| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2015 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed hypothesis test (upper tail, H₁: p > p₀) |
| Difficulty | Standard +0.3 This is a straightforward one-tailed binomial hypothesis test with clearly stated hypotheses (p > 0.25), small sample size allowing direct probability calculation, and standard 5% significance level. The question explicitly provides all parameters and requires only routine application of the binomial test procedure. Slightly above average difficulty due to being a hypothesis test rather than pure calculation, but still a standard S2 textbook exercise. |
| Spec | 2.01c Sampling techniques: simple random, opportunity, etc2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Avoids (reduces) bias, or "representative" or "allows calculations to be done" or "allows reliable estimates" | B1 | – unbiased (allow "fair"), – representative (allow "reliable"), – allows use of distribution. Both right and wrong: B1 |
| Total: 1 | *Not:* – all equally likely to be selected, – selections independent, – quick/easy/cheap, – random sample |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(B(18, 0.25)\) | M1 | \(B(18, 0.25)\) stated or used |
| \(H_0: p = 0.25\), \(H_1: p > 0.25\) | B2 | One error, B1; \(x\) or \(\bar{x}\) B0; \(\pi\): B2 |
| \(\alpha\): \(P(\geq 8) = 1 - P(\leq 7) = \mathbf{0.0569}\) | A1 | 0.0569 seen |
| \(> 0.05\) | A1 | Explicit comparison with 0.05 |
| \(\beta\): CR is \(\geq 9\), and \(8 < 9\), probability 0.0193 | A1dep*, *A1 | Correct CR and explicit comparison. 0.0193 explicitly seen |
| Do not reject \(H_0\). Insufficient evidence that proportion of 1s is greater than 25%. | M1, A1ft | Correct first conclusion. Contextualised, acknowledge uncertainty. FT on wrong CR/\(p\) |
| Total: 7 | *Any* symbol can get B2 if explicitly defined. Allow 0.9431 only if "\(> 0.95\)" and vice versa |
## Question 5:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Avoids (reduces) bias, or "representative" or "allows calculations to be done" or "allows reliable estimates" | B1 | – unbiased (allow "fair"), – representative (allow "reliable"), – allows use of distribution. Both right and wrong: B1 |
| **Total: 1** | | *Not:* – all equally likely to be selected, – selections independent, – quick/easy/cheap, – random sample |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $B(18, 0.25)$ | M1 | $B(18, 0.25)$ stated or used |
| $H_0: p = 0.25$, $H_1: p > 0.25$ | B2 | One error, B1; $x$ or $\bar{x}$ B0; $\pi$: B2 |
| $\alpha$: $P(\geq 8) = 1 - P(\leq 7) = \mathbf{0.0569}$ | A1 | 0.0569 seen |
| $> 0.05$ | A1 | Explicit comparison with 0.05 |
| $\beta$: CR is $\geq 9$, and $8 < 9$, probability 0.0193 | A1dep*, *A1 | Correct CR and explicit comparison. 0.0193 explicitly seen |
| Do not reject $H_0$. Insufficient evidence that proportion of 1s is greater than 25%. | M1, A1ft | Correct first conclusion. Contextualised, acknowledge uncertainty. FT on wrong CR/$p$ |
| **Total: 7** | | *Any* symbol can get B2 if explicitly defined. Allow 0.9431 only if "$> 0.95$" and vice versa |
---5 (i) State an advantage of using random numbers in selecting samples.\\
(ii) It is known that in analysing the digits in large sets of financial records, the probability that the leading digit is 1 is 0.25 . A random sample of 18 leading digits from a certain large set of financial records is obtained and it is found that 8 of the leading digits are 1 s . Test, at the $5 \%$ significance level, whether the probability that the leading digit is 1 in this set of records is greater than 0.25 .
\hfill \mbox{\textit{OCR S2 2015 Q5 [8]}}