| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2021 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Calculate CI for proportion |
| Difficulty | Standard +0.8 Part (a) is routine application of confidence interval formula for proportion. Part (b) requires understanding that confidence intervals are independent events and applying binomial probability with p=stated confidence level, which is a conceptual leap beyond standard textbook exercises that most students wouldn't encounter in typical practice. |
| Spec | 2.03e Model with probability: critiquing assumptions5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.25 \pm z\sqrt{\frac{0.25 \times 0.75}{140}}\) | M1 | Expression of correct form (allow M1 for just one side stated). Must be a \(z\)-value |
| \(z = 2.054\) or \(2.055\) | B1 | |
| \(0.175\) to \(0.325\) (3sf) | A1 | Must be an interval |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.90 \times 0.95 \times 0.01\) \(+ 0.90 \times 0.05 \times 0.99\) \(+ 0.10 \times 0.95 \times 0.99\) | M1 M1 | M1 for one correct triple product; M1 for all correct and added |
| \(0.147\) | A1 | SC If zero scored award B1 for a 2 or 3 term expression of the form \(0.90 \times 0.95\ [\times c]\) OE, \((0 < c \leqslant 1)\) |
| 3 |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.25 \pm z\sqrt{\frac{0.25 \times 0.75}{140}}$ | **M1** | Expression of correct form (allow M1 for just one side stated). Must be a $z$-value |
| $z = 2.054$ or $2.055$ | **B1** | |
| $0.175$ to $0.325$ (3sf) | **A1** | Must be an interval |
| | **3** | |
---
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.90 \times 0.95 \times 0.01$ $+ 0.90 \times 0.05 \times 0.99$ $+ 0.10 \times 0.95 \times 0.99$ | **M1 M1** | **M1** for one correct triple product; **M1** for all correct and added |
| $0.147$ | **A1** | SC If zero scored award **B1** for a 2 or 3 term expression of the form $0.90 \times 0.95\ [\times c]$ OE, $(0 < c \leqslant 1)$ |
| | **3** | |
---3 The probability that a certain spinner lands on red on any spin is $p$. The spinner is spun 140 times and it lands on red 35 times.
\begin{enumerate}[label=(\alph*)]
\item Find an approximate $96 \%$ confidence interval for $p$.\\
From three further experiments, Jack finds a 90\% confidence interval, a 95\% confidence interval and a 99\% confidence interval for $p$.
\item Find the probability that exactly two of these confidence intervals contain the true value of $p$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2021 Q3 [6]}}