| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 2 Statistics (Further AS Paper 2 Statistics) |
| Year | 2021 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Uniform Distribution |
| Type | Find parameter n from mean |
| Difficulty | Moderate -0.8 This is a straightforward application of the discrete uniform distribution formula for mean: (n+1)/2 = 8, giving n = 15. Parts (b) and (c) are direct substitutions into standard formulas. The question requires only recall of formulas with minimal algebraic manipulation, making it easier than average A-level content. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02e Discrete uniform distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{n+1}{2} = 8\) | M1 | Uses formula for mean of discrete uniform distribution to form an equation; condone one sign error |
| \(n + 1 = 16\), therefore \(n = 15\) | R1 | Completes rigorous algebraic proof by solving the equation to show \(n = 15\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(X > 4) = 11 \times \frac{1}{15} = \frac{11}{15}\) | B1 | Obtains correct value of \(P(X > 4)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\text{Var}(X) = \frac{15^2 - 1}{12}\) | M1 | Uses formula for variance of discrete uniform distribution |
| \(= \frac{56}{3}\) | A1 | Obtains \(\text{Var}(X) = \frac{56}{3}\) |
## Question 3(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{n+1}{2} = 8$ | M1 | Uses formula for mean of discrete uniform distribution to form an equation; condone one sign error |
| $n + 1 = 16$, therefore $n = 15$ | R1 | Completes rigorous algebraic proof by solving the equation to show $n = 15$ |
## Question 3(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(X > 4) = 11 \times \frac{1}{15} = \frac{11}{15}$ | B1 | Obtains correct value of $P(X > 4)$ |
## Question 3(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\text{Var}(X) = \frac{15^2 - 1}{12}$ | M1 | Uses formula for variance of discrete uniform distribution |
| $= \frac{56}{3}$ | A1 | Obtains $\text{Var}(X) = \frac{56}{3}$ |
---3 The random variable $X$ has a discrete uniform distribution and takes values $1,2,3 , \ldots , n$
The mean of $X$ is 8
3
\begin{enumerate}[label=(\alph*)]
\item Show that $n = 15$\\[0pt]
[2 marks]\\
LL\\
3
\item $\quad$ Find $\mathrm { P } ( X > 4 )$\\
3
\item Find the variance of $X$, giving your answer in exact form.
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 2 Statistics 2021 Q3 [5]}}