题目

Problem

In the colleges in three regions of a particular country, students are given individual targets to achieve. Their performance is measured against their individual target and graded as ‘above target’, ‘on target’ or ‘below target’. For a random sample of students from each of the three regions, the observed frequencies are summarised in the following table.

Performance	Region A	Region B	Region C	Total
Above target	62	41	44	147
On target	102	94	95	291
Below target	56	45	61	162
Total	220	180	200	600

Test, at the 10% significance level, whether performance is independent of region.

[7]

题目中文翻译

在某个国家三个地区的学院中，学生会被给予各自需要达到的个人目标。他们的表现会根据个人目标进行衡量，并被评为“高于目标”、“达到目标”或“低于目标”。从三个地区分别随机抽取学生，观测频数汇总如下表。

表现	地区 A	地区 B	地区 C	总计
高于目标	62	41	44	147
达到目标	102	94	95	291
低于目标	56	45	61	162
总计	220	180	200	600

在 10% 显著性水平下，检验表现是否与地区独立。

(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

展开

用行、列总数算独立假设下的期望频数。用三行三列确定卡方检验的自由度。

答题过程

展开

Test $H_0$ that performance relative to target is independent of region against $H_1$ that the variables are associated. Each expected frequency equals its row total times its column total divided by $600$ . The expected frequencies are

Performance	Region A	Region B	Region C
Above target	53.9	44.1	49.0
On target	106.7	87.3	97.0
Below target	59.4	48.6	54.0

All nine exceed $5$ . The test statistic is

\begin{align*} &\,\chi^2=\sum\frac{(O-E)^2}{E}\\ =&\,\frac{(62-53.9)^2}{53.9}\\ +&\,\frac{(41-44.1)^2}{44.1}+\cdots\\ =&\,4.07652\ldots. \end{align*}

There are $(3-1)(3-1)=4$ degrees of freedom. The upper $10\%$ critical value is $7.779$ . Since $4.07652\ldots<7.779$ , we fail to reject $H_0$ . There is insufficient evidence at the $10\%$ level that performance depends on region.

CIE 9231 2022 Nov Paper 42 Q2

题目

解答

解法一

思路

答题过程