题目
A manufacturer produces plates. The proportion of plates that are flawed is 45%, with flawed plates occurring independently.
A random sample of 10 of these plates is selected.
(a) Find the probability that the sample contains
(i) fewer than 2 flawed plates,
(ii) at least 6 flawed plates.
George believes that the proportion of flawed plates is not 45%. To assess his belief George takes a random sample of 120 plates. The random variable F represents the number of flawed plates found in the sample.
(b) Using a normal approximation, find the maximum number of plates, c, and the minimum number of plates, d, such that P(F ≤ c) ≤ 0.05 and P(F ≥ d) ≤ 0.05 where F ~ B(120, 0.45)
The manufacturer claims that, after a change to the production process, the proportion of flawed plates has decreased. A random sample of 30 plates, taken after the change to the production process, contains 8 flawed plates.
(c) Use a suitable hypothesis test, at the 5% level of significance, to assess the manufacturer’s claim. State your hypotheses clearly.
(Total for Question 3 is 15 marks)
题目中文翻译
一家制造商生产盘子。盘子有缺陷的比例为 45%，且缺陷盘子彼此独立出现。
随机抽取 10 个盘子。
(a) 求样本中：
(i) 少于 2 个有缺陷盘子的概率；
(ii) 至少 6 个有缺陷盘子的概率。
George 认为有缺陷盘子的比例不是 45%。为了检验他的看法，他抽取了 120 个盘子。随机变量 表示样本中有缺陷盘子的数量。
(b) 使用正态近似，求最大的盘子数 和最小的盘子数 ，使得
且
其中 。
制造商声称，在生产工艺改变后，有缺陷盘子的比例下降了。改变工艺后随机抽取 30 个盘子，其中 8 个有缺陷。
(c) 使用合适的假设检验，在 5% 显著性水平下检验制造商的说法，并清楚写出假设。
（第 3 题共 15 分）