| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2006 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Standard unbiased estimates calculation |
| Difficulty | Easy -1.2 This is a straightforward application of standard formulas for sample mean, standard deviation, and confidence intervals with no conceptual challenges. Students simply substitute given values into memorized formulas (mean = Σx/n, s = √[Σ(x-x̄)²/(n-1)], CI using t-distribution), then make basic comparisons. The question requires only routine recall and arithmetic, making it easier than average A-level content. |
| Spec | 2.02g Calculate mean and standard deviation2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| 3(a) | ||
| Mean \(= \frac{286.5}{50} = 5.73\) | B1 | CAO |
| Standard deviation \(= \sqrt{\frac{45.16}{49 \text{ or } 50}} = 0.95\) to \(0.961\) | B1 | AWFW |
| 2 | ||
| 3(b) | ||
| \(99\% \Rightarrow z = 2.57\) to \(2.58\) | B1 | AWFW, 2.5758 |
| CI for \(\mu\) is \(\bar{x} \pm z\times\frac{(\sigma \text{ or } s)}{\sqrt{n}}\) | M1 | Use of \(\sqrt{n}\) with \(n > 1\) |
| Thus \(5.73 \pm 2.5758 \times \frac{(0.95 \text{ to } 0.961)}{\sqrt{50}}\) | A1 | \(\checkmark\) on z and \(s^2 > 0\) but not on \(\bar{x}\). Accept only 50 or 49 for \(n\) |
| \(5.73 \pm (0.34\) to \(0.36)\) | \(\uparrow\) | Dependent |
| \(5.37\) to \(5.39, 6.07\) to \(6.09\) | A1 | AWFW |
| 4 | ||
| 3(c) | ||
| CI excludes both values of 5 and \(6\frac{1}{2}\) so Neither claim appears valid | B1\(\checkmark\), \(\uparrow\) | \(\checkmark\) on (b); OE. Dependent |
| B1\(\checkmark\) | \(\checkmark\) on (b); OE | |
| or CI excludes 5 so claim not valid and CI excludes \(6\frac{1}{2}\) so claim not valid | B1\(\checkmark\) | \(\checkmark\) on (b); OE |
| B1\(\checkmark\) | \(\checkmark\) on (b); OE | |
| 2 |
| **3(a)** |
|----------|
| Mean $= \frac{286.5}{50} = 5.73$ | B1 | CAO |
| Standard deviation $= \sqrt{\frac{45.16}{49 \text{ or } 50}} = 0.95$ to $0.961$ | B1 | AWFW |
| | | 2 |
| **3(b)** |
|----------|
| $99\% \Rightarrow z = 2.57$ to $2.58$ | B1 | AWFW, 2.5758 |
| CI for $\mu$ is $\bar{x} \pm z\times\frac{(\sigma \text{ or } s)}{\sqrt{n}}$ | M1 | Use of $\sqrt{n}$ with $n > 1$ |
| Thus $5.73 \pm 2.5758 \times \frac{(0.95 \text{ to } 0.961)}{\sqrt{50}}$ | A1 | $\checkmark$ on z and $s^2 > 0$ but not on $\bar{x}$. Accept only 50 or 49 for $n$ |
| $5.73 \pm (0.34$ to $0.36)$ | $\uparrow$ | Dependent |
| $5.37$ to $5.39, 6.07$ to $6.09$ | A1 | AWFW |
| | | 4 |
| **3(c)** |
|----------|
| CI excludes both values of 5 and $6\frac{1}{2}$ so Neither claim appears valid | B1$\checkmark$, $\uparrow$ | $\checkmark$ on (b); OE. Dependent |
| | B1$\checkmark$ | $\checkmark$ on (b); OE |
| or CI excludes 5 so claim not valid and CI excludes $6\frac{1}{2}$ so claim not valid | B1$\checkmark$ | $\checkmark$ on (b); OE |
| | B1$\checkmark$ | $\checkmark$ on (b); OE |
| | | 2 |
**Total for Q3: 8**
---3 When an alarm is raised at a market town's fire station, the fire engine cannot leave until at least five fire-fighters arrive at the station. The call-out time, $X$ minutes, is the time between an alarm being raised and the fire engine leaving the station.
The value of $X$ was recorded on a random sample of 50 occasions. The results are summarised below, where $\bar { x }$ denotes the sample mean.
$$\sum x = 286.5 \quad \sum ( x - \bar { x } ) ^ { 2 } = 45.16$$
\begin{enumerate}[label=(\alph*)]
\item Find values for the mean and standard deviation of this sample of 50 call-out times.
\item Hence construct a $99 \%$ confidence interval for the mean call-out time.
\item The fire and rescue service claims that the station's mean call-out time is less than 5 minutes, whereas a parish councillor suggests that it is more than $6 \frac { 1 } { 2 }$ minutes.
Comment on each of these claims.
\end{enumerate}
\hfill \mbox{\textit{AQA S1 2006 Q3 [8]}}