| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2008 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Calculate statistics from grouped frequency table |
| Difficulty | Moderate -0.3 This is a standard S1 grouped frequency question requiring routine calculations (mean, standard deviation from grouped data, confidence interval) and basic interpretation. The multi-part structure and confidence interval push it slightly above trivial, but all techniques are textbook applications with no novel problem-solving required. |
| Spec | 2.01c Sampling techniques: simple random, opportunity, etc2.02g Calculate mean and standard deviation5.05c Hypothesis test: normal distribution for population mean5.05d Confidence intervals: using normal distribution |
| Difference | Frequency |
| \(- 6 \leqslant x < - 4\) | 4 |
| \(- 4 \leqslant x < - 2\) | 9 |
| \(- 2 \leqslant x < 0\) | 13 |
| \(0 \leqslant x < 2\) | 27 |
| \(2 \leqslant x < 4\) | 21 |
| \(4 \leqslant x < 6\) | 15 |
| \(6 \leqslant x < 8\) | 7 |
| \(8 \leqslant x \leqslant 10\) | 4 |
| Total | 100 |
7 Vernon, a service engineer, is expected to carry out a boiler service in one hour.\\
One hour is subtracted from each of his actual times, and the resulting differences, $x$ minutes, for a random sample of 100 boiler services are summarised in the table.
\begin{center}
\begin{tabular}{|l|l|}
\hline
Difference & Frequency \\
\hline
$- 6 \leqslant x < - 4$ & 4 \\
\hline
$- 4 \leqslant x < - 2$ & 9 \\
\hline
$- 2 \leqslant x < 0$ & 13 \\
\hline
$0 \leqslant x < 2$ & 27 \\
\hline
$2 \leqslant x < 4$ & 21 \\
\hline
$4 \leqslant x < 6$ & 15 \\
\hline
$6 \leqslant x < 8$ & 7 \\
\hline
$8 \leqslant x \leqslant 10$ & 4 \\
\hline
Total & 100 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Calculate estimates of the mean and the standard deviation of these differences.\\
(4 marks)
\item Hence deduce, in minutes, estimates of the mean and the standard deviation of Vernon's actual service times for this sample.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Construct an approximate $98 \%$ confidence interval for the mean time taken by Vernon to carry out a boiler service.
\item Give a reason why this confidence interval is approximate rather than exact.
\end{enumerate}\item Vernon claims that, more often than not, a boiler service takes more than an hour and that, on average, a boiler service takes much longer than an hour.
Comment, with a justification, on each of these claims.
\end{enumerate}
\hfill \mbox{\textit{AQA S1 2008 Q7 [14]}}