| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2015 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Calculate CI for proportion |
| Difficulty | Challenging +1.2 Part (i) is a standard confidence interval for proportion calculation requiring the normal approximation formula. Part (ii) is more conceptually demanding, requiring students to recognize that confidence intervals have a (1-α) probability of containing the true parameter, then apply this to find P(exactly one succeeds) = 2(0.97)(0.03) = 0.0582. The conceptual leap in part (ii) elevates this above routine exercises, but it remains accessible to well-prepared S2 students. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{Var}(p_s) = \frac{0.22 \times (1-0.22)}{100}\) \(\left(= \frac{429}{250000}\ \text{or}\ 0.001716\right)\) | M1 | pq/100 |
| \(0.22 \pm z\sqrt{\frac{429}{250000}}\) | M1 | Expression of correct form with their variance; any \(z\) (must be a \(z\) value); accept one side only |
| \(z = 2.17\) or \(2.168/9\) or \(2.171\) | B1 | Seen |
| \(0.13(0)\) to \(0.31(0)\) (2 sf) | A1 [4] | Must be an interval |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(2 \times (1-0.97) \times 0.97 = 0.0582\) | M1, A1 [2] |
## Question 3:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Var}(p_s) = \frac{0.22 \times (1-0.22)}{100}$ $\left(= \frac{429}{250000}\ \text{or}\ 0.001716\right)$ | M1 | pq/100 |
| $0.22 \pm z\sqrt{\frac{429}{250000}}$ | M1 | Expression of correct form with their variance; any $z$ (must be a $z$ value); accept one side only |
| $z = 2.17$ or $2.168/9$ or $2.171$ | B1 | Seen |
| $0.13(0)$ to $0.31(0)$ (2 sf) | A1 [4] | Must be an interval |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2 \times (1-0.97) \times 0.97 = 0.0582$ | M1, A1 [2] | |
---3 A die is biased so that the probability that it shows a six on any throw is $p$.\\
(i) In an experiment, the die shows a six on 22 out of 100 throws. Find an approximate $97 \%$ confidence interval for $p$.\\
(ii) The experiment is repeated and another $97 \%$ confidence interval is found. Find the probability that exactly one of the two confidence intervals includes the true value of $p$.
\hfill \mbox{\textit{CAIE S2 2015 Q3 [6]}}