| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics Major (Further Statistics Major) |
| Year | 2021 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Sum of independent uniforms |
| Difficulty | Standard +0.3 This is a straightforward application of properties of sums of independent random variables. Students need to recognize that E(T) is the sum of means (5+25+3+28=61) and Var(T) is the sum of variances (100/12+4+36/12+16=23), then use the simulation data to estimate probabilities. While it involves multiple distributions, the calculations are routine and the simulation table guides the interpretation. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| A | B | C | D | E | F | G | H | |
| 1 | Waiting time morning | Journey time morning | Waiting time evening | Journey time evening | Total time | Total time t | Number \(\leqslant \mathbf { t }\) | |
| 2 | 0.89 | 20.78 | 1.88 | 26.30 | 49.86 | 46 | 0 | |
| 3 | 3.55 | 21.24 | 1.04 | 29.61 | 55.44 | 48 | 4 | |
| 4 | 2.13 | 21.83 | 2.40 | 28.64 | 55.00 | 50 | 13 | |
| 5 | 5.12 | 25.04 | 3.13 | 24.30 | 57.60 | 52 | 35 | |
| 6 | 4.03 | 27.49 | 2.19 | 30.81 | 64.52 | 54 | 57 | |
| 7 | 2.47 | 20.54 | 4.32 | 34.61 | 61.93 | 56 | 104 | |
| 8 | 3.21 | 26.93 | 3.78 | 27.66 | 61.58 | 58 | 156 | |
| 9 | 9.72 | 24.15 | 0.63 | 27.53 | 62.03 | 60 | 218 | |
| 10 | 1.59 | 28.45 | 0.08 | 35.87 | 65.99 | 62 | 288 | |
| 11 | 7.34 | 23.04 | 4.02 | 24.77 | 59.17 | 64 | 357 | |
| 12 | 1.04 | 24.69 | 1.66 | 31.95 | 59.33 | 66 | 408 | |
| 13 | 7.17 | 22.16 | 2.55 | 25.39 | 57.28 | 68 | 441 | |
| 14 | 5.20 | 26.97 | 2.41 | 30.05 | 64.62 | 70 | 475 | |
| 15 | 5.01 | 26.84 | 1.88 | 36.21 | 69.93 | 72 | 490 | |
| 16 | 3.76 | 26.03 | 2.21 | 30.96 | 62.96 | 74 | 496 | |
| 17 | 0.96 | 23.72 | 2.55 | 29.36 | 56.59 | 76 | 500 | |
| 18 | 8.64 | 24.97 | 2.82 | 26.39 | 62.82 | |||
| 19 | 0.59 | 20.82 | 4.57 | 31.41 | 57.38 | |||
| 20 | 9.85 | 23.68 | 5.54 | 29.92 | 68.99 | |||
| 01 |
| Answer | Marks | Guidance |
|---|---|---|
| 10 | (a) | 104 |
| Answer | Marks |
|---|---|
| 500 | B1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 10 | (b) | E(T) = 25 + 28 + 5 + 3 = 61 |
| Answer | Marks |
|---|---|
| P(W > 61) = 0.5 | B1 |
| Answer | Marks |
|---|---|
| [5] | 3.1a |
| Answer | Marks | Guidance |
|---|---|---|
| 1.1 | BC | |
| 10 | (c) | Because the mean is 61 and both the uniform and |
| Answer | Marks |
|---|---|
| close to 0.5 | E1 |
| Answer | Marks |
|---|---|
| [2] | 2.2b |
| 2.4 | For second mark must mention |
Question 10:
10 | (a) | 104
P(T ≤56)= =0.208
500
253
P(T >61)=1− =0.494
500 | B1
B1
[2] | 1.1
1.1
10 | (b) | E(T) = 25 + 28 + 5 + 3 = 61
Var(T)= 1 ×102+ 1 ×62+4+16
12 12
= 94 (= 31.333)
3
W ⁓ N(61, 31.333) so P(W ≤ 56) = 0.186
P(W > 61) = 0.5 | B1
M1
A1
B1
B1
[5] | 3.1a
1.1
1.1
3.3
1.1 | BC
10 | (c) | Because the mean is 61 and both the uniform and
Normal distributions are symmetrical so you
would expect the simulated probability to be very
close to 0.5 | E1
E1
[2] | 2.2b
2.4 | For second mark must mention
symmetrical10 Sarah takes a bus to work each weekday morning and returns each evening. The times in minutes that she has to wait for the bus in the morning and evening are modelled by uniform distributions over the intervals $[ 0,10 ]$ and $[ 0,6 ]$ respectively. The times in minutes for the bus journeys in the morning and evening are modelled by $\mathrm { N } ( 25,4 )$ and $\mathrm { N } ( 28,16 )$ respectively. You should assume that all of the times are independent. The total time in minutes that she takes for her two journeys, including the waiting times, is denoted by the random variable $T$.
