| Exam Board | AQA |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Calculate CI for proportion |
| Difficulty | Standard +0.3 This is a standard two-proportion confidence interval question requiring routine application of the normal approximation formula. While it involves multiple steps (calculating proportions, standard error, and constructing the interval), it follows a textbook procedure with no conceptual challenges. The interpretation in part (b) is straightforward. Slightly easier than average due to its mechanical nature. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| \(\hat{p}_M = \frac{264}{480} = \frac{11}{20}\) or 0.55 | B1 | Both CAO |
| \(\hat{p}_W = \frac{220}{500} = \frac{11}{25}\) or 0.44 | B1 | Both CAO |
| 95% \(\Rightarrow z = \mathbf{1.96}\) | B1 | AWRT (1.95996) |
| CI for \(p_M - p_W\) is | M1 | \((\hat{p}_M - \hat{p}_W) \pm (1.96 \text{ or } 1.64 \text{ to } 1.65)\sqrt{a}\) |
| \((\frac{0.55 \times 0.45}{480} + \frac{0.44 \times 0.56}{500})\sqrt{}\) | M1 AF1 | Expression for \(\sqrt{a}\); F on \(\hat{p}_M\) and \(\hat{p}_W\) and \(z\) |
| ie \(\mathbf{0.11 \pm 0.06}\) or \(\mathbf{(0.05, 0.17)}\) | A1 | CAO/AWRT (0.06224) |
| 6 |
| Answer | Marks | Guidance |
|---|---|---|
| CI \(> 0.025\) or LCL \(> 0.025\) | BF1 | F on CI providing CI \(> 0.025\); Dep on BF1 |
| Evidence to support the claim | Bdep1 | |
| 2 |
## Part (a)
$\hat{p}_M = \frac{264}{480} = \frac{11}{20}$ or **0.55** | B1 | Both CAO
$\hat{p}_W = \frac{220}{500} = \frac{11}{25}$ or **0.44** | B1 | Both CAO
95% $\Rightarrow z = \mathbf{1.96}$ | B1 | AWRT (1.95996)
CI for $p_M - p_W$ is | M1 | $(\hat{p}_M - \hat{p}_W) \pm (1.96 \text{ or } 1.64 \text{ to } 1.65)\sqrt{a}$
$(\frac{0.55 \times 0.45}{480} + \frac{0.44 \times 0.56}{500})\sqrt{}$ | M1 AF1 | Expression for $\sqrt{a}$; F on $\hat{p}_M$ and $\hat{p}_W$ and $z$
ie $\mathbf{0.11 \pm 0.06}$ or $\mathbf{(0.05, 0.17)}$ | A1 | CAO/AWRT (0.06224)
| | **6** |
## Part (b)
CI $> 0.025$ or LCL $> 0.025$ | BF1 | F on CI providing CI $> 0.025$; Dep on BF1
Evidence to **support the claim** | Bdep1 |
| | **2** |
**Notes:**
1. There must be a reference to 0.025 (OE) and a clear comparison with the answer to (a)
2. Accept answers suggesting that selections may not be random and/or independent or that based on 480 & 500 may not be representative or changes of opinions between opinion poll and referendum
---In advance of a referendum on independence, the regional assembly of an eastern province of a particular country carried out an opinion poll to assess the strength of the 'Yes' vote.
Of the 480 men polled, 264 indicated that they intended to vote 'Yes', and of the 500 women polled, 220 indicated that they intended to vote 'Yes'.
\begin{enumerate}[label=(\alph*)]
\item Construct an approximate 95\% confidence interval for the difference between the proportion of men who intend to vote 'Yes' and the proportion of women who intend to vote 'Yes'. [6 marks]
\item Comment on a claim that, in the forthcoming referendum, the percentage of men voting 'Yes' will exceed the percentage of women voting 'Yes' by at least 2.5 per cent. Justify your answer. [2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA S3 2016 Q1 [8]}}