| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2020 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Calculate CI for proportion |
| Difficulty | Moderate -0.8 This is a straightforward binomial confidence interval question requiring standard formula application (p̂ ± z√(p̂(1-p̂)/n)) and basic interpretation. The calculation is routine with given values, and the interpretation simply requires checking if 1/6 lies within the interval—no novel insight or complex reasoning needed. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{56}{300} \pm z \times \sqrt{\frac{\frac{56}{300} \times \frac{244}{300}}{300}}\) | M1 | For expression of the correct form. Must be a \(z\) value |
| \(z = 2.054\) or \(2.055\) | B1 | |
| \(0.14(0)\) to \(0.233\) (3sf) or \(0.141\) to \(0.233\) (3sf) | A1 | Must be an interval |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{6}\) \(( = 0.167)\); this is within confidence interval, so no reason to believe die is biased. | B1 FT | Note if confidence interval set up with \(\frac{1}{6}, \frac{56}{300}\) it should be the value used here. FT *their* confidence interval. Not definite, e.g. not 'Die not biased'. |
| 1 |
## Question 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{56}{300} \pm z \times \sqrt{\frac{\frac{56}{300} \times \frac{244}{300}}{300}}$ | M1 | For expression of the correct form. Must be a $z$ value |
| $z = 2.054$ or $2.055$ | B1 | |
| $0.14(0)$ to $0.233$ (3sf) or $0.141$ to $0.233$ (3sf) | A1 | Must be an interval |
| | **3** | |
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## Question 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{6}$ $( = 0.167)$; this is within confidence interval, so no reason to believe die is biased. | B1 FT | Note if confidence interval set up with $\frac{1}{6}, \frac{56}{300}$ it should be the value used here. FT *their* confidence interval. Not definite, e.g. not 'Die not biased'. |
| | **1** | |2 A six-sided die has faces marked $1,2,3,4,5,6$. When the die is thrown 300 times it shows a six on 56 throws.
\begin{enumerate}[label=(\alph*)]
\item Calculate an approximate $96 \%$ confidence interval for the probability that the die shows a six on one throw.
\item Maroulla claims that the die is biased.
Use your answer to part (a) to comment on this claim.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2020 Q2 [4]}}