题目
Rowan believes that 35% of type A vacuum tubes shatter when exposed to alternating high and low temperatures.
Rowan takes a random sample of 15 of these type A vacuum tubes and uses a two-tailed test, at the 5% level of significance, to test his belief.
(a) Give two assumptions, in context, that Rowan needs to make for a binomial distribution to be a suitable model for the number of these type A vacuum tubes that shatter when exposed to alternating high and low temperatures.
(b) Using a binomial distribution, find the critical region for the test.
You should state the probability of rejection in each tail, which should be as close as possible to 0.025
(c) Find the actual level of significance of the test based on your critical region from part (b)
Rowan records that in the latest batch of 15 type A vacuum tubes exposed to alternating high and low temperatures, 4 of them shattered.
(d) With reference to part (b), comment on Rowan’s belief. Give a reason for your answer.
Rowan changes to type B vacuum tubes. He takes a random sample of 40 type B vacuum tubes and finds that 8 of them shatter when exposed to alternating high and low temperatures.
(e) Test, at the 5% level of significance, whether or not there is evidence that the proportion of type B vacuum tubes that shatter when exposed to alternating high and low temperatures is lower than 35%.
You should state your hypotheses clearly.
(Total for Question 3 is 12 marks)
题目中文翻译
罗文认为 35% 的 A 型真空管在交替高低温环境下会破裂。
罗文随机抽取 15 个 A 型真空管作为样本，并使用双尾检验在 5% 的显著性水平下检验他的假设。
(a) 给出罗文需要做的两个假设（在上下文中），以便二项分布能成为在交替高低温环境下破裂的 A 型真空管数量的合适模型。
(b) 使用二项分布，求该检验的临界区域。
你应说明每个尾部的拒绝概率，该概率应尽可能接近 0.025
(c) 根据 (b) 中的临界区域，求该检验的实际显著性水平
罗文记录在最新一批 15 个暴露于交替高低温环境下的 A 型真空管中，有 4 个破裂。
(d) 参考 (b)，评论罗文的假设。给出你答案的理由。
罗文改为 B 型真空管。他随机抽取 40 个 B 型真空管作为样本，发现其中 8 个在交替高低温环境下破裂。
(e) 在 5% 的显著性水平下，检验是否有证据表明在交替高低温环境下破裂的 B 型真空管比例低于 35%。
你应清楚地陈述你的假设。
（第 3 题共 12 分）