| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 2 Statistics (Further AS Paper 2 Statistics) |
| Year | 2020 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Uniform Distribution |
| Type | Modelling assumptions and refinements |
| Difficulty | Moderate -0.8 This is a straightforward question on discrete uniform distribution requiring only direct formula application for E(X), Var(X), and a simple probability calculation. Part (d) asks for basic interpretation of data versus model expectations. All parts are standard textbook exercises with no problem-solving or novel insight required, making it easier than average even for Further Maths. |
| Spec | 5.02e Discrete uniform distribution |
| \(x\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Frequency | 103 | 63 | 84 | 110 | 74 | 41 | 85 | 240 |
| Answer | Marks | Guidance |
|---|---|---|
| 3(a) | Obtains E(X) = 4.5 OE | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| 3(b) | Obtains Var (X) = 5.25 OE | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| 3(c) | Obtains P(X ≥ 6) = 0.375 OE | 1.1b |
| Answer | Marks |
|---|---|
| 3(d) | Evaluates the results and suggests |
| Answer | Marks | Guidance |
|---|---|---|
| approximately equal. | 3.5a | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| frequencies. | 3.5c | E1 |
| Total | 5 | |
| Q | Marking Instructions | AO |
Question 3:
--- 3(a) ---
3(a) | Obtains E(X) = 4.5 OE | 1.1b | B1 | 8+1
E(X) = =4.5
2
--- 3(b) ---
3(b) | Obtains Var (X) = 5.25 OE | 1.1b | B1 | 82 −1
Var (X) = =5.25
12
--- 3(c) ---
3(c) | Obtains P(X ≥ 6) = 0.375 OE | 1.1b | B1 | 1
P(X ≥ 6) = 3× =0.375
8
--- 3(d) ---
3(d) | Evaluates the results and suggests
or implies the dice is biased or that
a uniform distribution is not suitable
because the frequencies are not
approximately equal. | 3.5a | E1 | The dice appears to be biased.
The random variable X would be
modelled with a discrete random
variable where the probabilities are
estimated using relative
frequencies.
Explains that the probabilities
would be estimated using relative
frequencies. | 3.5c | E1
Total | 5
Q | Marking Instructions | AO | Marks | Typical SolutionThe random variable $X$ represents the value on the upper face of an eight-sided dice after it has been rolled. The faces are numbered 1 to 8
The random variable $X$ is modelled by a discrete uniform distribution with $n = 8$
\begin{enumerate}[label=(\alph*)]
\item Find E$(X)$
[1 mark]
\item Find Var$(X)$
[1 mark]
\item Find P$(X \geq 6)$
[1 mark]
\item The dice was rolled 800 times and the results below were obtained.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
$x$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
Frequency & 103 & 63 & 84 & 110 & 74 & 41 & 85 & 240 \\
\hline
\end{tabular}
\end{center}
State, with a reason, how you would refine the model for the random variable $X$.
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 2 Statistics 2020 Q3 [5]}}