题目

Problem

A machine fills bottles with mineral water. The machine is checked every day to ensure that it is working correctly. On a particular day a random sample of 100 bottles is taken. The volume of water, x millilitres, for each bottle is measured and each measurement is coded using

y=x-1000

The results are summarised below

\sum y=847 \qquad \sum y^2=13\ 510.09

(a) (i) Show that the value of the unbiased estimate of the mean of x is 1008.47 (ii) Calculate the unbiased estimate of the variance of x

The machine was initially set so that the volume of water in a bottle had a mean value of 1010 millilitres. Later, a test at the 5% significance level is used to determine whether or not the mean volume of water in a bottle has changed. If it has changed then the machine is stopped and reset.

(b) Write down suitable null and alternative hypotheses for a 2-tailed test.

(d) Using your answer to part (a) and your critical region found in part (c), comment on whether or not the machine needs to be stopped and reset. Give a reason for your answer.

(e) Explain why the use of $\sigma^2 \approx s^2$ is reasonable in this situation.

(4)

(1)

(4)

(2)

(1)

(Total for Question 1 is 12 marks)

题目中文翻译

一台机器正在灌装矿泉水瓶。机器每天都会检查，以确保它正常工作。某天随机抽取了 100 个瓶子。记录每个瓶中的水量 $x$ 毫升，并使用

y=x-1000

进行编码。

结果汇总如下：

\sum y=847 \qquad \sum y^2=13\ 510.09

(a) (i) 证明 $x$ 的无偏均值估计为 1008.47。 (ii) 求 $x$ 的无偏方差估计。

机器最初被设定为每瓶水的平均容量为 1010 毫升。后来在 5% 显著性水平下进行检验，以判断每瓶水的平均容量是否已经改变。如果改变了，机器就会停止并重新设定。

(b) 写出合适的零假设和备择假设，用于双侧检验。

(d) 利用 (a) 小题的答案和 (c) 小题得到的临界域，说明机器是否需要停止并重新设定。给出理由。

(e) 解释为什么在这种情况下使用 $\sigma^2 \approx s^2$ 是合理的。

IAL 2023 Jan S3 Q1

题目

解答