题目

Problem

There are three bus companies in a city. The council is investigating whether the buses reliably arrive at their destination on time. The results from random samples of buses from each company are summarised in the following table.

		Bus company
		A	B	C	Total
Arrival	Early	22	22	10	54
Arrival	On time	30	52	42	124
Arrival	Late	28	26	18	72
	Total	80	100	70	250

Test, at the 5% significance level, whether the reliability of buses is independent of bus company.

[7]

题目中文翻译

某城市有三家公交公司。市政部门正在调查公交车是否能可靠地准时到达目的地。从每家公司公交车的随机样本得到的结果概括如下表。

		公交公司
		A	B	C	合计
到达情况	提前	22	22	10	54
到达情况	准时	30	52	42	124
到达情况	晚点	28	26	18	72
	合计	80	100	70	250

在 5% 显著性水平下，检验公交车的可靠性是否与公交公司相互独立。

(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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在独立假设下，从每个到达类别的总数和每家公司的总数求期望频数。将九格的偏差合成一个卡方统计量。

答题过程

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Test $H_0$ that arrival category is independent of bus company against $H_1$ that the variables are associated. The expected count for an early bus from company A is $54(80)/250=17.28$ . The full expected table is

Arrival	A	B	C
Early	17.28	21.60	15.12
On time	39.68	49.60	34.72
Late	23.04	28.80	20.16

All expected counts exceed $5$ . The contributions $(O-E)^2/E$ , in the same cell order, are

Arrival	A	B	C
Early	1.289259	0.007407	1.733757
On time	2.361452	0.116129	1.526452
Late	1.067778	0.272222	0.231429

Summing the unrounded contributions gives

\begin{align*} &\,\chi^2=\sum\frac{(O-E)^2}{E}\\ =&\,\frac{(22-17.28)^2}{17.28}+\frac{(22-21.60)^2}{21.60}+\cdots\\ =&\,8.60588\ldots. \end{align*}

The degrees of freedom are $(3-1)(3-1)=4$ . The upper $5\%$ critical value is $9.488$ . Since the statistic is smaller, we fail to reject $H_0$ . There is insufficient evidence at the $5\%$ level that arrival reliability depends on bus company.

CIE 9231 2024 June Paper 43 Q3

题目

解答

解法一

思路

答题过程