| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2017 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sum of Poisson processes |
| Type | Hypothesis test on Poisson rate |
| Difficulty | Standard +0.3 This is a straightforward S2 Poisson question requiring standard applications: calculating P(X=0), using independence for multiple periods, binomial-Poisson combination, and a one-tailed hypothesis test. All parts follow textbook methods with no novel insight required, though part (c) requires recognizing the binomial structure and part (e) involves cumulative probability calculation, making it slightly above average routine difficulty. |
| 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 = lambda5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(X \sim \text{Po}(\frac{1}{4})\), \(P(X = 0) = e^{-1/4} = 0.778800\ldots\) | B1, B1 | 1st B1 for writing or using \(\text{Po}(\frac{1}{4})\). May be implied by correct answer or by awrt 0.78 |
| (b) \([(P(X \geq 1))^3] = (1 - '0.7788')^3 = 0.010823\ldots\) | M1, A1 | M1 for \((1 - 0.779)^3\) or \((1 - \text{'their (a)'})^3\) |
| (c) \(Y \sim B(7, 0.7788\ldots)\), \(P(Y = 5) = 7C5(0.7788)^5(1 - 0.7788)^2 = 0.294386\ldots\) awrt 0.294 | B1ft, M1, A1 | B1ft for writing or using \(B(7, \text{'their a'})\). May be implied by M1 scored. M1 for correct binomial expression for \(P(Y=5)\) (ft their value of \(p\)). Allow \(\binom{7}{2}\) etc or 21. May be implied by correct answer but if \(p \neq 0.779\) or better we must see expression |
| (d) \(H_0: \mu = 8\) or \(\lambda = 0.25\); \(H_1: \mu < 8\) or \(\lambda < 0.25\) | B1 | B1 for both hypotheses correct. Must use \(\lambda\) or \(\mu\) for either 8 or 0.25. [Use of \(<\) is B0] |
| (e) \(W \sim \text{Po}(8)\), \(P(W \leq 3) = 0.0424 (< 0.05)\), \(P(W \leq 4) = 0.0996 (> 0.05)\). Largest possible value of \(f\) is 3 | B1, M1, A1 | B1 for writing \(\text{Po}(8)\) can be awarded if seen in (d) (may be implied e.g. by scoring M1). M1 for using \(\text{Po}(8)\) to find a lower-tail critical region. Need to see one of the given probability statements or implied by \(\text{Po}(8)\) and \(f = 3\) seen. A1 for \([f] = 3\) but allow \(f \leq 3\). Correct answer only scores 3/3 |
**(a)** $X \sim \text{Po}(\frac{1}{4})$, $P(X = 0) = e^{-1/4} = 0.778800\ldots$ | B1, B1 | 1st B1 for writing or using $\text{Po}(\frac{1}{4})$. May be implied by correct answer or by awrt 0.78
**(b)** $[(P(X \geq 1))^3] = (1 - '0.7788')^3 = 0.010823\ldots$ | M1, A1 | M1 for $(1 - 0.779)^3$ or $(1 - \text{'their (a)'})^3$
**(c)** $Y \sim B(7, 0.7788\ldots)$, $P(Y = 5) = 7C5(0.7788)^5(1 - 0.7788)^2 = 0.294386\ldots$ awrt **0.294** | B1ft, M1, A1 | B1ft for writing or using $B(7, \text{'their a'})$. May be implied by M1 scored. M1 for correct binomial expression for $P(Y=5)$ (ft their value of $p$). Allow $\binom{7}{2}$ etc or 21. May be implied by correct answer but if $p \neq 0.779$ or better we must see expression
**(d)** $H_0: \mu = 8$ or $\lambda = 0.25$; $H_1: \mu < 8$ or $\lambda < 0.25$ | B1 | B1 for both hypotheses correct. Must use $\lambda$ or $\mu$ for either 8 or 0.25. [Use of $<$ is B0]
**(e)** $W \sim \text{Po}(8)$, $P(W \leq 3) = 0.0424 (< 0.05)$, $P(W \leq 4) = 0.0996 (> 0.05)$. Largest possible value of $f$ is 3 | B1, M1, A1 | B1 for writing $\text{Po}(8)$ can be awarded if seen in (d) (may be implied e.g. by scoring M1). M1 for using $\text{Po}(8)$ to find a lower-tail critical region. Need to see one of the given probability statements or implied by $\text{Po}(8)$ and $f = 3$ seen. A1 for $[f] = 3$ but allow $f \leq 3$. Correct answer only scores 3/3
**Total: 11**
---\begin{enumerate}
\item At a particular junction on a train line, signal failures are known to occur randomly at a rate of 1 every 4 days.\\
(a) Find the probability that there are no signal failures on a randomly selected day.\\
(b) Find the probability that there is at least 1 signal failure on each of the next 3 days.\\
(c) Find the probability that in a randomly selected 7 -day week, there are exactly 5 days with no signal failures.
\end{enumerate}
Repair works are carried out on the line. After these repair works, the number, $f$, of signal failures in a 32-day period is recorded.
A test is carried out, at the $5 \%$ level of significance, to determine whether or not there has been a decrease in the rate of signal failures following the repair works.\\
(d) State the hypotheses for this test.\\
(e) Find the largest value of $f$ for which the null hypothesis should be rejected.
\hfill \mbox{\textit{Edexcel S2 2017 Q1 [11]}}