| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2011 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | One-tailed test (increase or decrease) |
| Difficulty | Standard +0.3 This is a standard Poisson hypothesis test with straightforward critical region calculation and Type I error identification. Part (iv) requires applying the normal approximation to Poisson, which is routine for S2. All parts follow textbook procedures with no novel problem-solving required, making it slightly easier than average A-level difficulty. |
| Spec | 2.04d Normal approximation to binomial5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| (i) mean = 6.3 | B1 | B1 for 6.3 |
| \(P(X \leq 1) = e^{-6.3}(1 + 6.3) = 0.0134\) | M1 | Allow incorrect \(\lambda\) in both probs |
| \(P(X \leq 2) = e^{-6.3}(1 + 6.3 + \frac{6.3^2}{2}) = 0.0498\) | M1A1 | |
| CR is \(X \leq 1\) | A1 | A1 for both values |
| [5] | ||
| (ii) P(Type I error) = P(\(X \leq 1\)) = 0.0134 | B1 | |
| [1] | ||
| (iii) \(H_0: \lambda = 6.3\) \(H_1: \lambda < 6.3\) | B1 | Can be scored in (i). Accept \(\lambda = 2.1\)(per month) |
| 3 not in CR | M1 | or P(\(X \leq 3\)) = 0.126 > 0.02 |
| No evidence mean no. of injuries has decreased | A1 | Correct conclusion |
| [3] | ||
| (iv) \(N(25.2, 25.2)\) | B2 | B1 for N & \(\mu = 25.2\), B1 for \(\sigma^2 = 25.2\) |
| May be implied | ||
| \(\frac{19.5 - 25.2}{\sqrt{25.2}} (= -1.135)\) | M1 | Allow with wrong or no cc or no \(\sqrt{}\) |
| \(\Phi("-1.135") = 1 - \Phi("1.135")\) | M1 | Correct area |
| \(= 0.128\) (3 sfs) | A1 | |
| [5] |
**(i)** mean = 6.3 | B1 | B1 for 6.3
$P(X \leq 1) = e^{-6.3}(1 + 6.3) = 0.0134$ | M1 | Allow incorrect $\lambda$ in both probs
$P(X \leq 2) = e^{-6.3}(1 + 6.3 + \frac{6.3^2}{2}) = 0.0498$ | M1A1 |
CR is $X \leq 1$ | A1 | A1 for both values
| [5] |
**(ii)** P(Type I error) = P($X \leq 1$) = 0.0134 | B1 |
| [1] |
**(iii)** $H_0: \lambda = 6.3$ $H_1: \lambda < 6.3$ | B1 | Can be scored in (i). Accept $\lambda = 2.1$(per month)
3 not in CR | M1 | or P($X \leq 3$) = 0.126 > 0.02
No evidence mean no. of injuries has decreased | A1 | Correct conclusion
| [3] |
**(iv)** $N(25.2, 25.2)$ | B2 | B1 for N & $\mu = 25.2$, B1 for $\sigma^2 = 25.2$
| | May be implied
$\frac{19.5 - 25.2}{\sqrt{25.2}} (= -1.135)$ | M1 | Allow with wrong or no cc or no $\sqrt{}$
$\Phi("-1.135") = 1 - \Phi("1.135")$ | M1 | Correct area
$= 0.128$ (3 sfs) | A1 |
| [5] |6 The number of injuries per month at a certain factory has a Poisson distribution. In the past the mean was 2.1 injuries per month. New safety procedures are put in place and the management wishes to use the next 3 months to test, at the $2 \%$ significance level, whether there are now fewer injuries than before, on average.\\
(i) Find the critical region for the test.\\
(ii) Find the probability of a Type I error.\\
(iii) During the next 3 months there are a total of 3 injuries. Carry out the test.\\
(iv) Assuming that the mean remains 2.1 , calculate an estimate of the probability that there will be fewer than 20 injuries during the next 12 months.
\hfill \mbox{\textit{CAIE S2 2011 Q6 [14]}}