| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Find minimum sample size |
| Difficulty | Standard +0.3 This is a straightforward application of confidence interval width formula requiring algebraic manipulation to find n, plus standard interpretations of confidence intervals. Part (a) involves routine rearrangement of width = 2 × z × σ/√n, parts (b) and (c) test basic understanding of sampling and confidence interval interpretation using binomial probability. Slightly above average due to the multi-part nature and need to recall the width formula, but all techniques are standard S2 material with no novel problem-solving required. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(z \times \sqrt{\frac{11.2}{n}} = 1.4076 \div 2\) | M1 | Any \(z\), but must be a \(z\). |
| \(z = 1.881\) or \(1.882\) | B1 | |
| \(\left[n = \left(\frac{1.881}{0.7038}\right)^2 \times 11.2\right]\), \(n = 80\) | A1 | Must be a whole number. |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Jan, Feb and March not typical of whole year. | B1 | Or, e.g., weather is different at different times of year. |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.94^3 \times 0.06 \times 4\) | M1 | |
| \(= 0.199\) (3 sf) | A1 | |
| Total: 2 |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $z \times \sqrt{\frac{11.2}{n}} = 1.4076 \div 2$ | M1 | Any $z$, but must be a $z$. |
| $z = 1.881$ or $1.882$ | B1 | |
| $\left[n = \left(\frac{1.881}{0.7038}\right)^2 \times 11.2\right]$, $n = 80$ | A1 | Must be a whole number. |
| **Total: 3** | | |
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Jan, Feb and March not typical of whole year. | B1 | Or, e.g., weather is different at different times of year. |
| **Total: 1** | | |
## Question 4(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.94^3 \times 0.06 \times 4$ | M1 | |
| $= 0.199$ (3 sf) | A1 | |
| **Total: 2** | | |4 A certain train journey takes place every day throughout the year. The time taken, in minutes, for the journey is normally distributed with variance 11.2.
\begin{enumerate}[label=(\alph*)]
\item The mean time for a random sample of $n$ of these journeys was found. A $94 \%$ confidence interval for the population mean time was calculated and was found to have a width of 1.4076 minutes, correct to 4 decimal places.
Find the value of $n$.
\item A passenger noted the times for 50 randomly chosen journeys in January, February and March.
Give a reason why this sample is unsuitable for use in finding a confidence interval for the population mean time.
\item A researcher took 4 random samples and a $94 \%$ confidence interval for the population mean was found from each sample.
Find the probability that exactly 3 of these confidence intervals contain the true value of the population mean.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2023 Q4 [6]}}