single proportion |
| Difficulty | Moderate -0.8 This is a straightforward application of the standard confidence interval formula for a single proportion. Part (a) requires substituting values into a memorized formula (p̂ ± z*√(p̂(1-p̂)/n)), and part (b) tests basic conceptual understanding that narrower intervals come from lower confidence levels or larger samples. No problem-solving or novel insight required—purely routine recall and application. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left[\frac{49}{140} = 0.35\right]\) | ||
| \(0.35 \pm z\sqrt{\frac{0.35(1-0.35)}{140}}\) | M1 | Use of formula of correct form, ft *their* \(\frac{49}{140}\), any \(z\) (not a probability) |
| \(z = 2.326\) | B1 | Accept 2.326 to 2.329 |
| Confidence interval \(= 0.256\) to \(0.444\) (3 sf) | A1 | Must be an interval |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Find a smaller percentage confidence interval / lower level of confidence | B1 | ISW if 2 reasons given. Just saying 'use smaller \(z\)' scores B0. Accept a correct example e.g. 90% (even if not qualified with statement) |
| [1] |
## Question 1:
### Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left[\frac{49}{140} = 0.35\right]$ | | |
| $0.35 \pm z\sqrt{\frac{0.35(1-0.35)}{140}}$ | **M1** | Use of formula of correct form, ft *their* $\frac{49}{140}$, any $z$ (not a probability) |
| $z = 2.326$ | **B1** | Accept 2.326 to 2.329 |
| Confidence interval $= 0.256$ to $0.444$ (3 sf) | **A1** | Must be an interval |
| | **[3]** | |
### Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Find a smaller **percentage** confidence interval / lower **level** of confidence | **B1** | ISW if 2 reasons given. Just saying 'use smaller $z$' scores B0. Accept a correct example e.g. 90% (even if not qualified with statement) |
| | **[1]** | |
---1 Anita carried out a survey of 140 randomly selected students at her college. She found that 49 of these students watched a TV programme called Bunch.
\begin{enumerate}[label=(\alph*)]
\item Calculate an approximate $98 \%$ confidence interval for the proportion, $p$, of students at Anita's college who watch Bunch.\\
Carlos says that the confidence interval found in (a) is not useful because it is too wide.
\item Without calculation, explain briefly how Carlos can use the results of Anita's survey to find a narrower confidence interval for $p$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2023 Q1 [4]}}