题目

Problem

A mobile phone company offers an insurance policy to its customers when they purchase a mobile phone. The company conducted a survey on the age of the customers and whether or not claims were made.

A random sample of 1200 customers from this company was investigated for 2020 and the results are shown in the table below.

Age	Claim made in 2020	No claim made in 2020	Total
17 – 20 years	24	176	200
21 – 50 years	48	652	700
51 years and over	14	286	300
Total	86	1114	1200

The data are to be used to determine whether or not making a claim is independent of age.

(a) Calculate the expected frequencies for the age group 51 years and over that (i) made a claim in 2020 (ii) did not make a claim in 2020

(2)

The 4 classes of customers aged between 17 and 50 give a value of

\sum \frac{(O-E)^2}{E}=7.123

correct to 3 decimal places.

(b) Test, at the 1% level of significance, whether or not making a claim is independent of age. Show your working clearly, stating your hypotheses, the degrees of freedom, the test statistic and the critical value used.

(7)

(Total for Question 3 is 9 marks)

题目中文翻译

手机公司在客户购买手机时提供保险。公司对客户年龄以及是否提出理赔进行了调查。

从该公司随机抽取了 1200 名 2020 年的客户，结果如下表所示。

年龄	2020 年提出理赔	2020 年未提出理赔	总计
17 至 20 岁	24	176	200
21 至 50 岁	48	652	700
51 岁及以上	14	286	300
总计	86	1114	1200

这些数据将用于判断是否“提出理赔”与“年龄”相互独立。

(a) 计算 51 岁及以上年龄组中 (i) 2020 年提出理赔的期望频数 (ii) 2020 年未提出理赔的期望频数

17 至 50 岁这 4 个类别的

\sum \frac{(O-E)^2}{E}=7.123

，精确到 3 位小数。

(b) 在 1% 显著性水平下，检验提出理赔是否与年龄独立。请清楚写出你的工作过程、原假设、自由度、检验统计量和所用临界值。

解答

(a)(i)

解法一

思路

展开

在“提出理赔”与“年龄”独立的假设下，某格的期望频数等于该格所在行总数乘所在列总数，再除以总样本数。这里使用 51 岁及以上的行总数 300 与提出理赔的列总数 86。

答题过程

展开

The expected frequency for customers aged 51 years and over who made a claim is

\begin{align*} E =&\,\frac{300\times86}{1200}\\ =&\,\boxed{21.5}. \end{align*}

(a)(ii)

解法一

思路

展开

同样使用独立性假设下的期望频数公式，这次改用未提出理赔的列总数 1114。

答题过程

展开

The expected frequency for customers aged 51 years and over who did not make a claim is

\begin{align*} E =&\,\frac{300\times1114}{1200}\\ =&\,\boxed{278.5}. \end{align*}

(b)

解法一

思路

展开

使用卡方独立性检验。题目已给出前四格对统计量的总贡献，只需计算 51 岁及以上两格的贡献并相加。列联表有 3 个年龄组和 2 种理赔状态，所以自由度为 $(3-1)(2-1)$ 。

答题过程

展开

The hypotheses are:

\begin{aligned} H_0:&\ \text{Making a claim and age are independent},\\ H_1:&\ \text{Making a claim and age are associated}. \end{aligned}

For the two classes corresponding to customers aged 51 years and over, the contributions to the test statistic are

\begin{align*} \frac{(14-21.5)^2}{21.5} =&\,2.6162\ldots,\\ \frac{(286-278.5)^2}{278.5} =&\,0.20197\ldots. \end{align*}

Therefore,

\begin{align*} \chi^2 =&\,7.123+2.6162\ldots\\ &\,\hspace{2pt}+0.20197\ldots\\ =&\,9.941\ldots\\ =&\,9.94. \end{align*}

The number of degrees of freedom is

\nu=(3-1)(2-1)=2.

At the $1\%$ significance level, the critical value is

\chi^2_2(0.01)=9.210.

Since $9.94>9.210$ , the test statistic lies in the critical region, so $H_0$ is rejected. There is sufficient evidence at the $1\%$ significance level that making a claim is not independent of age.

IAL 2023 Jan S3 Q3

题目

解答

(a)(i)

解法一

思路

答题过程

(a)(ii)

解法一

思路

答题过程

(b)

解法一

思路

答题过程