| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2020 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Two independent Poisson sums |
| Difficulty | Standard +0.3 This is a straightforward application of standard Poisson distribution techniques. Part (a) requires basic understanding of Poisson conditions (2 marks of discussion). Parts (b) use direct Poisson probability calculations with given parameters. Part (c) applies the sum of independent Poisson distributions property (Po(5) + Po(7.2) = Po(12.2)), then calculates cumulative probabilities. Parts (d-e) test understanding of independence assumptions. All techniques are routine for Further Statistics with no novel problem-solving required, making it slightly easier than average A-level maths difficulty. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02n Sum of Poisson variables: is Poisson |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (a) | Any reason for independence (or not) |
| Answer | Marks |
|---|---|
| they mean | B1 |
| Answer | Marks |
|---|---|
| [2] | 3.5b |
| 3.5b | “Events occur independently and at constant average rate”: B0 |
| Answer | Marks | Guidance |
|---|---|---|
| (b) | (i) | 0.146(223) BC |
| Answer | Marks |
|---|---|
| [2] | 3.4 |
| 1.1 | Correct method stated or implied |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) | 0.133(372) BC | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 1.1 | 0.068: M1A0 |
| Answer | Marks |
|---|---|
| (c) | Po(12.2) |
| Answer | Marks |
|---|---|
| = 0.604(224) BC | M1 |
| Answer | Marks |
|---|---|
| [3] | 3.3 |
| Answer | Marks |
|---|---|
| 3.4 | Stated or implied |
| Answer | Marks | Guidance |
|---|---|---|
| (d) | Sales of CD players and integrated systems need | |
| to be independent | B1 | |
| [1] | 1.1 | Need “independent” or “not related” clearly referred to the two |
| Answer | Marks |
|---|---|
| (e) | If a customer buys a CD player they probably |
| Answer | Marks | Guidance |
|---|---|---|
| as well | B1 | |
| [1] | 3.5b | Any reason for non-independence of sales of CD players and |
| Answer | Marks |
|---|---|
| Exx | α: May buy both so not independent: B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 6:
6 | (a) | Any reason for independence (or not)
… and for constant average rate (or not), in each
case without misunderstanding of what
they mean | B1
B1
[2] | 3.5b
3.5b | “Events occur independently and at constant average rate”: B0
SC: Mere assertion of both, properly contextualised: B1
SC: Variance = 4.67 which is closer to 5: B1
SC: Considers only the assumptions given in the question: B0
(b) | (i) | 0.146(223) BC | M1
A1
[2] | 3.4
1.1 | Correct method stated or implied
Correct answer only, awrt 0.146
(ii) | 0.133(372) BC | M1
A1
[2] | 1.1
1.1 | 0.068: M1A0
(treat 0.1337 as a slip, i.e. give A1 BOD)
(c) | Po(12.2)
P(≤ 15) – P(≤ 9) [= 0.8296 – 0.2253]
= 0.604(224) BC | M1
M1
A1
[3] | 3.3
1.1
3.4 | Stated or implied
Allow P(≤ 16) or P(≤ 10), e.g. 0.503 or 0.662 (M1M1A0)
Allow this M1 also from λ = 7.2 (0.187, 0.110, 0.189)
Correct answer only, awrt 0.604
(d) | Sales of CD players and integrated systems need
to be independent | B1
[1] | 1.1 | Need “independent” or “not related” clearly referred to the two
types of machine. Not just “purchases independent” or
“distributions independent”
(e) | If a customer buys a CD player they probably
won’t (or will) buy an integrated system
as well | B1
[1] | 3.5b | Any reason for non-independence of sales of CD players and
integrated sound systems
Can get B0B1 provided they are focussing on independence
Exx | α: May buy both so not independent: B0
β: Often bought together: B1
γ: Misunderstanding of context, e.g. CDs/CD players, or assuming that integrated systems don’t include CD players: can get B1
Question | Answer | Marks | AO | GuidanceThe numbers of CD players sold in a shop on three consecutive weekends were 7, 6 and 2. It may be assumed that sales of CD players occur randomly and that nobody buys more than one CD player at a time. The number of CD players sold on a randomly chosen weekend is denoted by $X$.
\begin{enumerate}[label=(\alph*)]
\item How appropriate is the Poisson distribution as a model for $X$? [2]
\end{enumerate}
Now assume that a Poisson distribution with mean 5 is an appropriate model for $X$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find
\begin{enumerate}[label=(\roman*)]
\item P$(X = 6)$, [2]
\item P$(X \geqslant 8)$. [2]
\end{enumerate}
\end{enumerate}
The number of integrated sound systems sold in a weekend at the same shop can be assumed to have the distribution Po(7.2).
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the probability that on a randomly chosen weekend the total number of CD players and integrated sound systems sold is between 10 and 15 inclusive. [3]
\item State an assumption needed for your answer to part (c) to be valid. [1]
\item Give a reason why the assumption in part (d) may not be valid in practice. [1]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2020 Q6 [11]}}