题目
Problem
A machine produces packets of cereal with weights, X kg, such that X ~ N(μ, σ^2).
A random sample of these packets is taken and a 95% confidence interval for μ is found to be (509.1572, 510.8428).
(a) Find a 90% confidence interval for μ based on the same sample of packets. You must show your working and give your confidence limits to one decimal place.
(b) Calculate the probability that at least 1 of these intervals will contain μ.
(8)
(Total for Question 3 is 8 marks)
题目中文翻译
某机器生产一批批谷物包装，重量 X kg，且 X ~ N(μ, σ^2)。
从这些包装中抽取一个随机样本，得到 μ 的 95% 置信区间为 (509.1572, 510.8428)。
(a) 根据同一样本，求 μ 的 90% 置信区间。你必须写出计算过程，并将置信区间端点保留到小数点后 1 位。
(b) 计算有至少 1 个这样的区间包含 μ 的概率。