The spreadsheet below shows the first 20 rows of a simulation of 500 return journeys. It also shows in column H the numbers of values of $T$ that are less than or equal to the corresponding values in column G. For example, there are 156 out of the 500 simulated values of $T$ which are less than or equal to 58 minutes. All of the times have been rounded to 2 decimal places.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
& A & B & C & D & E & F & G & H \\
\hline
1 & Waiting time morning & Journey time morning & Waiting time evening & Journey time evening & Total time & & Total time t & Number $\leqslant \mathbf { t }$ \\
\hline
2 & 0.89 & 20.78 & 1.88 & 26.30 & 49.86 & & 46 & 0 \\
\hline
3 & 3.55 & 21.24 & 1.04 & 29.61 & 55.44 & & 48 & 4 \\
\hline
4 & 2.13 & 21.83 & 2.40 & 28.64 & 55.00 & & 50 & 13 \\
\hline
5 & 5.12 & 25.04 & 3.13 & 24.30 & 57.60 & & 52 & 35 \\
\hline
6 & 4.03 & 27.49 & 2.19 & 30.81 & 64.52 & & 54 & 57 \\
\hline
7 & 2.47 & 20.54 & 4.32 & 34.61 & 61.93 & & 56 & 104 \\
\hline
8 & 3.21 & 26.93 & 3.78 & 27.66 & 61.58 & & 58 & 156 \\
\hline
9 & 9.72 & 24.15 & 0.63 & 27.53 & 62.03 & & 60 & 218 \\
\hline
10 & 1.59 & 28.45 & 0.08 & 35.87 & 65.99 & & 62 & 288 \\
\hline
11 & 7.34 & 23.04 & 4.02 & 24.77 & 59.17 & & 64 & 357 \\
\hline
12 & 1.04 & 24.69 & 1.66 & 31.95 & 59.33 & & 66 & 408 \\
\hline
13 & 7.17 & 22.16 & 2.55 & 25.39 & 57.28 & & 68 & 441 \\
\hline
14 & 5.20 & 26.97 & 2.41 & 30.05 & 64.62 & & 70 & 475 \\
\hline
15 & 5.01 & 26.84 & 1.88 & 36.21 & 69.93 & & 72 & 490 \\
\hline
16 & 3.76 & 26.03 & 2.21 & 30.96 & 62.96 & & 74 & 496 \\
\hline
17 & 0.96 & 23.72 & 2.55 & 29.36 & 56.59 & & 76 & 500 \\
\hline
18 & 8.64 & 24.97 & 2.82 & 26.39 & 62.82 & & & \\
\hline
19 & 0.59 & 20.82 & 4.57 & 31.41 & 57.38 & & & \\
\hline
20 & 9.85 & 23.68 & 5.54 & 29.92 & 68.99 & & & \\
\hline
01 & & & & & & & & \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Use the spreadsheet output to estimate each of the following.
\begin{itemize}
\item $\mathrm { P } ( T \leqslant 56 )$
\item $\mathrm { P } ( T > 61 )$
\item The random variable $W$ is Normally distributed with the same mean and variance as $T$. Find each of the following.
\item $\mathrm { P } ( W \leqslant 56 )$
\item $\mathrm { P } ( W > 61 )$
\item Explain why, if many more journeys were used in the simulation, you would expect $\mathrm { P } ( T > 61 )$ to be extremely close to $\mathrm { P } ( W > 61 )$.
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics Major 2021 Q10 [9]}}