| Exam Board | OCR MEI |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modelling and Hypothesis Testing |
| Type | Queueing and service simulation |
| Difficulty | Easy -1.2 This is a standard D1 simulation question requiring routine application of random number allocation rules and table completion. Parts (i)-(iii) involve straightforward probability-to-random-number mapping, (iv) is mechanical table filling, and (v) is basic arithmetic. No problem-solving insight or novel reasoning required—purely procedural execution of a well-practiced technique. |
| Spec | 2.01d Select/critique sampling: in context |
| Time taken (mins) | 1 | 1.5 | 2 | 2.5 |
| Probability | \(\frac { 3 } { 10 }\) | \(\frac { 2 } { 5 }\) | \(\frac { 1 } { 5 }\) | \(\frac { 1 } { 10 }\) |
| Time taken (mins) | 1 | 1.5 | 2 | 2.5 | 3 |
| Probability | \(\frac { 1 } { 7 }\) | \(\frac { 2 } { 7 }\) | \(\frac { 2 } { 7 }\) | \(\frac { 1 } { 7 }\) | \(\frac { 1 } { 7 }\) |
| Answer | Marks |
|---|---|
| (i) Grouping example: 0–6 petrol; 7–9 other | B1 |
| (ii) Time grouping example: 0–2 min 1 min; 3–6 1.5 mins; 7–8 2 mins; 9 2.5 mins | M1; A1 |
| (iii) Two-digit grouping example: 00–13 1 min; 14–41 1.5 mins; 42–69 2 mins; 70–83 2.5 mins; 84–97 3 mins; 98, 99 reject. Two digits gives fewer rejects | M1 some rejected; A1 2 rejected; A1; B1 |
| (iv) Simulation table with customer arrivals, types (F/N), service times, and times in shop | B1 arrival times; M1 types; M1 service start; M1 service duration; M1 service end; M1 time in shop; A1 |
| (v) Average time in shop: \(\frac{24.5}{10} = 2.45\) mins | M1 A1 |
**(i)** Grouping example: 0–6 petrol; 7–9 other | B1 |
**(ii)** Time grouping example: 0–2 min 1 min; 3–6 1.5 mins; 7–8 2 mins; 9 2.5 mins | M1; A1 |
**(iii)** Two-digit grouping example: 00–13 1 min; 14–41 1.5 mins; 42–69 2 mins; 70–83 2.5 mins; 84–97 3 mins; 98, 99 reject. Two digits gives fewer rejects | M1 some rejected; A1 2 rejected; A1; B1 |
**(iv)** Simulation table with customer arrivals, types (F/N), service times, and times in shop | B1 arrival times; M1 types; M1 service start; M1 service duration; M1 service end; M1 time in shop; A1 |
**(v)** Average time in shop: $\frac{24.5}{10} = 2.45$ mins | M1 A1 |6 Answer part (iv) of this question on the insert provided.
There are two types of customer who use the shop at a service station. $70 \%$ buy fuel, the other $30 \%$ do not. There is only one till in operation.\\
(i) Give an efficient rule for using one-digit random numbers to simulate the type of customer arriving at the service station.
Table 6.1 shows the distribution of time taken at the till by customers who are buying fuel.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | }
\hline
Time taken (mins) & 1 & 1.5 & 2 & 2.5 \\
\hline
Probability & $\frac { 3 } { 10 }$ & $\frac { 2 } { 5 }$ & $\frac { 1 } { 5 }$ & $\frac { 1 } { 10 }$ \\
\hline
\end{tabular}
\end{center}
\section*{Table 6.1}
(ii) Specify an efficient rule for using one-digit random numbers to simulate the time taken at the till by customers purchasing fuel.
Table 6.2 shows the distribution of time taken at the till by customers who are not buying fuel.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
Time taken (mins) & 1 & 1.5 & 2 & 2.5 & 3 \\
\hline
Probability & $\frac { 1 } { 7 }$ & $\frac { 2 } { 7 }$ & $\frac { 2 } { 7 }$ & $\frac { 1 } { 7 }$ & $\frac { 1 } { 7 }$ \\
\hline
\end{tabular}
\end{center}
\section*{Table 6.2}
(iii) Specify an efficient rule for using two-digit random numbers to simulate the time taken at the till by customers not buying fuel.
What is the advantage in using two-digit random numbers instead of one-digit random numbers in this part of the question?
The table in the insert shows a partially completed simulation study of 10 customers arriving at the till.\\
(iv) Complete the table using the random numbers which are provided.\\
(v) Calculate the mean total time spent queuing and paying.
\hfill \mbox{\textit{OCR MEI D1 2006 Q6 [16]}}