| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2006 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Single sample confidence interval t-distribution |
| Difficulty | Standard +0.3 This is a straightforward t-distribution confidence interval question with clear data and standard procedures. While it requires knowing the t-distribution formula and interpretation, it's a routine textbook exercise with no conceptual challenges—slightly easier than average since the calculations are simple with n=9 and the data is clean. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| \(\bar{x} = 8.0\) | B1 | |
| \(S = 2.121\) | B1 | |
| \(\nu = 8\) | B1 | |
| \(t = 1.860\) | B1√ |
| Answer | Marks | Guidance |
|---|---|---|
| \[= 8 \pm 1.860\left(\frac{2.121}{3}\right)\] | M1 | |
| \[= 8 \pm 1.315\] | A1ft | |
| \[= (6.68, 9.32)\] | A1 | 7 marks |
| Answer | Marks | Guidance |
|---|---|---|
| The Headteacher's claim seems to be slightly optimistic | E1ft | |
| because value of 5 outside the confidence interval | E1ft | 2 marks |
**3(a)**
$\bar{x} = 8.0$ | B1 | |
$S = 2.121$ | B1 | |
$\nu = 8$ | B1 | |
$t = 1.860$ | B1√ | | (on their ν)
$90\%$ confidence interval for $\mu$:
$$= 8 \pm 1.860\left(\frac{2.121}{3}\right)$$ | M1 | |
$$= 8 \pm 1.315$$ | A1ft | |
$$= (6.68, 9.32)$$ | A1 | 7 marks | (6.68 to 6.69, 9.31 to 9.32)
**3(b)**
The Headteacher's claim seems to be slightly optimistic | E1ft | | Headteacher's claim isn't supported by the evidence **and**
because value of 5 outside the confidence interval | E1ft | 2 marks | It appears that the mean time to see a mathematics teacher is greater than 5 minutes
**Question 3 Total: 9 marks**
---3 The time, $T$ minutes, that parents have to wait before seeing a mathematics teacher at a school parents' evening can be modelled by a normal distribution with mean $\mu$ and standard deviation $\sigma$.
At a recent parents' evening, a random sample of 9 parents was asked to record the times that they waited before seeing a mathematics teacher.
The times, in minutes, are
$$\begin{array} { l l l l l l l l l }
5 & 12 & 10 & 8 & 7 & 6 & 9 & 7 & 8
\end{array}$$
\begin{enumerate}[label=(\alph*)]
\item Construct a $90 \%$ confidence interval for $\mu$.
\item Comment on the headteacher's claim that the mean time that parents have to wait before seeing a mathematics teacher is 5 minutes.
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2006 Q3 [9]}}