Hint
Use identical observation durations when comparing null and alternative means. Use the upper tail under the null mean.
Worked solution
(a)
\(P_{\lambda=2.25}(X\ge 6)=0.0273653540633016\)
Answer: \(0.0273653540633016\).
Answer: \(\approx 0.0273653540633016\)
(b)
\(P_{\lambda=4.25}(X\ge 6)=0.255060947622092\)
Answer: \(0.255060947622092\).
Answer: \(\approx 0.255060947622092\)
(c)
\(1-\operatorname{power}=0.744939052377908\)
Answer: \(0.744939052377908\).
Answer: \(\approx 0.744939052377908\)
(d)
The p-value is \(0.0273653540633016\). Since it is at most 0.05, reject \(\displaystyle H_{0}\): there is evidence that the Poisson rate exceeds its null value. This does not give the probability that \(\displaystyle H_{0}\) is true.
Answer:
The p-value is \(0.0273653540633016\). Since it is at most 0.05, reject \(\displaystyle H_{0}\): there is evidence that the Poisson rate exceeds its null value. This does not give the probability that \(\displaystyle H_{0}\) is true.
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.
(a) · 2 marks
- 1 mark: Use the upper tail under the null mean.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: A high count is tested with an upper, not lower, tail.
(b) · 2 marks
- 1 mark: Keep the region fixed and change the mean to the alternative.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not report the target level as the actual size without calculation.
(c) · 2 marks
- 1 mark: Use the alternative probability of the noncritical region.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: A high count is tested with an upper, not lower, tail.
(d) · 1 marks
- 1 mark: Compare the relevant p-value with 0.05 and state the contextual conclusion without claiming the null is proved.
Alternative methods: A comparison with the correct critical value and the same valid contextual conclusion is acceptable.
Common error: A p-value is not the probability that the null hypothesis is true.
Total: 7 marks.