| Exam Board | Edexcel |
|---|---|
| Module | FS1 AS (Further Statistics 1 AS) |
| Year | 2019 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sum of Poisson processes |
| Type | Normal approximation to summed Poisson |
| Difficulty | Standard +0.8 This is a comprehensive multi-part Poisson question requiring: (a) basic Poisson probability, (b) scaling the parameter, (c) comparing two schemes using both Poisson and binomial approximations with clear justification, and (d) conducting a formal hypothesis test with critical region. The conceptual demand of part (c) comparing schemes and part (d) setting up a one-tailed test with correct critical region elevates this above routine exercises, though the techniques are standard for Further Statistics 1. |
| Spec | 5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Part | Answer/Working | Marks |
| (a) | [\(X\) = number of errors in 100-word piece] \(X \sim \text{Po}(1.7)\); \(P(X < 2) = P(X \leq 1) = 0.49324...\) awrt 0.493 | M1, A1 |
| (b) | [\(R\) = number of errors in the article] \(R \sim \text{Po}(4.25)\); \(P(R = 5) = 0.16482...\) awrt 0.165 | M1, A1 |
| (c) | Scheme A: Let \(A \sim \text{B}(40, e^{-1.7})\) or \(\text{B}(40, 0.18268...)\); \(P(A > 10) = 1 - P(A \leq 10) = 0.0995591...\) awrt 0.0996. Scheme B: Let \(B \sim \text{Po}(40 \times 1.7)\) or \(\text{Po}(68)\); \(P(B < 56) = P(B \leq 55) = 0.061133...\) awrt 0.061. So choose scheme A (since the probability of a bonus is greater) | M1, M1, A1, M1, A1 |
| (d) | \(H_0: \lambda = 1.7\) (or \(\mu = 8.5\)); \(H_1: \lambda < 1.7\) (or \(\mu < 8.5\)). [\(E\) = no. of errors in the piece of work] \(E \sim \text{Po}(8.5)\); \(P(E \leq 3) = 0.0301\) or \(P(E \leq 4) = 0.0744\). So critical region is \(E \leq 3\) | B1, M1, A1, A1 |
| (13 marks) |
| Part | Answer/Working | Marks | Guidance |
|------|---|---|---|
| (a) | [$X$ = number of errors in 100-word piece] $X \sim \text{Po}(1.7)$; $P(X < 2) = P(X \leq 1) = 0.49324...$ awrt **0.493** | M1, A1 | M1 for selecting the correct Poisson distribution. A1 for awrt 0.493 |
| (b) | [$R$ = number of errors in the article] $R \sim \text{Po}(4.25)$; $P(R = 5) = 0.16482...$ awrt **0.165** | M1, A1 | M1 for selecting the correct Poisson distribution. A1 for awrt 0.165 |
| (c) | **Scheme A**: Let $A \sim \text{B}(40, e^{-1.7})$ or $\text{B}(40, 0.18268...)$; $P(A > 10) = 1 - P(A \leq 10) = 0.0995591...$ awrt **0.0996**. **Scheme B**: Let $B \sim \text{Po}(40 \times 1.7)$ or $\text{Po}(68)$; $P(B < 56) = P(B \leq 55) = 0.061133...$ awrt **0.061**. So choose scheme A (since the probability of a bonus is greater) | M1, M1, A1, M1, A1 | 1st M1 for choosing a correct model for scheme A i.e. B(40, P(X = 0)), where $X \sim \text{Po}(1.7)$. Allow use of awrt 0.183 for P(X = 0)... 0.183 gives awrt 0.101 and scores M1M1A0M1A1. 2nd M1 for 1 – P(A ≤ 10). 1st A1 for awrt 0.0996 [NB use of 0.183 will give awrt 0.101 and scores M1M1A0M1A1]. 3rd M1 for selecting a correct Poisson model for scheme B i.e. Po(40 × 1.7) or better. 2nd A1 for a correct conclusion based on comparing two probs: awrt 0.1 vs 0.061 or better. So can allow 0.1 > 0.061 leading to choosing A [Probably scores M1M1A0M1A1]. |
| (d) | $H_0: \lambda = 1.7$ (or $\mu = 8.5$); $H_1: \lambda < 1.7$ (or $\mu < 8.5$). [$E$ = no. of errors in the piece of work] $E \sim \text{Po}(8.5)$; $P(E \leq 3) = 0.0301$ or $P(E \leq 4) = 0.0744$. So critical region is $E \leq 3$ | B1, M1, A1, A1 | B1 for both hypotheses in terms of $\lambda$ or $\mu$ (can be interchanged). M1 for selecting Po(8.5) (sight of or use of e.g. may be implied by 1st A1). 1st A1 for some evidence of correct use of Po(8.5) i.e. either of these probs (2dp or better). May be implied by a correct critical region. 2nd A1 for a correct critical region. Allow $E < 4$ and allow any letter for $E$. Two different regions (e.g. from 2 tail test) is 2nd A0 |
| | | **(13 marks)** | |\begin{enumerate}
\item Andreia's secretary makes random errors in his work at an average rate of 1.7 errors every 100 words.\\
(a) Find the probability that the secretary makes fewer than 2 errors in the next 100 -word piece of work.
\end{enumerate}
Andreia asks the secretary to produce a 250 -word article for a magazine.\\
(b) Find the probability that there are exactly 5 errors in this article.
Andreia offers the secretary a choice of one of two bonus schemes, based on a random sample of 40 pieces of work each consisting of 100 words.
In scheme $\mathbf { A }$ the secretary will receive the bonus if more than 10 of the 40 pieces of work contain no errors.
In scheme $\mathbf { B }$ the bonus is awarded if the total number of errors in all 40 pieces of work is fewer than 56\\
(c) Showing your calculations clearly, explain which bonus scheme you would advise the secretary to choose.
Following the bonus scheme, Andreia randomly selects a single 500 -word piece of work from the secretary to test if there is any evidence that the secretary's rate of errors has decreased.\\
(d) Stating your hypotheses clearly and using a $5 \%$ level of significance, find the critical region for this test.
\hfill \mbox{\textit{Edexcel FS1 AS 2019 Q3 [13]}}