| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2006 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | One-tail z-test (lower tail) |
| Difficulty | Moderate -0.3 This is a straightforward one-sample z-test with known variance following a standard template: state hypotheses, calculate test statistic using given values (σ=9, n=35, x̄=61.5), compare to critical value, and conclude. Part (b) requires recalling the definition of Type I error in context. The question is slightly easier than average because it's a direct application of a standard procedure with no complications, though it does require proper hypothesis test structure. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: \mu = 65\) | B1 | |
| \(H_1: \mu < 65\) | ||
| \[\bar{X} \sim N\left(65, \frac{81}{35}\right)\] | ||
| \(z_{\text{crit}} = -1.6449\) | B1 | |
| \[z = \frac{61.5 - 65}{9/\sqrt{35}} = -2.30\] | M1A1 | |
| Reject \(H_0\) at 5% level of significance | A1√ | |
| Evidence to suggest students may be under-achieving | E1 | 6 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Reject \(H_0\) when \(H_0\) true ⇒ Conclude that students are under-achieving when in fact they are not | E1, E1 | 2 marks |
**6(a)**
$H_0: \mu = 65$ | B1 | | 1-tailed test
$H_1: \mu < 65$ | |
$$\bar{X} \sim N\left(65, \frac{81}{35}\right)$$ | |
$z_{\text{crit}} = -1.6449$ | B1 | |
$$z = \frac{61.5 - 65}{9/\sqrt{35}} = -2.30$$ | M1A1 | | for σ√n used
Reject $H_0$ at 5% level of significance | A1√ | | (on their z-values)
Evidence to suggest students may be under-achieving | E1 | 6 marks
**6(b)**
Reject $H_0$ when $H_0$ true ⇒ Conclude that students are under-achieving when in fact they are not | E1, E1 | 2 marks
**Question 6 Total: 8 marks**
---6 In previous years, the marks obtained in a French test by students attending Topnotch College have been modelled satisfactorily by a normal distribution with a mean of 65 and a standard deviation of 9 .
Teachers in the French department at Topnotch College suspect that this year their students are, on average, underachieving.
In order to investigate this suspicion, the teachers selected a random sample of 35 students to take the French test and found that their mean score was 61.5.
\begin{enumerate}[label=(\alph*)]
\item Investigate, at the $5 \%$ level of significance, the teachers' suspicion.
\item Explain, in the context of this question, the meaning of a Type I error.
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2006 Q6 [8]}}