| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2012 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Direct binomial probability calculation |
| Difficulty | Moderate -0.3 This is a straightforward application of binomial distribution formulas with clearly stated parameters. Part (a) requires direct use of binomial probability calculations (including cumulative probabilities), which are standard S1 techniques. Part (b) involves basic mean/variance formulas and a simple hypothesis commentary comparing observed vs expected values. While multi-part with several calculations, each step follows routine procedures without requiring problem-solving insight or novel approaches—slightly easier than the average A-level question due to its procedural nature. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.01a Permutations and combinations: evaluate probabilities |
| Colour | White | Yellow | Green | Blue | Red | Purple |
| Proportion | 0.15 | 0.15 | 0.20 | 0.15 | 0.22 | 0.13 |
6 A bin contains a very large number of paper clips of different colours. The proportion of each colour is shown in the table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
Colour & White & Yellow & Green & Blue & Red & Purple \\
\hline
Proportion & 0.15 & 0.15 & 0.20 & 0.15 & 0.22 & 0.13 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Packets are filled from the bin. Each filled packet contains exactly 30 paper clips which may be considered to be a random sample.
Use binomial distributions to determine the probability that a filled packet contains:
\begin{enumerate}[label=(\roman*)]
\item exactly 2 purple paper clips;
\item a total of more than 10 red or purple paper clips;
\item at least 5 but at most 10 green paper clips.
\end{enumerate}\item Jumbo packets are also filled from the bin. Each filled jumbo packet contains exactly 100 paper clips.
\begin{enumerate}[label=(\roman*)]
\item Assuming that the number of paper clips in a jumbo packet may be considered to be a random sample, calculate the mean and the variance of the number of red paper clips in a filled jumbo packet.
\item It is claimed that the proportion of red paper clips in the bin is greater than 0.22 and that jumbo packets do not contain random samples of paper clips.
An analysis of the number of red paper clips in each of a random sample of 50 filled jumbo packets resulted in a mean of 22.1 and a standard deviation of 4.17.
Comment, with numerical justification, on each of the two claims.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S1 2012 Q6 [14]}}