Hint
The mean must be sufficiently large for the normal approximation. Match both Poisson mean and variance.
Worked solution
(a)
\(\operatorname{Var}(Y)=\lambda=54\)
Answer: \(54\).
Answer: \(54\)
(b)
\(\displaystyle \Phi (\frac{55.5- 54}{\sqrt{54}})= 0.580871756807087\)
Answer: \(0.580871756807087\).
Answer: \(\approx 0.580871756807087\)
(c)
\(\displaystyle e^{- 54}\sum_{j= 0}^{55} \frac{54^{j}}{j!}= 0.589327180322753\)
Answer: \(0.589327180322753\).
Answer: \(\approx 0.589327180322753\)
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: Match both Poisson mean and variance.
- 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 use λ as the standard deviation.
(b) · 2 marks
- 1 mark: Use k+0.5 and standard deviation \(\displaystyle \sqrt{\lambda }\).
- 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 silently report an exact Poisson answer when a normal approximation is requested.
(c) · 2 marks
- 1 mark: Use the finite exact Poisson cumulative sum.
- 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 use λ as the standard deviation.
Total: 6 marks.