| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2017 |
| Session | October |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Finding minimum stock level for demand |
| Difficulty | Standard +0.3 This is a straightforward application of Poisson distribution with standard bookwork techniques. Part (a) requires P(X≥10) using tables, (b) is simple multiplication, (c) involves trial-and-error with tables to find appropriate stock level, and (d) uses normal approximation for hypothesis testing. All parts follow standard S2 procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.05e Hypothesis test for normal mean: known variance5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(P(B\geq 10)=1-P(B\leq 9)=1-0.7166=0.2834\), awrt 0.283 | M1 A1 | M1 for writing or using \(1-P(B\leq 9)\); A1 awrt 0.283 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| Expected number of weeks \(=0.2834\times 50=14.2\), accept 14 | M1 A1 | M1 for their (a) \(\times 50\); A1 awrt 14 (isw if 15 follows from awrt 14.2) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(P(B\geq n)<0.04\) where \(B\sim Po(8)\) | ||
| \(P(B>12)=0.0638\) | M1 | M1 for any of these three lines |
| \(P(B>13)=0.0342\) | ||
| \(\therefore 13\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(H_0: \lambda=8(80)\); \(H_1: \lambda>8(80)\) | B1 | Both hypotheses; allow \(\lambda\) or \(\mu\), 8 or 80 |
| \(Y\sim N(80,80)\) | M1M1 | M1 using Normal with mean 80; M1 using Normal with mean = variance |
| \(P(Y\geq 95)=P\!\left(Z>\frac{94.5-80}{\sqrt{80}}\right)\) | M1 dM1 | M1 standardising with \(\pm\frac{(95\ \text{or}\ 95.5\ \text{or}\ 94.5)-\text{mean}}{\text{sd}}\); dM1 continuity correction \(95\pm0.5\) |
| \(=P(Z>1.62)=0.0526\) | A1 | Correct standardisation and tail; award for \(Z>\frac{94.5-80}{\sqrt{80}}\) or \(Z>\) awrt 1.62 |
| Do not reject \(H_0\) | M1 | Correct statement; do not allow contradictory non-contextual statements |
| There is no evidence that reducing the price of a *Birdscope* has increased demand | A1cso | Must contain word demand; ft their probability/CR and \(H_1\) |
## Question 3:
### Part (a):
| Working | Marks | Guidance |
|---------|-------|----------|
| $P(B\geq 10)=1-P(B\leq 9)=1-0.7166=0.2834$, awrt **0.283** | M1 A1 | M1 for writing or using $1-P(B\leq 9)$; A1 awrt 0.283 |
### Part (b):
| Working | Marks | Guidance |
|---------|-------|----------|
| Expected number of weeks $=0.2834\times 50=14.2$, accept 14 | M1 A1 | M1 for their (a) $\times 50$; A1 awrt 14 (isw if 15 follows from awrt 14.2) |
### Part (c):
| Working | Marks | Guidance |
|---------|-------|----------|
| $P(B\geq n)<0.04$ where $B\sim Po(8)$ | | |
| $P(B>12)=0.0638$ | M1 | M1 for any of these three lines |
| $P(B>13)=0.0342$ | | |
| $\therefore 13$ | A1 | |
### Part (d):
| Working | Marks | Guidance |
|---------|-------|----------|
| $H_0: \lambda=8(80)$; $H_1: \lambda>8(80)$ | B1 | Both hypotheses; allow $\lambda$ or $\mu$, 8 or 80 |
| $Y\sim N(80,80)$ | M1M1 | M1 using Normal with mean 80; M1 using Normal with mean = variance |
| $P(Y\geq 95)=P\!\left(Z>\frac{94.5-80}{\sqrt{80}}\right)$ | M1 dM1 | M1 standardising with $\pm\frac{(95\ \text{or}\ 95.5\ \text{or}\ 94.5)-\text{mean}}{\text{sd}}$; dM1 continuity correction $95\pm0.5$ |
| $=P(Z>1.62)=0.0526$ | A1 | Correct standardisation and tail; award for $Z>\frac{94.5-80}{\sqrt{80}}$ or $Z>$ awrt 1.62 |
| Do not reject $H_0$ | M1 | Correct statement; do not allow contradictory non-contextual statements |
| There is no evidence that reducing the price of a *Birdscope* has increased **demand** | A1cso | Must contain word **demand**; ft their probability/CR and $H_1$ |
---3. In a shop, the weekly demand for Birdscope cameras is modelled by a Poisson distribution with mean 8
The shop has 9 Birdscope cameras in stock at the start of each week.
A week is selected at random.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that the demand for Birdscope cameras cannot be met in this particular week.
In a year, there are 50 weeks in which Birdscope cameras can be sold.
\item Find the expected number of weeks in the year that the shop will not be able to meet the demand for Birdscope cameras.
\item Find the number of Birdscope cameras the shop should stock at the beginning of each week if it wants the estimated number of weeks in the year in which demand cannot be met to be less than 2
The shop increases its stock and reduces the price of Birdscope cameras in order to increase demand. A random sample of 10 weeks is selected and it is found that, in the 10 weeks, a total of 95 Birdscope cameras were sold.
Given that there were no weeks when the shop was unable to meet the demand for Birdscope cameras,
\item use a suitable approximation to test whether or not the demand for Birdscope cameras has increased following the price reduction. You should state your hypotheses clearly and use a 5\% level of significance.
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2017 Q3 [14]}}