三家不同的公交公司 $A$、$B$ 和 $C$ 记录各自服务的可靠性。每趟公交行程被记录为准点、晚点或取消。准点的公交服务被视为可靠。对每家公司随机抽取公交车辆的记录，汇总如下表。

(a) 在 $5\%$ 的显著性水平下，检验服务的可靠性是否与公交公司相互独立。

(b) 进一步的证据表明，与另外两家公司相比，其中一家公司可能特别不可靠。利用你在 2(a) 中的计算，指出最可能是哪家公司。

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。

先按独立假设算九格期望频数。因为可靠指准时，将晚点和取消合成同一个不准时类别。

Test $H_0$ that service reliability is independent of bus company against $H_1$ that it is associated with bus company. The original expected counts, using row total times column total divided by $100$, are

The expected count $3.50$ is below $5$. Combine late and cancelled into the not-on-time category. Its observed counts are $18,17,16$, and its expected counts are $12.75,19.89,18.36$. All pooled expected frequencies exceed $5$. The statistic is

There are $(2-1)(3-1)=2$ degrees of freedom. The upper $5\%$ critical value is $5.991$. Since the statistic is smaller, we fail to reject $H_0$. There is insufficient evidence at the $5\%$ level that reliability is associated with bus company.

比较各公司的不准时班次与独立假设下的期望值。找出实际频数高于期望频数的公司。

Company A appears least reliable. It had $18$ late or cancelled journeys against an expected $12.75$. For companies B and C, the observed numbers of late or cancelled journeys were below their expected counts.