| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics B AS (Further Statistics B AS) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Distribution |
| Type | Normal approximation to sum |
| Difficulty | Standard +0.3 This question involves understanding exponential distributions and simulation, but requires only basic interpretation of given data. Part (a) is conceptual explanation, part (b) is simple counting from a table (13 out of 20 values exceed 52), and part (c) appears incomplete but likely asks about simulation methodology. No complex calculations or novel insights needed—straightforward application of understanding exponential distributions as waiting times and basic probability estimation from simulation data. |
| Spec | 5.02i Poisson distribution: random events model |
| A | B | C | D | |
| 1 | \(\mathrm { X } _ { 1 }\) | \(\mathrm { X } _ { 2 }\) | \(\mathrm { X } _ { 3 }\) | T |
| 2 | 10.9 | 21.5 | 5.3 | 37.7 |
| 3 | 23.9 | 52.4 | 85.3 | 161.6 |
| 4 | 5.2 | 10.4 | 24.0 | 39.6 |
| 5 | 2.9 | 14.4 | 0.8 | 18.1 |
| 6 | 9.0 | 43.3 | 49.7 | 102.0 |
| 7 | 0.4 | 16.2 | 12.4 | 29.0 |
| 8 | 44.1 | 39.5 | 22.1 | 105.7 |
| 9 | 9.2 | 43.6 | 13.9 | 66.7 |
| 10 | 40.4 | 10.9 | 6.1 | 57.4 |
| 11 | 3.2 | 54.8 | 15.7 | 73.7 |
| 12 | 5.3 | 6.1 | 1.6 | 13.0 |
| 13 | 20.5 | 28.9 | 22.9 | 72.3 |
| 14 | 37.3 | 2.1 | 28.6 | 68.0 |
| 15 | 7.1 | 13.6 | 50.1 | 70.8 |
| 16 | 18.6 | 2.0 | 9.3 | 29.9 |
| 17 | 9.0 | 1.2 | 49.9 | 60.1 |
| 18 | 1.9 | 9.5 | 69.8 | 81.2 |
| 19 | 9.0 | 2.1 | 10.4 | 21.5 |
| 20 | 28.7 | 1.4 | 93.8 | 123.9 |
| 21 | 1.8 | 2.9 | 34.8 | 39.5 |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (a) | X , X and X are the times for which 3 individual |
| Answer | Marks | Guidance |
|---|---|---|
| third failure | E1 | |
| [1] | 2.4 | Must mention sum or total |
| 5 | (b) | 1 2 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 0 | B1 | |
| [1] | 1.1 | |
| 5 | (c) | By using more rows in the spreadsheet |
| [1] | 1.1 | Allow ‘increase the sample size’ |
| 5 | (d) | E(Mean of 50 values) = 60 |
| Answer | Marks |
|---|---|
| P(Mean > 52) = 0.9488 | B1 |
| Answer | Marks |
|---|---|
| [5] | 1.1 |
Question 5:
5 | (a) | X , X and X are the times for which 3 individual
1 2 3
keyboards last so their sum T is the total time up to the
third failure | E1
[1] | 2.4 | Must mention sum or total
5 | (b) | 1 2
Estimate = = 0 .6
2 0 | B1
[1] | 1.1
5 | (c) | By using more rows in the spreadsheet | E1
[1] | 1.1 | Allow ‘increase the sample size’
5 | (d) | E(Mean of 50 values) = 60
Var(T) = 1200
1 2 0 0
Var(Mean of 50 values) =
5 0
So mean of 50 values of T ~ approx N(60, 24)
P(Mean > 52) = 0.9488 | B1
M1
A1
M1
A1
[5] | 1.1
1.1a
1.1
3.3
1.15 Layla works at an internet café. Each terminal at the café has its own keyboard, and keyboards need to be replaced whenever faults develop.
Layla knows that the number of weeks for which a keyboard lasts before it needs to be replaced can be modelled by the random variable $X$, which has an exponential distribution with mean 20 and variance 400 . She wants to investigate how likely it is that the keyboard at a terminal will need to be replaced at least 3 times within a year (taken as being a period of 52 weeks).
Layla designs the simulation shown in the spreadsheet below. Each of the 20 rows below the heading row consists of 3 values of $X$ together with their sum $T$. All of the values in the spreadsheet have been rounded to 1 decimal place.
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
& A & B & C & D \\
\hline
1 & $\mathrm { X } _ { 1 }$ & $\mathrm { X } _ { 2 }$ & $\mathrm { X } _ { 3 }$ & T \\
\hline
2 & 10.9 & 21.5 & 5.3 & 37.7 \\
\hline
3 & 23.9 & 52.4 & 85.3 & 161.6 \\
\hline
4 & 5.2 & 10.4 & 24.0 & 39.6 \\
\hline
5 & 2.9 & 14.4 & 0.8 & 18.1 \\
\hline
6 & 9.0 & 43.3 & 49.7 & 102.0 \\
\hline
7 & 0.4 & 16.2 & 12.4 & 29.0 \\
\hline
8 & 44.1 & 39.5 & 22.1 & 105.7 \\
\hline
9 & 9.2 & 43.6 & 13.9 & 66.7 \\
\hline
10 & 40.4 & 10.9 & 6.1 & 57.4 \\
\hline
11 & 3.2 & 54.8 & 15.7 & 73.7 \\
\hline
12 & 5.3 & 6.1 & 1.6 & 13.0 \\
\hline
13 & 20.5 & 28.9 & 22.9 & 72.3 \\
\hline
14 & 37.3 & 2.1 & 28.6 & 68.0 \\
\hline
15 & 7.1 & 13.6 & 50.1 & 70.8 \\
\hline
16 & 18.6 & 2.0 & 9.3 & 29.9 \\
\hline
17 & 9.0 & 1.2 & 49.9 & 60.1 \\
\hline
18 & 1.9 & 9.5 & 69.8 & 81.2 \\
\hline
19 & 9.0 & 2.1 & 10.4 & 21.5 \\
\hline
20 & 28.7 & 1.4 & 93.8 & 123.9 \\
\hline
21 & 1.8 & 2.9 & 34.8 & 39.5 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Explain why $T$ represents the number of weeks after which the third keyboard at a terminal will need to be replaced.
\item Use the information in the spreadsheet to write down an estimate of $\mathrm { P } ( T > 52 )$.
\item Explain how you could obtain a more reliable estimate of $\mathrm { P } ( T > 52 )$.
\item The internet café has 50 terminals. You are given that faults in keyboards occur independently of each other.
Determine an estimate of the probability that the mean number of weeks before which the third keyboard at a terminal needs to be replaced is more than 52 .
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics B AS 2022 Q5 [8]}}