| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2008 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Find sample size for test |
| Difficulty | Standard +0.3 This is a straightforward S2 hypothesis test question requiring standard binomial test procedures. Part (i) is routine calculation with given parameters, part (ii) requires conceptual understanding of symmetry in binomial tests, and part (iii) involves iterative calculation to find a critical value. While multi-part, each component uses standard A-level techniques without requiring novel insight or complex problem-solving. |
| Spec | 2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0: p = 0.65\) OR \(p \geq 0.65\); \(H_1: p < 0.65\) | B2 | Both hypotheses correctly stated; [One error (but not \(r\), \(x\) or \(\bar{x}\)): B1] |
| \(B(12, 0.65)\) | M1 | \(B(12, 0.65)\) stated or implied |
| \(\alpha\): \(P(\leq 6) = 0.2127\); Compare 0.10 | A1, B1 | Correct probability from tables, *not* \(P(=6)\); Explicit comparison with 0.10 |
| \(\beta\): Critical region \(\leq 5\); \(6>5\); Probability 0.0846 | B1, A1 | Critical region \(\leq 5\) or \(\leq 6\) or \(\{\leq 4\} \cap \{\geq 11\}\) and compare 6; Correct probability |
| Do not reject \(H_0\); Insufficient evidence that proportion of population in favour is not at least 65% | M1\(\sqrt{}\), A1\(\sqrt{}\) 7 | Correct comparison and conclusion, needs correct distribution, correct tail, like-with-like; Interpret in context, e.g. "consistent with claim". [SR: N(7.8, 2.73): can get B2M1A0B1M0: 4 ex 7] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Insufficient evidence to reject claim; test and \(p/q\) symmetric | B1\(\sqrt{}\), B1 2 | Same conclusion as part (i), don't need context; Valid relevant reason, e.g. "same as (i)" |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(R \sim B(2n, 0.65)\), \(P(R \leq n) > 0.15\) | M1 | \(B(2n, 0.65)\), \(P(R \leq n) > 0.15\) stated or implied |
| \(B(18, 0.65)\), \(p = 0.1391\) | A1, A1 | Any probability in list below seen; \(p = 0.1391\) picked out (not just in a list of \(> 2\)) |
| Therefore \(n = 9\) | A1 4 | Final answer \(n = 9\) only. [SR \( |
# Question 8:
## Part (i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: p = 0.65$ OR $p \geq 0.65$; $H_1: p < 0.65$ | B2 | Both hypotheses correctly stated; [One error (but not $r$, $x$ or $\bar{x}$): B1] |
| $B(12, 0.65)$ | M1 | $B(12, 0.65)$ stated or implied |
| $\alpha$: $P(\leq 6) = 0.2127$; Compare 0.10 | A1, B1 | Correct probability from tables, *not* $P(=6)$; Explicit comparison with 0.10 |
| $\beta$: Critical region $\leq 5$; $6>5$; Probability 0.0846 | B1, A1 | Critical region $\leq 5$ or $\leq 6$ or $\{\leq 4\} \cap \{\geq 11\}$ and compare 6; Correct probability |
| Do not reject $H_0$; Insufficient evidence that proportion of population in favour is not at least 65% | M1$\sqrt{}$, A1$\sqrt{}$ **7** | Correct comparison and conclusion, needs correct distribution, correct tail, like-with-like; Interpret in context, e.g. "consistent with claim". [SR: N(7.8, 2.73): can get B2M1A0B1M0: 4 ex 7] |
## Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Insufficient evidence to reject claim; test and $p/q$ symmetric | B1$\sqrt{}$, B1 **2** | Same conclusion as part (i), don't need context; Valid relevant reason, e.g. "same as (i)" |
## Part (iii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $R \sim B(2n, 0.65)$, $P(R \leq n) > 0.15$ | M1 | $B(2n, 0.65)$, $P(R \leq n) > 0.15$ stated or implied |
| $B(18, 0.65)$, $p = 0.1391$ | A1, A1 | Any probability in list below seen; $p = 0.1391$ picked out (not just in a list of $> 2$) |
| Therefore $n = 9$ | A1 **4** | Final answer $n = 9$ only. [SR $<n$: M1A0, $n=4$, 0.1061 A1A0]; Table of values given in scheme. |8 Consultations are taking place as to whether a site currently in use as a car park should be developed as a shopping mall. An agency acting on behalf of a firm of developers claims that at least $65 \%$ of the local population are in favour of the development. In a survey of a random sample of 12 members of the local population, 6 are in favour of the development.\\
(i) Carry out a test, at the $10 \%$ significance level, to determine whether the result of the survey is consistent with the claim of the agency.\\
(ii) A local residents' group claims that no more than $35 \%$ of the local population are in favour of the development. Without further calculations, state with a reason what can be said about the claim of the local residents' group.\\
(iii) A test is carried out, at the $15 \%$ significance level, of the agency's claim. The test is based on a random sample of size $2 n$, and exactly $n$ of the sample are in favour of the development. Find the smallest possible value of $n$ for which the outcome of the test is to reject the agency's claim.\\[0pt]
[4]
4
\hfill \mbox{\textit{OCR S2 2008 Q8 [13]}}