题目

Problem

The continuous random variable X is normally distributed with

X \sim N(\mu,5^2)

A random sample of 10 observations of X is taken and $\bar X$ denotes the sample mean.

(a) Show that a 90% confidence interval for $\mu$ , in terms of $\bar x$ , is given by

(\bar x - 2.60,\ \bar x + 2.60)

The continuous random variable Y is normally distributed with

Y \sim N(\mu,3^2)

A random sample of 20 observations of Y are taken and $\bar Y$ denotes the sample mean.

(b) Find a 95% confidence interval for $\mu$ , in terms of $\bar y$

(c) Given that X and Y are independent, (i) find the distribution of $\bar X - \bar Y$ (ii) calculate the probability that the two confidence intervals from part (a) and part (b) do not overlap.

(3)

(7)

(Total for Question 5 is 13 marks)

题目中文翻译

连续随机变量 $X$ 服从正态分布，

X \sim N(\mu,5^2)

抽取了 $X$ 的 10 个观测值的随机样本，并用 $\bar X$ 表示样本均值。

(a) 证明以 $\bar x$ 表示的 $\mu$ 的 90% 置信区间为

(\bar x - 2.60,\ \bar x + 2.60)

连续随机变量 $Y$ 服从正态分布，

Y \sim N(\mu,3^2)

抽取了 $Y$ 的 20 个观测值的随机样本，并用 $\bar Y$ 表示样本均值。

(b) 求以 $\bar y$ 表示的 $\mu$ 的 95% 置信区间。

IAL 2023 June S3 Q5

题目

解答