| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Single probability inequality |
| Difficulty | Moderate -0.8 This is a straightforward application of the normal approximation to the binomial distribution with clear parameters (n=100, p=0.24). The question requires only routine application of the continuity correction and standard normal tables, with no conceptual challenges or multi-step reasoning beyond the standard textbook procedure. |
| Spec | 2.01c Sampling techniques: simple random, opportunity, etc2.04d Normal approximation to binomial |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Allocate 4-digit number to each DVD | B1 | "DVD" & "4 digits/1 to 9000/sequentially" etc must be mentioned *somewhere*. *Not* allocate "random" numbers, unless subsequently sorted |
| Select using random numbers | B1 | Mention random numbers |
| Ignore random numbers outside range | B1 | Unbiased method, mention of "outside range" or "repeats". If "pick random numbers in range 1 to 9000", must mention repeats |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(B(100, 0.24) \approx N(24, 18.24)\) | M1, A1 | N(attempt at \(np\)). Both parameters correct. Allow 18.24/100 A1 but then M0A0 |
| \(\Phi\!\left(\dfrac{19.5-24}{\sqrt{18.24}}\right) = \Phi(-1.0537)\) | M1, A1 | Standardise with their \(np\) and \(\sqrt{npq}\) or \(npq\). Both cc correct and \(\sqrt{npq}\) used. Allow cc/\(\sqrt{}\) errors |
| \(= \mathbf{0.1461}\) | A1 | Answer, a.r.t. 0.146 |
## Question 3:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Allocate 4-digit number to each DVD | B1 | "DVD" & "4 digits/1 to 9000/sequentially" etc must be mentioned *somewhere*. *Not* allocate "random" numbers, unless subsequently sorted |
| Select using random numbers | B1 | Mention random numbers |
| Ignore random numbers outside range | B1 | Unbiased method, mention of "outside range" or "repeats". If "pick random numbers in range 1 to 9000", must mention repeats |
**[3 marks]**
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $B(100, 0.24) \approx N(24, 18.24)$ | M1, A1 | N(attempt at $np$). Both parameters correct. Allow 18.24/100 A1 but then M0A0 |
| $\Phi\!\left(\dfrac{19.5-24}{\sqrt{18.24}}\right) = \Phi(-1.0537)$ | M1, A1 | Standardise with their $np$ and $\sqrt{npq}$ or $npq$. Both cc correct and $\sqrt{npq}$ used. Allow cc/$\sqrt{}$ errors |
| $= \mathbf{0.1461}$ | A1 | Answer, a.r.t. 0.146 |
**[5 marks]**
---3 A factory produces 9000 music DVDs each day. A random sample of 100 such DVDs is obtained.\\
(i) Explain how to obtain this sample using random numbers.\\
(ii) Given that $24 \%$ of the DVDs produced by the factory are classical, use a suitable approximation to find the probability that, in the sample of 100 DVDs, fewer than 20 are classical.
\hfill \mbox{\textit{OCR S2 2013 Q3 [8]}}