题目

Problem

A business accepts cash, bank cards or mobile apps as payment methods.

The manager wishes to test whether or not there is an association between the payment amount and the payment method used.

The manager takes a random sample of 240 payments and records the payment amount and the payment method used. The manager’s results are shown in the table.

Payment method	Under £50	£50 to £150	Over £150
Cash	23	19	18
Bank card	21	32	31
Mobile app	16	39	41

(a) calculate the expected frequencies for the payment amount under £50 that (i) use cash (ii) use a bank card (iii) use a mobile app

(3)

Given that for the other 6 classes

\sum \frac{(O-E)^2}{E}=2.4048

to 4 decimal places,

(b) test, at the 5% level of significance, whether or not there is evidence for an association between the payment amount and the payment method used. You should state the hypotheses, the test statistic, the degrees of freedom and the critical value used for this test.

(7)

(Total for Question 2 is 10 marks)

题目中文翻译

一家商家接受现金、银行卡或移动应用付款。

经理想检验付款金额与付款方式之间是否存在关联。

经理随机抽取了 240 笔付款，并记录付款金额和所使用的付款方式。结果如下表所示。

付款方式	低于 £50	£50 到 £150	高于 £150
现金	23	19	18
银行卡	21	32	31
移动应用	16	39	41

(a) 计算低于 £50 的付款金额对应的期望频数，分别是 (i) 使用现金 (ii) 使用银行卡 (iii) 使用移动应用

已知其余 6 个格子的

\sum \frac{(O-E)^2}{E}=2.4048

，精确到 4 位小数，

(b) 在 5% 显著性水平下，检验是否有证据表明付款金额与付款方式有关联。你应写出原假设、检验统计量、自由度和临界值。

解答

(a)

解法一

思路

展开

独立性检验中，每个格子的期望频数为“所在行总数乘所在列总数，再除以总数”。先求三种付款方式的行总数，以及低于 £50 这一列的总数。

答题过程

展开

The row totals for cash, bank card and mobile app are respectively

60,\qquad84,\qquad96,

and the column total for payments under £50 is $60$ . Therefore,

\begin{align*} E_{\text{cash}} =&\,\frac{60\times60}{240}\\ =&\,\boxed{15},\\ E_{\text{bank card}} =&\,\frac{84\times60}{240}\\ =&\,\boxed{21},\\ E_{\text{mobile app}} =&\,\frac{96\times60}{240}\\ =&\,\boxed{24}. \end{align*}

(b)

解法一

思路

展开

使用卡方独立性检验。先把 (a) 中三个格子对检验统计量的贡献与题目给出的其余六格贡献相加，再根据 $3\times3$ 列联表求自由度并与 $5\%$ 临界值比较。

答题过程

展开

The hypotheses are:

\begin{aligned} H_0:&\ \text{The payment amount and payment method are independent},\\ H_1:&\ \text{The payment amount and payment method are associated}. \end{aligned}

For the three classes corresponding to payments under £50, the contributions to the test statistic are

\begin{align*} \frac{(23-15)^2}{15} =&\,4.2667\ldots,\\ \frac{(21-21)^2}{21} =&\,0,\\ \frac{(16-24)^2}{24} =&\,2.6667\ldots. \end{align*}

Hence

\begin{align*} \chi^2 =&\,2.4048+4.2667\ldots\\ &\,\hspace{2pt}+0+2.6667\ldots\\ =&\,9.3381\ldots\\ =&\,9.34. \end{align*}

The number of degrees of freedom is

\nu=(3-1)(3-1)=4.

At the $5\%$ significance level, the critical value is

\chi^2_4(0.05)=9.488.

Since $9.34<9.488$ , the test statistic does not lie in the critical region, so $H_0$ is not rejected. There is insufficient evidence at the $5\%$ significance level of an association between the payment amount and the payment method used.

IAL 2023 June S3 Q2

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程