题目

Problem

A random sample of 50 values of the continuous random variable $X$ was taken. These values are summarised in the following table.

Interval	$1 \leq x < 1.5$	$1.5 \leq x < 2$	$2 \leq x < 2.5$	$2.5 \leq x < 3$	$3 \leq x < 3.5$	$3.5 \leq x \leq 4$
Observed frequency	3	3	8	11	13	12

It is required to test the goodness of fit of the distribution with probability density function f given by

f(x) = \begin{cases} \dfrac{1}{24}\bigg(\dfrac{4}{x^2} + x^2\bigg), & 1 \leq x \leq 4,\\ 0, & \text{otherwise.} \end{cases}

The expected frequencies, correct to 4 decimal places, are given in the following table.

Interval	$1 \leq x < 1.5$	$1.5 \leq x < 2$	$2 \leq x < 2.5$	$2.5 \leq x < 3$	$3 \leq x < 3.5$	$3.5 \leq x \leq 4$
Expected frequency	4.4271	$a$	6.1285	8.4549	$b$	14.9678

(a) Show that $a = 4.6007$ and find the value of $b$ .

[3]

(b) Carry out a goodness of fit test, at the 10% significance level, to test whether f is a satisfactory model for the data.

[6]

题目中文翻译

从连续随机变量 $X$ 中随机抽取 50 个值。这些值汇总在下表中。

区间	$1 \leq x < 1.5$	$1.5 \leq x < 2$	$2 \leq x < 2.5$	$2.5 \leq x < 3$	$3 \leq x < 3.5$	$3.5 \leq x \leq 4$
观察频数	3	3	8	11	13	12

现在要检验具有以下概率密度函数 f 的分布的拟合优度：

f(x) = \begin{cases} \dfrac{1}{24}\bigg(\dfrac{4}{x^2} + x^2\bigg), & 1 \leq x \leq 4,\\ 0, & \text{otherwise.} \end{cases}

下表给出了期望频数，结果精确到 4 位小数。

区间	$1 \leq x < 1.5$	$1.5 \leq x < 2$	$2 \leq x < 2.5$	$2.5 \leq x < 3$	$3 \leq x < 3.5$	$3.5 \leq x \leq 4$
期望频数	4.4271	$a$	6.1285	8.4549	$b$	14.9678

(a) 证明 $a = 4.6007$ ，并求 $b$ 的值。

(b) 在 10% 显著性水平下进行拟合优度检验，检验 f 是否是该数据的一个令人满意的模型。

CIE 9231 2023 June Paper 41 Q3