Standard +0.3 This is a straightforward one-tailed Poisson hypothesis test with clearly stated parameters. Students must set up H₀: λ=8 vs H₁: λ<8, calculate P(X≤3) using tables, and compare to 5%. While it requires understanding of hypothesis testing framework, the calculation is direct with no conceptual tricks, making it slightly easier than average A-level material.
Richard regularly travels to work on a ferry. Over a long period of time, Richard has found that the ferry is late on average 2 times every week. The company buys a new ferry to improve the service. In the 4-week period after the new ferry is launched, Richard finds the ferry is late 3 times and claims the service has improved. Assuming that the number of times the ferry is late has a Poisson distribution, test Richard's claim at the 5\% level of significance. State your hypotheses clearly.
[6]
\(H_0: \lambda = 8\) or \(\mu = 2\); \(H_1: \lambda < 8\) or \(\mu < 2\)
B1, B1
Under \(H_0\), \(X \sim \text{Po}(8)\)
M1
\(P(X \leq 3) = 0.0424\)
A1
CR \(X \leq 3\)
M1A1ft
\(0.0424 < 0.05\), Reject \(H_0\). Richard's claim is supported.
Notes:
- B1 for \(H_0\) correct. Must use \(\lambda\) or \(\mu\) and 8 or 2.
- B1 for \(H_1\) correct. Must use \(\lambda\) or \(\mu\) and 8 or 2.
- M1 for writing or using Po(8) – may be implied by correct CR.
- A1 awrt 0.0424 or CR \(X \leq 3\).
- M1 need \(p<0.5\) and: correct statement using their Probability and 0.05 if one tail test or correct statement using their Probability and 0.025 if two tail test (condone a comparison with 0.05 instead of 0.025 for a two tail test). Do not allow non-contextual conflicting statements eg "significant" and "accept \(H_0\)".
- A1 ft correct contextual statement followed through from "their prob".
- Either a comment on whether Richard's claim was correct or on whether the service has improved.
- NB if a correct contextual statement only is given for their probability then award M1 A1.
- \(p>0.5\): They may compare with 0.95 (one tail method) or 0.975 (two tail method). Probability is 0.9576.
[6 marks total]
$H_0: \lambda = 8$ or $\mu = 2$; $H_1: \lambda < 8$ or $\mu < 2$ | B1, B1 |
Under $H_0$, $X \sim \text{Po}(8)$ | M1 |
$P(X \leq 3) = 0.0424$ | A1 |
CR $X \leq 3$ | M1A1ft |
$0.0424 < 0.05$, Reject $H_0$. Richard's claim is supported. |
**Notes:**
- B1 for $H_0$ correct. Must use $\lambda$ or $\mu$ and 8 or 2.
- B1 for $H_1$ correct. Must use $\lambda$ or $\mu$ and 8 or 2.
- M1 for writing or using Po(8) – may be implied by correct CR.
- A1 awrt 0.0424 or CR $X \leq 3$.
- M1 need $p<0.5$ and: correct statement using their Probability and 0.05 if one tail test or correct statement using their Probability and 0.025 if two tail test (condone a comparison with 0.05 instead of 0.025 for a two tail test). Do not allow non-contextual conflicting statements eg "significant" and "accept $H_0$".
- A1 ft correct contextual statement followed through from "their prob".
- Either a comment on whether Richard's claim was correct or on whether the service has improved.
- NB if a correct contextual statement only is given for their probability then award M1 A1.
- $p>0.5$: They may compare with 0.95 (one tail method) or 0.975 (two tail method). Probability is 0.9576.
**[6 marks total]**
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Richard regularly travels to work on a ferry. Over a long period of time, Richard has found that the ferry is late on average 2 times every week. The company buys a new ferry to improve the service. In the 4-week period after the new ferry is launched, Richard finds the ferry is late 3 times and claims the service has improved. Assuming that the number of times the ferry is late has a Poisson distribution, test Richard's claim at the 5\% level of significance. State your hypotheses clearly.
[6]
\hfill \mbox{\textit{Edexcel S2 2011 Q4 [6]}}