题目
The number of water fleas, in 100 ml of pond water, has a Poisson distribution with mean 7
(a) Find the probability that a sample of 100 ml of the pond water does not contain exactly 4 water fleas.
Aja collects 5 separate samples, each of 100 ml, of the pond water.
(b) Find the probability that exactly 1 of these samples contains exactly 4 water fleas.
Using a normal approximation, the probability that more than 3 water fleas will be found in a random sample of n ml of the pond water is 0.9394 correct to 4 significant figures.
(c) (i) Show that
(ii) Hence find the value of n
After the pond has been cleaned, the number of water fleas in a 100 ml random sample of the pond water is 15
(d) Using a suitable test, at the 1% level of significance, assess whether or not there is evidence that the number of water fleas per 100 ml of the pond water has increased. State your hypotheses clearly.
(Total for Question 3 is 17 marks)
题目中文翻译
每 100 ml 池塘水中的水蚤数量服从均值为 7 的泊松分布。
(a) 求一份 100 ml 池塘水样本中不恰好含有 4 只水蚤的概率。
(b) Aja 取了 5 份独立的 100 ml 池塘水样本。求其中恰好 1 份样本恰好含有 4 只水蚤的概率。
使用正态近似，随机取 ml 的池塘水样本时，含有多于 3 只水蚤的概率为 0.9394，精确到 4 位有效数字。
(c) (i) 证明
(ii) 因而求 的值。
池塘清理后，随机取 100 ml 池塘水样本，发现有 15 只水蚤。
(d) 使用合适的检验，在 1% 显著性水平下判断是否有证据表明每 100 ml 池塘水中的水蚤数量增加了，并清楚写出假设。
（第 3 题共 17 分）