| Exam Board | Edexcel |
|---|---|
| Module | FS1 AS (Further Statistics 1 AS) |
| Year | 2018 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sum of Poisson processes |
| Type | Multi-period repeated application |
| Difficulty | Standard +0.3 This is a straightforward Further Statistics 1 question testing standard Poisson distribution techniques: basic probability calculations, sum of independent Poissons, binomial probability with given p, and a simple hypothesis test. All parts follow textbook procedures with no novel insight required, though the multi-part structure and hypothesis test elevate it slightly above the most routine questions. |
| Spec | 5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling5.02m Poisson: mean = variance = lambda5.02n Sum of Poisson variables: is Poisson5.05c Hypothesis test: normal distribution for population mean |
| VILU SIHI NI IIIUM ION OC | VGHV SIHILNI IMAM ION OO | VJYV SIHI NI JIIYM ION OC |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(H \geq 2) = 0.1558\), awrt 0.156 | B1 | awrt 0.156 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H \sim Po(0.7)\), \(G \sim Po(3)\); \(Y = H + G \rightarrow Y \sim Po(3.7)\) | M1 | For combining distributions and use of Po(3.7) |
| \(P(Y \leq 3) = 0.494^*\) | A1cso* | \(P(Y \leq 3) = 0.494\); need to see \(P(Y \leq 3)\) or \(P(Y < 4)\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(K \sim B(6, 0.494)\) | M1 | Setting up model B(6, 0.494) |
| \(P(K \geq 5) = 1 - P(K \leq 4)\) | M1 | Using \(1 - P(K \leq 4)\) |
| \(= 1 - 0.896\ldots = 0.1039\ldots\), awrt 0.104 | A1 | awrt 0.104 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: \lambda = \text{"3.7"}\); \(H_1: \lambda > \text{"3.7"}\) | B1ft | Both hypotheses correct using \(\lambda\) or \(\mu\); ft "3.7" from part (b) |
| \(J \sim Po(7.4)\) | B1ft | Realising Po(\(2 \times\) "their 3.7") to be used |
| Method 1: \(P(J \geq 14) = 1 - P(J \leq 13) = 1 - 0.9804\ldots = 0.0195\ldots\) | M1 | Writing/using \(1-P(J \leq 13)\) or \(1-P(J < 14)\) |
| Method 2: \(P(J \geq 12) = 0.0735\ldots\); \(P(J \geq 13) = 0.0391\ldots\); CR \(J \geq 13\) | M1 | Finding CR; awrt 0.0195 or CR \(J \geq 13\) or \(J > 12\) |
| \(0.0195 < 0.05\) or \(14 \geq 13\); Reject \(H_0\). Evidence at 5% significance level that number of heaters brought in total from two supermarkets has increased | A1 | Fully correct solution and correct inference in context |
# Question 2:
## Part (a)
| $P(H \geq 2) = 0.1558$, awrt **0.156** | B1 | awrt 0.156 |
## Part (b)
| $H \sim Po(0.7)$, $G \sim Po(3)$; $Y = H + G \rightarrow Y \sim Po(3.7)$ | M1 | For combining distributions and use of Po(3.7) |
| $P(Y \leq 3) = 0.494^*$ | A1cso* | $P(Y \leq 3) = 0.494$; need to see $P(Y \leq 3)$ or $P(Y < 4)$ |
## Part (c)
| $K \sim B(6, 0.494)$ | M1 | Setting up model B(6, 0.494) |
| $P(K \geq 5) = 1 - P(K \leq 4)$ | M1 | Using $1 - P(K \leq 4)$ |
| $= 1 - 0.896\ldots = 0.1039\ldots$, awrt **0.104** | A1 | awrt 0.104 |
## Part (d)
| $H_0: \lambda = \text{"3.7"}$; $H_1: \lambda > \text{"3.7"}$ | B1ft | Both hypotheses correct using $\lambda$ or $\mu$; ft "3.7" from part (b) |
| $J \sim Po(7.4)$ | B1ft | Realising Po($2 \times$ "their 3.7") to be used |
| **Method 1:** $P(J \geq 14) = 1 - P(J \leq 13) = 1 - 0.9804\ldots = 0.0195\ldots$ | M1 | Writing/using $1-P(J \leq 13)$ or $1-P(J < 14)$ |
| **Method 2:** $P(J \geq 12) = 0.0735\ldots$; $P(J \geq 13) = 0.0391\ldots$; CR $J \geq 13$ | M1 | Finding CR; awrt 0.0195 or CR $J \geq 13$ or $J > 12$ |
| $0.0195 < 0.05$ or $14 \geq 13$; Reject $H_0$. Evidence at 5% significance level that number of heaters brought in total from two supermarkets has **increased** | A1 | Fully correct solution and correct inference in context |\begin{enumerate}
\item The number of heaters, $H$, bought during one day from Warmup supermarket can be modelled by a Poisson distribution with mean 0.7\\
(a) Calculate $\mathrm { P } ( H \geqslant 2 )$
\end{enumerate}
The number of heaters, $G$, bought during one day from Pumraw supermarket can be modelled by a Poisson distribution with mean 3, where $G$ and $H$ are independent.\\
(b) Show that the probability that a total of fewer than 4 heaters are bought from these two supermarkets in a day is 0.494 to 3 decimal places.\\
(c) Calculate the probability that a total of fewer than 4 heaters are bought from these two supermarkets on at least 5 out of 6 randomly chosen days.
December was particularly cold. Two days in December were selected at random and the total number of heaters bought from these two supermarkets was found to be 14\\
(d) Test whether or not the mean of the total number of heaters bought from these two supermarkets had increased. Use a $5 \%$ level of significance and state your hypotheses clearly.
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VILU SIHI NI IIIUM ION OC & VGHV SIHILNI IMAM ION OO & VJYV SIHI NI JIIYM ION OC \\
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\hfill \mbox{\textit{Edexcel FS1 AS 2018 Q2 [11]}}