| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2012 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Confidence interval for single proportion |
| Difficulty | Moderate -0.3 This is a straightforward two-part question combining standard confidence interval calculation for a proportion (using normal approximation) and a routine normal distribution inverse problem. Both parts require only direct application of formulas with no problem-solving insight, making it slightly easier than average. |
| Spec | 2.04f Find normal probabilities: Z transformation5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.14 \pm zs\) | M1 | |
| \(z = 1.96\) | B1 | Or /99 |
| \(s^2 = 0.14 \times 0.86/100\) | A1 | \((0.0716, 0.208(4))\) from /99, A0 |
| \((0.072(0), 0.2080)\) | A1 | |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Equate \(\phi((500-\mu)/\sqrt{50})\) to \(0.14\) | M1 | |
| \(\Rightarrow (500-\mu)/\sqrt{50} = -1.0803\) | A1 | |
| \(\Rightarrow \mu = 508\) (3SF) | A1 | |
| [3] |
# Question 1:
## Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.14 \pm zs$ | M1 | |
| $z = 1.96$ | B1 | Or /99 |
| $s^2 = 0.14 \times 0.86/100$ | A1 | $(0.0716, 0.208(4))$ from /99, A0 |
| $(0.072(0), 0.2080)$ | A1 | |
| | **[4]** | |
## Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Equate $\phi((500-\mu)/\sqrt{50})$ to $0.14$ | M1 | |
| $\Rightarrow (500-\mu)/\sqrt{50} = -1.0803$ | A1 | |
| $\Rightarrow \mu = 508$ (3SF) | A1 | |
| | **[3]** | |
---1 A machine fills packets of flour whose nominal weights are 500 g . Each of a random sample of 100 packets was weighed and 14 packets weighed less than 500 g . The population proportion of packets that weigh less than 500 g is denoted by $p$.\\
(i) Calculate an approximate $95 \%$ confidence interval for $p$.\\
(ii) The weights of the packets, in grams, are normally distributed with mean $\mu$ and variance 50 . Assuming that $p = 0.14$, calculate the value of $\mu$.
\hfill \mbox{\textit{OCR S3 2012 Q1 [7]}}