| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | State meaning of Type I error |
| Difficulty | Moderate -0.5 This is a straightforward hypothesis testing question requiring standard definitions and Poisson probability calculations. Part (i) asks for a textbook definition in context, part (ii) is a direct calculation of P(X ≤ 1|λ=4.8), and part (iii) is P(X ≥ 2|λ=0.9). All are routine applications of learned procedures with no problem-solving or novel insight required, making it slightly easier than average. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Mean is 4.8 but ≤ 1 breakdown | B1 [1] | Accept reduction when none has occurred |
| (ii) \(e^{-4.8}(1 + 4.8) = 0.0477\) | M1 A1 [2] | Poisson attempt at P(0) + P(1) |
| (iii) \(P(X > 1) = 1 - e^{-0.9}(1 + 0.9) = 0.228\) (3 sfs) | M1 M1 A1 [3] | Attempt correct probability for Type II error; Allow any \(\lambda\) except 4.8; \(1 - \text{P}(0)(\text{P}(1))\) using Poisson; As final answer |
**(i)** Mean is 4.8 but ≤ 1 breakdown | B1 [1] | Accept reduction when none has occurred
**(ii)** $e^{-4.8}(1 + 4.8) = 0.0477$ | M1 A1 [2] | Poisson attempt at P(0) + P(1)
**(iii)** $P(X > 1) = 1 - e^{-0.9}(1 + 0.9) = 0.228$ (3 sfs) | M1 M1 A1 [3] | Attempt correct probability for Type II error; Allow any $\lambda$ except 4.8; $1 - \text{P}(0)(\text{P}(1))$ using Poisson; As final answer4 At a power plant, the number of breakdowns per year has a Poisson distribution. In the past the mean number of breakdowns per year has been 4.8. Following some repairs, the management carry out a hypothesis test at the 5\% significance level to determine whether this mean has decreased. If there is at most 1 breakdown in the following year, they will conclude that the mean has decreased.\\
(i) State what is meant by a Type I error in this context.\\
(ii) Find the probability of a Type I error.\\
(iii) Find the probability of a Type II error if the mean is now 0.9 breakdowns per year.
\hfill \mbox{\textit{CAIE S2 2010 Q4 [6]}}