题目

Problem

A shop selling electrical goods has a team of three salespeople: Avril, Ben and Charlie. The manager wishes to investigate whether the salespeople are equally successful at selling particular types of items. The following table gives a record of a random sample of 250 sales of laptops, cameras and televisions, with the number sold by each of the three salespeople.

Salesperson	Laptop	Camera	Television	Total
Avril	31	40	24	95
Ben	23	45	29	97
Charlie	21	25	12	58
Total	75	110	65	250

Test, at the 10% significance level, whether there is independence between the type of item sold and the salesperson.

[7]

题目中文翻译

一家销售电器商品的商店有三名销售人员：Avril、Ben 和 Charlie。经理希望调查这些销售人员在销售特定类型商品方面是否同样成功。下表记录了随机抽取的 250 次笔记本电脑、相机和电视销售，并给出了三名销售人员各自售出的数量。

销售人员	笔记本电脑	相机	电视	总计
Avril	31	40	24	95
Ben	23	45	29	97
Charlie	21	25	12	58
总计	75	110	65	250

在 10% 显著性水平下，检验售出商品类型与销售人员之间是否独立。

(7)

原卷通用作答要求

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

解答

解法一

思路

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根据各销售员的销售总数与各商品总数，求独立假设下的期望频数。再比较卡方统计量和临界值。

答题过程

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Let $H_0$ be that item type and salesperson are independent, and $H_1$ that they are associated. Each expected frequency is its row total times its column total divided by $250$ . For example, Avril’s expected laptop count is $95(75)/250=28.5$ . The expected frequencies are

Salesperson	Laptop	Camera	Television
Avril	28.50	41.80	24.70
Ben	29.10	42.68	25.22
Charlie	17.40	25.52	15.08

All nine expected counts exceed $5$ . The statistic is

\begin{align*} &\,\chi^2=\sum\frac{(O-E)^2}{E}\\ =&\,\frac{(31-28.5)^2}{28.5}+\frac{(40-41.8)^2}{41.8}+\cdots\\ =&\,3.67249\ldots. \end{align*}

The degrees of freedom are $(3-1)(3-1)=4$ , so the upper $10\%$ critical value is $7.779$ . Since $3.67249\ldots<7.779$ , we fail to reject $H_0$ . There is insufficient evidence at the $10\%$ level of an association between item type and salesperson.

CIE 9231 2025 June Paper 44 Q3

题目

解答

解法一

思路

答题过程