题目
In a peat bog, Common Spotted-orchids occur at a mean rate of 4.5 per m^2
(a) Give an assumption, not already stated, that is required for the number of Common Spotted-orchids per m^2 of the peat bog to follow a Poisson distribution.
Given that the number of Common Spotted-orchids in 1 m^2 of the peat bog can be modelled by a Poisson distribution,
(b) find the probability that in a randomly selected 1 m^2 of the peat bog
(i) there are exactly 6 Common Spotted-orchids,
(ii) there are fewer than 10 but more than 4 Common Spotted-orchids.
Juan believes that by introducing a new management scheme the number of Common Spotted-orchids in the peat bog will increase. After three years under the new management scheme, a randomly selected 2 m^2 of the peat bog contains 11 Common Spotted-orchids.
(c) Using a 5% significance level assess Juan’s belief. State your hypotheses clearly.
Assuming that in the peat bog, Common Spotted-orchids still occur at a mean rate of 4.5 per m^2
(d) use a normal approximation to find the probability that in a randomly selected 20 m^2 of the peat bog there are fewer than 70 Common Spotted-orchids.
Following a period of dry weather, the probability that there are fewer than 70 Common Spotted-orchids in a randomly selected 20 m^2 of the peat bog is 0.012
A random sample of 200 non-overlapping 20 m^2 areas of the peat bog is taken.
(e) Using a suitable approximation, calculate the probability that at most 1 of these areas contains fewer than 70 Common Spotted-orchids.
(Total for Question 4 is 16 marks)
题目中文翻译
在一个泥炭沼泽中，斑点兰的平均出现率为每平方米 4.5 株。
(a) 写出一条未在题干中说明、但使每平方米斑点兰数量服从泊松分布所需的假设。
已知该泥炭沼泽中 1 m^2 的斑点兰数量可用泊松分布建模。
(b) 求随机选取的 1 m^2 沼泽中：
(i) 恰好有 6 株斑点兰的概率；
(ii) 多于 4 株但少于 10 株斑点兰的概率。
Juan 认为通过引入新的管理方案，沼泽中的斑点兰数量会增加。经过三年新的管理方案后，随机选取的 2 m^2 沼泽中有 11 株斑点兰。
(c) 在 5% 显著性水平下检验 Juan 的看法，并清楚写出假设。
假设泥炭沼泽中的斑点兰仍以每平方米 4.5 株的平均速率出现。
(d) 使用正态近似，求随机选取的 20 m^2 沼泽中斑点兰少于 70 株的概率。
在一段干旱天气后，随机选取的 20 m^2 沼泽中斑点兰少于 70 株的概率为 0.012。
随机抽取 200 个互不重叠的 20 m^2 区域。
(e) 使用合适的近似，求其中至多 1 个区域含有少于 70 株斑点兰的概率。
（第 4 题共 16 分）