| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics Major (Further Statistics Major) |
| Session | Specimen |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | One-tailed test (increase or decrease) |
| Difficulty | Standard +0.3 This is a straightforward application of Poisson distribution with standard calculations: stating conditions (recall), finding probabilities using tables/calculator, and combining Poisson distributions. Part (iv) requires recognizing that Poisson rates add (λ=4.5) and using binomial probability, but these are routine techniques for Further Statistics students with no novel problem-solving required. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling5.02m Poisson: mean = variance = lambda |
| Answer | Marks | Guidance |
|---|---|---|
| 8 | (i) | The neutrons that are detected must occur |
| Answer | Marks |
|---|---|
| and at a constant average rate. | E1 |
| Answer | Marks |
|---|---|
| [2] | 3.3 |
| 3.3 | For randomly, independently |
| Answer | Marks | Guidance |
|---|---|---|
| 8 | (ii) | (A) |
| (B) | P(0) = 0.333 |
| Answer | Marks |
|---|---|
| P(at least 60) = 1 – 0.214 = 0.786 | B1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks |
|---|---|
| 3.4 | N |
| Answer | Marks | Guidance |
|---|---|---|
| 8 | (iii) | P(more than 8 neutrons) |
| Answer | Marks |
|---|---|
| = 0.00243 | B1 |
| Answer | Marks |
|---|---|
| [3] | 3.4 |
| Answer | Marks |
|---|---|
| 1.1 | M |
| Answer | Marks | Guidance |
|---|---|---|
| 8 | (iv) | New (cid:79)(cid:32)3.4(cid:14)1.1(cid:32)4.5 |
| Answer | Marks |
|---|---|
| = 0.337 | B1 |
| Answer | Marks |
|---|---|
| [4] | C |
| Answer | Marks |
|---|---|
| 1.1 | BC |
Question 8:
8 | (i) | The neutrons that are detected must occur
randomly, independently
and at a constant average rate. | E1
E1
[2] | 3.3
3.3 | For randomly, independently
For constant average rate or
uniform rate but not constant
rate
8 | (ii) | (A)
(B) | P(0) = 0.333
λ = 66
P(at least 60) = 1 – 0.214 = 0.786 | B1
M1
A1
[3] | 1.1
3.3
3.4 | N
BEC
BC
8 | (iii) | P(more than 8 neutrons)
= 1 – 0.999997573 = 0.000002427
Expected number = 1000 × 0.000002427
= 0.00243 | B1
M1
A1
[3] | 3.4
I
1.1a
1.1 | M
BC
8 | (iv) | New (cid:79)(cid:32)3.4(cid:14)1.1(cid:32)4.5
P(No alarm triggered in 1 second) = 0.95974
P(At least one in 10 pds) = 1 – (0.95974)10
= 0.337 | B1
E
B1
M1
A1
[4] | C
3.1b
3.4
1.1a
1.1 | BC
FT from here if (cid:79)(cid:32)3.4 used8 Natural background radiation consists of various particles, including neutrons. A detector is used to count the number of neutrons per second at a particular location.
\begin{enumerate}[label=(\roman*)]
\item State the conditions required for a Poisson distribution to be a suitable model for the number of neutrons detected per second.
The number of neutrons detected per second due to background radiation only is modelled by a Poisson distribution with mean 1.1.
\item Find the probability that the detector detects\\
(A) no neutrons in a randomly chosen second,\\
(B) at least 60 neutrons in a randomly chosen period of 1 minute.
A neutron source is switched on. It emits neutrons which should all be contained in a protective casing. The detector is used to check whether any neutrons have not been contained; these are known as stray neutrons.
If the detector detects more than 8 neutrons in a period of 1 second, an alarm will be triggered in case this high reading is due to stray neutrons.
\item Suppose that there are no stray neutrons and so the neutrons detected are all due to the background radiation. Find the expected number of times the alarm is triggered in 1000 randomly chosen periods of 1 second.
\item Suppose instead that stray neutrons are being produced at a rate of 3.4 per second in addition to the natural background radiation. Find the probability that at least one alarm will be triggered in 10 randomly chosen periods of 1 second. You should assume that all stray neutrons produced are detected.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics Major Q8 [12]}}