| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Session | Specimen |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Calculate Type II error probability |
| Difficulty | Challenging +1.2 This is a comprehensive binomial hypothesis testing question covering one-tailed tests, Type I and Type II errors, and requires working backwards from power calculations. While it involves multiple parts and understanding of error types, the techniques are standard S2 content with straightforward binomial probability calculations using tables. The conceptual demand is moderate but execution is methodical rather than requiring novel insight. |
| Spec | 2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(H_0: \rho = 0.15\); \(H_1: \rho > 0.15\) | B1 | For correct statement of null hypothesis |
| B1 | For correct statement of alt hypothesis |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{P}(L \geq 5) = 1 - 0.9761 = 0.0239\) | M1 | For correct distribution stated or implied |
| M1 | For calculation of relevant tail probability, or finding the critical region | |
| This is significant, since \(0.0239 < 0.05\) | A1 | For correct value 0.0239 or region \(l \geq 5\) |
| M1 | For comparing tail probability with 0.05 or observed value with critical region |
| Answer | Marks | Guidance |
|---|---|---|
| Accept the suggestion that the proportion of mathematicians who are left-handed is more than 15% | A1 | For stating or implying rejection of \(H_0\) |
| A1 | For stating the outcome in context | |
| (iii) \(\text{P}_I = \text{P}(L \text{ in critical region}) = 0.0239\) | M1 | For evaluating P(reject \(H_0\)) |
| A1 | For correct answer 0.0239 or equivalent | |
| (iii) \(\text{P}_{II} = \text{P}(L < 4 \mid p = 0.2) = 0.9274\) | M1 | For evaluating P(accept \(H_0\)) with \(p = 0.2\) |
| A1 | For correct probability |
| Answer | Marks | Guidance |
|---|---|---|
| So the proportion is between 67% and 70% | M1 | For relevant use of tables |
| A1 | For an appropriate conclusion |
(i) $H_0: \rho = 0.15$; $H_1: \rho > 0.15$ | B1 | For correct statement of null hypothesis
| B1 | For correct statement of alt hypothesis
Under $H_0$, number left-handed $L \sim \text{B}(12, 0.15)$
$\text{P}(L \geq 5) = 1 - 0.9761 = 0.0239$ | M1 | For correct distribution stated or implied
| M1 | For calculation of relevant tail probability, or finding the critical region
This is significant, since $0.0239 < 0.05$ | A1 | For correct value 0.0239 or region $l \geq 5$
| M1 | For comparing tail probability with 0.05 or observed value with critical region
Hence $H_0$ is rejected
Accept the suggestion that the proportion of mathematicians who are left-handed is more than 15% | A1 | For stating or implying rejection of $H_0$
| A1 | For stating the outcome in context
(iii) $\text{P}_I = \text{P}(L \text{ in critical region}) = 0.0239$ | M1 | For evaluating P(reject $H_0$)
| A1 | For correct answer 0.0239 or equivalent
(iii) $\text{P}_{II} = \text{P}(L < 4 \mid p = 0.2) = 0.9274$ | M1 | For evaluating P(accept $H_0$) with $p = 0.2$
| A1 | For correct probability
(iv) $\text{P}_d = 0.0188$ for $p = 2$ and $0.0095$ for $p = 0.7$
So the proportion is between 67% and 70% | M1 | For relevant use of tables
| A1 | For an appropriate conclusion
**Total: 14 marks**8 The proportion of left-handed adults in a country is known to be $15 \%$. It is suggested that for mathematicians the proportion is greater than $15 \%$. A random sample of 12 members of a university mathematics department is taken, and it is found to include five who are left-handed.\\
(i) Stating your hypotheses, test whether the suggestion is justified, using a significance level as close to $5 \%$ as possible.\\
(ii) In fact the significance test cannot be carried out at a significance level of exactly $5 \%$. State the probability of making a Type I error in the test.\\
(iii) Find the probability of making a Type II error in the test for the case when the proportion of mathematicians who are left-handed is actually $20 \%$.\\
(iv) Determine, as accurately as the tables of cumulative binomial probabilities allow, the actual proportion of mathematicians who are left-handed for which the probability of making a Type II error in the test is 0.01 .
\hfill \mbox{\textit{OCR S2 Q8 [14]}}