| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Type I/II errors and power of test |
| Type | Find minimum sample size for Type II error constraint |
| Difficulty | Challenging +1.2 This is a standard S4 hypothesis testing question requiring knowledge of critical regions for normal distributions and Type II error calculations. Part (a) involves routine application of z-tables and standardization (finding critical value -2.326 and converting to x-bar scale). Part (b) requires setting up and solving an inequality involving both null and alternative hypotheses, which is more challenging but still a well-practiced S4 technique. The question is harder than average A-level due to being Further Maths content and requiring algebraic manipulation with sample size n, but it follows standard S4 patterns without requiring novel insight. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Under \(H_0\): \(\bar{X} \sim N\left(202, \frac{4}{n}\right)\) | B1 | Correct distribution stated |
| Critical region: \(\bar{x} < 202 - 2.3263 \times \frac{2}{\sqrt{n}}\) | M1 | Correct z-value for 1% one-tailed |
| \(\bar{x} < 202 - \frac{4.6526}{\sqrt{n}}\) | A1 | Correct critical region |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Type II error: \(P(\bar{X} \geq 202 - \frac{4.6526}{\sqrt{n}} \mid \mu = 200) < 0.05\) | M1 | Setting up type II error condition |
| \(P\left(Z \geq \frac{202 - \frac{4.6526}{\sqrt{n}} - 200}{\frac{2}{\sqrt{n}}}\right) < 0.05\) | M1 | Standardising with \(\mu = 200\) |
| \(\frac{2\sqrt{n} - 4.6526}{2} \cdot \frac{\sqrt{n}}{\sqrt{n}} \Rightarrow \sqrt{n} - 2.3263 < 1.6449\) | M1 | Correct rearrangement |
| \(\sqrt{n} > 1.6449 + 2.3263 = 3.9712\) | A1 | Correct inequality |
| \(n > 15.77...\) | A1 | |
| Minimum value of \(n = 16\) | A1 |
# Question 7:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Under $H_0$: $\bar{X} \sim N\left(202, \frac{4}{n}\right)$ | B1 | Correct distribution stated |
| Critical region: $\bar{x} < 202 - 2.3263 \times \frac{2}{\sqrt{n}}$ | M1 | Correct z-value for 1% one-tailed |
| $\bar{x} < 202 - \frac{4.6526}{\sqrt{n}}$ | A1 | Correct critical region |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Type II error: $P(\bar{X} \geq 202 - \frac{4.6526}{\sqrt{n}} \mid \mu = 200) < 0.05$ | M1 | Setting up type II error condition |
| $P\left(Z \geq \frac{202 - \frac{4.6526}{\sqrt{n}} - 200}{\frac{2}{\sqrt{n}}}\right) < 0.05$ | M1 | Standardising with $\mu = 200$ |
| $\frac{2\sqrt{n} - 4.6526}{2} \cdot \frac{\sqrt{n}}{\sqrt{n}} \Rightarrow \sqrt{n} - 2.3263 < 1.6449$ | M1 | Correct rearrangement |
| $\sqrt{n} > 1.6449 + 2.3263 = 3.9712$ | A1 | Correct inequality |
| $n > 15.77...$ | A1 | |
| Minimum value of $n = 16$ | A1 | |
---7. A machine produces bricks. The lengths, $x \mathrm {~mm}$, of the bricks are distributed $\mathrm { N } \left( \mu , 2 ^ { 2 } \right)$. At the start of each week a random sample of $n$ bricks is taken to check the machine is working correctly.\\
A test is then carried out at the $1 \%$ level of significance with
$$\mathrm { H } _ { 0 } : \mu = 202 \text { and } \mathrm { H } _ { 1 } : \mu < 202$$
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $n$, the critical region of the test.
The probability of a type II error, when $\mu = 200$, is less than 0.05
\item Find the minimum value of $n$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 2013 Q7 [9]}}