Standard +0.3 This is a straightforward one-tailed Poisson hypothesis test with clear parameters. Students must set up H₀: λ = 0.32 vs H₁: λ > 0.32, calculate P(X ≥ 6) where X ~ Po(3.2), and compare to 1%. The calculation is routine using tables or calculator, requiring only standard application of the Poisson test procedure with no conceptual complications or multi-step reasoning.
4 The number of floods in a certain river plain is known to have a Poisson distribution. It is known that up until 10 years ago the mean number of floods per year was 0.32 . During the last 10 years there were 6 floods. Test at the \(1 \%\) significance level whether there is evidence of an increase in the mean number of floods per year.
Both correct: B2. One error e.g. wrong/no/different symbols, or two-tail: B1. But \(x\), \(\bar{x}\), \(r\), \(t\) etc: B0. E(\(X\)), words: B1
\(R \sim \text{Po}(3.2)\)
M1
Stated or implied, e.g. N(3.2, 3.2)
\(\alpha\): \(P(R \geq 6) = 0.1054 > 0.01\)
A1
\([0.105, 0.106]\) before rounding. Explicit comparison with 0.01. \(P(=6)\) or \((\leq 6)\) or \(> 6\) or normal: no more marks, maximum B2M1
\(\beta\): \(CR \geq 9\) and \(6 < 9\), with probability 0.0057
A1 A1
\(CR \geq 9\) stated; allow \(CV = 9\) if comparison ft. 0.0057 or 0.9943 seen, and 6 compared
Do not reject \(H_0\). Insufficient evidence of an increase in the number of floods.
M1 A1 ft
Consistent first conclusion. Conclusion mentions "floods", "evidence". *Not* "evidence of no increase". Needs correct method and like-with-like comparison, but 0.01 needn't be explicit
Total: 7
# Question 4:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: \lambda = 3.2$ (or 0.32) [Allow $\mu$]; $H_1: \lambda > 3.2$ (or 0.32) [Allow $\mu$] | B2 | Both correct: B2. One error e.g. wrong/no/different symbols, or two-tail: B1. But $x$, $\bar{x}$, $r$, $t$ etc: B0. E($X$), words: B1 |
| $R \sim \text{Po}(3.2)$ | M1 | Stated or implied, e.g. N(3.2, 3.2) |
| $\alpha$: $P(R \geq 6) = 0.1054 > 0.01$ | A1 | $[0.105, 0.106]$ before rounding. Explicit comparison with 0.01. $P(=6)$ or $(\leq 6)$ or $> 6$ or normal: no more marks, maximum B2M1 |
| $\beta$: $CR \geq 9$ and $6 < 9$, with probability 0.0057 | A1 A1 | $CR \geq 9$ stated; allow $CV = 9$ if comparison ft. 0.0057 or 0.9943 seen, and 6 compared |
| Do not reject $H_0$. Insufficient evidence of an increase in the number of floods. | M1 A1 ft | Consistent first conclusion. Conclusion mentions "floods", "evidence". *Not* "evidence of no increase". Needs correct method and like-with-like comparison, but 0.01 needn't be explicit |
| | **Total: 7** | |
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4 The number of floods in a certain river plain is known to have a Poisson distribution. It is known that up until 10 years ago the mean number of floods per year was 0.32 . During the last 10 years there were 6 floods. Test at the $1 \%$ significance level whether there is evidence of an increase in the mean number of floods per year.
\hfill \mbox{\textit{OCR S2 2013 Q4 [7]}}