| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics Major (Further Statistics Major) |
| Year | 2020 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Simulation and spreadsheet problems |
| Difficulty | Standard +0.3 This is a straightforward spreadsheet simulation question requiring basic counting from given data (part a), a standard improvement suggestion (part b), and applying the Central Limit Theorem to find a probability (part c). All techniques are routine for Further Statistics students, with no novel problem-solving required. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.05a Sample mean distribution: central limit theorem |
| 1 | A | B | C |
| 1 | X | Y | \(X - 2 Y\) |
| 2 | 6 | 6 | -6 |
| 3 | 5 | 4 | -3 |
| 4 | 8 | 1 | 6 |
| 5 | 6 | 5 | -4 |
| 6 | 6 | 3 | 0 |
| 7 | 8 | 1 | 6 |
| 8 | 6 | 4 | -2 |
| 9 | 5 | 4 | -3 |
| 10 | 7 | 4 | -1 |
| 11 | 8 | 3 | 2 |
| 12 | 6 | 2 | 2 |
| 13 | 5 | 1 | 3 |
| 14 | 6 | 1 | 4 |
| 15 | 5 | 4 | -3 |
| 16 | 7 | 2 | 3 |
| 17 | 5 | 2 | 1 |
| 18 | 4 | 4 | -4 |
| 19 | 5 | 0 | 5 |
| 20 | 5 | 1 | 3 |
| 21 | 4 | 2 | 0 |
| nn |
| Answer | Marks | Guidance |
|---|---|---|
| 10 | (a) | Estimate of P(X − 2Y > 0) is 0.5 |
| Estimate of P(X − 2Y > 1) is 0.45 | B1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 10 | (b) | By using more rows in the spreadsheet |
| [1] | 2.4 | Condone ‘Run more simulations’, ‘take more samples’ or |
| Answer | Marks | Guidance |
|---|---|---|
| 10 | (c) | E(W) = 0 |
| Answer | Marks |
|---|---|
| P(W > 1) = P(Normal > 1.01) = 0.0380 | B1 |
| Answer | Marks |
|---|---|
| [9] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | For Normal |
Question 10:
10 | (a) | Estimate of P(X − 2Y > 0) is 0.5
Estimate of P(X − 2Y > 1) is 0.45 | B1
B1
[2] | 1.1
1.1
10 | (b) | By using more rows in the spreadsheet | E1
[1] | 2.4 | Condone ‘Run more simulations’, ‘take more samples’ or
‘increasing the number of values sampled’ o.e.
10 | (c) | E(W) = 0
Var(X)=20×0.3×0.7
= 4.2
Var(Y)=3
Var(X −2Y)=4.2+22×3
=16.2
( 0,16.2)
Distribution is N
50
P(W > 1) = P(Normal > 1.01) = 0.0380 | B1
M1
A1
B1
M1
A1
M1
M1
A1
[9] | 1.1
3.3
1.1
1.1
3.4
1.1
3.3
1.1
1.1 | For Normal
For parameters
(Omitting cc gives 0.0395)10 The discrete random variables $X$ and $Y$ have distributions as follows: $X \sim \mathrm {~B} ( 20,0.3 )$ and $Y \sim \operatorname { Po } ( 3 )$.
The spreadsheet in Fig. 10 shows a simulation of the distributions of $X$ and $Y$. Each of the 20 rows below the heading row consists of a value of $X$, a value of $Y$, and the value of $X - 2 Y$.
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
1 & A & B & C \\
\hline
1 & X & Y & $X - 2 Y$ \\
\hline
2 & 6 & 6 & -6 \\
\hline
3 & 5 & 4 & -3 \\
\hline
4 & 8 & 1 & 6 \\
\hline
5 & 6 & 5 & -4 \\
\hline
6 & 6 & 3 & 0 \\
\hline
7 & 8 & 1 & 6 \\
\hline
8 & 6 & 4 & -2 \\
\hline
9 & 5 & 4 & -3 \\
\hline
10 & 7 & 4 & -1 \\
\hline
11 & 8 & 3 & 2 \\
\hline
12 & 6 & 2 & 2 \\
\hline
13 & 5 & 1 & 3 \\
\hline
14 & 6 & 1 & 4 \\
\hline
15 & 5 & 4 & -3 \\
\hline
16 & 7 & 2 & 3 \\
\hline
17 & 5 & 2 & 1 \\
\hline
18 & 4 & 4 & -4 \\
\hline
19 & 5 & 0 & 5 \\
\hline
20 & 5 & 1 & 3 \\
\hline
21 & 4 & 2 & 0 \\
\hline
nn & & & \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 10}
\end{center}
\end{table}
\begin{enumerate}[label=(\alph*)]
\item Use the spreadsheet to estimate each of the following.
\begin{itemize}
\item $\mathrm { P } ( X - 2 Y > 0 )$
\item $\mathrm { P } ( X - 2 Y > 1 )$
\item How could the estimates in part (a) be improved?
\end{itemize}
The mean of 50 values of $X - 2 Y$ is denoted by the random variable $W$.
\item Calculate an estimate of $\mathrm { P } ( W > 1 )$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics Major 2020 Q10 [12]}}