题目
The volume of water in a bottle has a normal distribution with unknown mean, μ millilitres, and known standard deviation, σ millilitres.
A random sample of 150 of the bottles of water gave a 95% confidence interval for μ of (327.84, 329.76)
(a) Using the confidence interval given, test whether or not μ = 328 State your hypotheses clearly and write down the significance level you have used.
A second random sample, of 200 of these bottles of water, had a mean volume of 328 millilitres.
(b) Calculate a 98% confidence interval for μ based on this second sample. You must show all steps in your working. (Solutions relying entirely on calculator technology are not acceptable.)
Using five different random samples of 200 of these bottles of water, five 98% confidence intervals for μ are to be found.
(c) Calculate the probability that more than 3 of these intervals will contain μ
题目中文翻译
瓶中水的容量服从均值未知、标准差已知的正态分布，均值为 μ 毫升，标准差为 σ 毫升。
对 150 个水瓶的随机样本给出的 μ 的 95% 置信区间为 (327.84, 329.76)
(a) 利用给出的置信区间，检验 μ = 328 是否成立。 请清楚写出你的原假设，并写明所用的显著性水平。
第二个随机样本包含这 200 个水瓶，样本平均容量为 328 毫升。
(b) 基于这个第二个样本，求 μ 的 98% 置信区间。 你必须写出所有步骤。 （完全依赖计算器技术的解法不予接受。）
从这 200 个水瓶中抽取五个不同的随机样本， 将得到五个 μ 的 98% 置信区间。
(c) 求这五个区间中多于 3 个包含 μ 的概率。