题目
Jim farms oysters in a particular lake. He knows from past experience that 5% of young oysters do not survive to be harvested.
In a random sample of 30 young oysters, the random variable X represents the number that do not survive to be harvested.
(a) Write down a suitable model for the distribution of X.
(b) State an assumption that has been made for the model in part (a).
(c) Find the probability that
(i) exactly 24 young oysters do survive to be harvested,
(ii) at least 3 young oysters do not survive to be harvested.
A second random sample, of 200 young oysters, is taken.
The probability that at least n of these young oysters do not survive to be harvested is more than 0.8
(d) Using a suitable approximation, find the maximum value of n.
Jim believes that the level of salt in the lake water has changed and it has altered the survival rate of his oysters. He takes a random sample of 25 young oysters and places them in the lake.
When Jim harvests the oysters, he finds that 21 do survive to be harvested.
(e) Use a suitable test, at the 5% level of significance, to assess whether or not there is evidence that the proportion of oysters not surviving to be harvested is more than 5%. State your hypotheses clearly.
(Total for Question 1 is 14 marks)
题目中文翻译
Jim 在某个湖里养殖牡蛎。他根据以往经验知道，年轻牡蛎中有 5% 无法存活到收获。
在一个随机抽取的 30 只年轻牡蛎样本中，随机变量 表示无法存活到收获的数量。
(a) 写下 分布的合适模型。
(b) 写出 (a) 中模型所作的一个假设。
(c) 求以下概率：
(i) 恰好 24 只年轻牡蛎能够存活到收获；
(ii) 至少 3 只年轻牡蛎无法存活到收获。
再随机抽取 200 只年轻牡蛎。
至少有 只无法存活到收获的概率大于 0.8。
(d) 使用合适的近似，求 的最大值。
Jim 认为湖水盐度发生了变化，从而改变了牡蛎的存活率。他随机取 25 只年轻牡蛎放入湖中。
当他收获这些牡蛎时，发现有 21 只能够存活。
(e) 使用合适的检验，在 5% 显著性水平下判断是否有证据表明无法存活到收获的牡蛎比例大于 5%，并清楚写出假设。
（第 1 题共 14 分）