| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | One-tailed test (increase or decrease) |
| Difficulty | Standard +0.8 This is a comprehensive Poisson hypothesis testing question requiring understanding of Type I/II errors, critical region determination at 10% significance, and probability calculations under both null and alternative hypotheses. While the individual calculations are standard for Further Maths Statistics, the multi-part nature, conceptual understanding of error types in context, and Type II error calculation (requiring work under a specific alternative hypothesis) elevate this above routine exercises. |
| 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) Type I error is made when we say the number of white blood cells has decreased when it hasn't. | B1 | Correct and relating to question |
| \(P(0) = e^{-5.2} = 0.005516\), \(P(1) = e^{-5.2}(5.2) = 0.02868\), \(\Sigma < 0.10\), \(P(2) = e^{-5.2}(5.2^2/2) = 0.07458\), \(\Sigma \geq 0.10\) | M1, M1* | Evaluating at least 2 of \(P(X = 0, 1, 2)\); Comparing their \(\Sigma \geq 3\) probs with 10% (must be \(\Sigma\) probs) |
| \(P(\text{Type I error}) = 0.0342\) | A1 dep [4] | Correct answer, dep on previous M |
| (ii) \(H_0: \lambda = 5.2\), \(H_1: \lambda < 5.2\) | B1 | Both hypotheses correct |
| \(P(0+1+2) = 0.1087 > 10\%\), 2 not in C Region. Accept \(H_0\). Not enough evidence to say the number of blood cells has decreased. | M1, A1 [3] | Stating 2 is not in the critical region from above, or evaluating P(0, 1, 2) and comparing with 10% again; Correct conclusion no contradictions |
| (iii) \(P(\text{Type II error}) = 1 - P(0, 1) = 1 - e^{-4.1}(1 + 4.1)\) | B1, M1 | Identifying correct area; (indep) Some form of (Poisson) expression with mean 4.1 |
| \(= 0.915\) | A1 [3] | Correct answer |
**(i)** Type I error is made when we say the number of white blood cells has decreased when it hasn't. | B1 | Correct and relating to question
$P(0) = e^{-5.2} = 0.005516$, $P(1) = e^{-5.2}(5.2) = 0.02868$, $\Sigma < 0.10$, $P(2) = e^{-5.2}(5.2^2/2) = 0.07458$, $\Sigma \geq 0.10$ | M1, M1* | Evaluating at least 2 of $P(X = 0, 1, 2)$; Comparing their $\Sigma \geq 3$ probs with 10% (must be $\Sigma$ probs)
$P(\text{Type I error}) = 0.0342$ | A1 dep [4] | Correct answer, dep on previous M
**(ii)** $H_0: \lambda = 5.2$, $H_1: \lambda < 5.2$ | B1 | Both hypotheses correct
$P(0+1+2) = 0.1087 > 10\%$, 2 not in C Region. Accept $H_0$. Not enough evidence to say the number of blood cells has decreased. | M1, A1 [3] | Stating 2 is not in the critical region from above, or evaluating P(0, 1, 2) and comparing with 10% again; Correct conclusion no contradictions
**(iii)** $P(\text{Type II error}) = 1 - P(0, 1) = 1 - e^{-4.1}(1 + 4.1)$ | B1, M1 | Identifying correct area; (indep) Some form of (Poisson) expression with mean 4.1
$= 0.915$ | A1 [3] | Correct answer7 A hospital patient's white blood cell count has a Poisson distribution. Before undergoing treatment the patient had a mean white blood cell count of 5.2. After the treatment a random measurement of the patient's white blood cell count is made, and is used to test at the $10 \%$ significance level whether the mean white blood cell count has decreased.\\
(i) State what is meant by a Type I error in the context of the question, and find the probability that the test results in a Type I error.\\
(ii) Given that the measured value of the white blood cell count after the treatment is 2 , carry out the test.\\
(iii) Find the probability of a Type II error if the mean white blood cell count after the treatment is actually 4.1.
\hfill \mbox{\textit{CAIE S2 2010 Q7 [10]}}