题目

Problem

A scientist is investigating the size of shells at various beach locations. She selects four beach locations and takes a random sample of shells from each of these beaches. She classifies each shell as large or small. Her results are summarised in the following table.

Size of shell	A	B	C	D	Total
Large	68	69	96	81	314
Small	28	55	64	39	186
Total	96	124	160	120	500

Test, at the 10% significance level, whether the size of shell is independent of the beach location.

[7]

题目中文翻译

一名科学家正在研究不同海滩地点的贝壳大小。她选择四个海滩地点，并从每个海滩随机抽取贝壳样本。她将每个贝壳分类为大或小。她的结果汇总在下表中。

贝壳大小	A	B	C	D	总计
大	68	69	96	81	314
小	28	55	64	39	186
总计	96	124	160	120	500

在 10% 显著性水平下，检验贝壳大小是否与海滩地点相互独立。

(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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由行总数和列总数求独立假设下的期望频数。四处地点、两种大小对应三个自由度。

答题过程

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Test $H_0$ that shell size is independent of beach location against $H_1$ that they are associated. Each expected count is its row total times its column total divided by $500$ . The expected frequencies are

Size	A	B	C	D
Large	60.288	77.872	100.480	75.360
Small	35.712	46.128	59.520	44.640

All exceed $5$ . The test statistic is

\begin{align*} &\,\chi^2=\sum\frac{(O-E)^2}{E}\\ =&\,\frac{(68-60.288)^2}{60.288}+\frac{(69-77.872)^2}{77.872}\\ +&\,\cdots\\ =&\,7.04073\ldots. \end{align*}

There are $(2-1)(4-1)=3$ degrees of freedom. The upper $10\%$ critical value is $6.251$ . Since the statistic exceeds it, we reject $H_0$ . There is sufficient evidence at the $10\%$ level that shell size is associated with beach location.

CIE 9231 2022 June Paper 43 Q2

题目

解答

解法一

思路

答题过程