| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Type I/II errors and power of test |
| Type | Simultaneous critical region and Type II error |
| Difficulty | Standard +0.3 This is a standard hypothesis testing question on the binomial distribution covering critical regions, Type I and Type II errors. Part (i) requires finding critical values from binomial tables (routine S2 skill). Parts (ii) and (iii) test understanding of Type II error definition with straightforward probability calculations. While multi-part, each component follows textbook procedures without requiring novel insight or complex reasoning. |
| Spec | 2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working: P(≤ 4) = 0.0473 | Marks: B1 | Guidance: |
| Answer/Working: Therefore CR is ≤ 4 | Marks: M1 | Guidance: |
| Answer/Working: P(Type I error) = \(\frac{4.73\%}{1}\) | Marks: A1, 3 | Guidance: Answer, an 0.0473 or 4.73%, must be stated |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working: B(10, 0.4) and find P(> 4) | Marks: M1 | Guidance: Must be this, not isw, \(J\)on (i) |
| Answer/Working: 1 − P(≤ 4) | Marks: M1 | Guidance: Allow for 0.6177 or 0.1622 |
| Answer/Working: = 0.3669 | Marks: A1, 3 | Guidance: Answer, art 0.367 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working: 0.5 × 0.3669 | Marks: M1 | Guidance: 0.5 × (ii) |
| Answer/Working: = 0.18345 | Marks: A1✓, 2 | Guidance: Ans correct to 3 SF, e.g. 0.184 from 0.367 |
### (i)
**Answer/Working:** P(≤ 4) = 0.0473 | **Marks:** B1 | **Guidance:**
**Answer/Working:** Therefore CR is ≤ 4 | **Marks:** M1 | **Guidance:**
**Answer/Working:** P(Type I error) = $\frac{4.73\%}{1}$ | **Marks:** A1, 3 | **Guidance:** Answer, an 0.0473 or 4.73%, must be stated
### (ii)
**Answer/Working:** B(10, 0.4) and find P(> 4) | **Marks:** M1 | **Guidance:** Must be this, not isw, $J$on (i)
**Answer/Working:** 1 − P(≤ 4) | **Marks:** M1 | **Guidance:** Allow for 0.6177 or 0.1622
**Answer/Working:** = 0.3669 | **Marks:** A1, 3 | **Guidance:** Answer, art 0.367
### (iii)
**Answer/Working:** 0.5 × 0.3669 | **Marks:** M1 | **Guidance:** 0.5 × (ii)
**Answer/Working:** = 0.18345 | **Marks:** A1✓, 2 | **Guidance:** Ans correct to 3 SF, e.g. 0.184 from 0.367
---The random variable $R$ has the distribution B(10, $p$). The null hypothesis H$_0$: $p = 0.7$ is to be tested against the alternative hypothesis H$_1$: $p < 0.7$, at a significance level of 5%.
\begin{enumerate}[label=(\roman*)]
\item Find the critical region for the test and the probability of making a Type I error. [3]
\item Given that $p = 0.4$, find the probability that the test results in a Type II error. [3]
\item Given that $p$ is equally likely to take the values 0.4 and 0.7, find the probability that the test results in a Type II error. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR S2 2010 Q8 [8]}}