| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2019 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | CI from raw data list |
| Difficulty | Standard +0.3 Part (i) is a standard confidence interval calculation requiring the assumption of normality and use of the t-distribution (or z if assuming known σ). Part (ii) requires understanding that confidence intervals have a 3% chance of not containing μ, then applying basic probability (0.03² = 0.0009). While this tests conceptual understanding of confidence intervals beyond mere calculation, it's still a straightforward application once the concept is grasped, making it slightly easier than average for A-level Further Maths Statistics. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Assume population is normally distributed | B1 | |
| \(\bar{x} = 25.9\) | B1 | Allow \(\frac{259}{10}\) |
| \(z = 2.17\) | B1 | |
| \('25.9' \pm z \times \frac{3}{\sqrt{10}}\) | M1 | Must have correct form and \(z\) |
| \(23.8\) to \(28.0\) (3 sf) | A1 | CWO |
| 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.03^2 \ (= 0.0009)\) | B1 | |
| 1 |
**Question 3:**
**Part (i):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Assume population is normally distributed | B1 | |
| $\bar{x} = 25.9$ | B1 | Allow $\frac{259}{10}$ |
| $z = 2.17$ | B1 | |
| $'25.9' \pm z \times \frac{3}{\sqrt{10}}$ | M1 | Must have correct form and $z$ |
| $23.8$ to $28.0$ (3 sf) | A1 | CWO |
| | **5** | |
**Part (ii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.03^2 \ (= 0.0009)$ | B1 | |
| | **1** | |3 The times, in minutes, taken by competitors to complete a puzzle have mean $\mu$ and standard deviation 3 . The times taken by a random sample of 10 competitors are noted and the results are given below.\\
$\begin{array} { l l l } 25.2 & 26.8 & 18.5 \end{array}$\\
25.5\\
30.1\\
$28.9 \quad 27.0$\\
$26.1 \quad 26.0$\\
24.9\\
(i) Stating a necessary assumption, calculate a $97 \%$ confidence interval for $\mu$.\\
(ii) Two more random samples, each of 10 competitors, are taken. Their times are used to calculate two more $97 \%$ confidence intervals for $\mu$. Find the probability that neither of these intervals contains the true value of $\mu$.\\
\hfill \mbox{\textit{CAIE S2 2019 Q3 [6]}}