一位研究者调查了在三个驾驶考试中心之一参加驾驶考试的考生结果。

从 620 名考生中抽得一个随机样本，结果如下。

(a) 在 5% 的显著性水平下检验考生的驾驶考试结果与驾驶考试中心之间是否存在关联。请清楚写出你的过程。你应写出原假设、期望频数、检验统计量和临界值。

研究者决定对考生驾驶考试结果做进一步调查。

(b) 说明你会建议进一步调查哪个驾驶考试中心，并给出理由。

这是一个 $2\times3$ 列联表的卡方独立性检验。先在“考试中心与结果相互独立”的原假设下，用“行总计乘列总计再除以总人数”求每格期望频数；然后计算六格对 $\chi^2$ 的贡献，确定自由度、临界值并作出有语境的结论。

Let $H_0$ be the hypothesis that driving test centre and result are independent.

Let $H_1$ be the hypothesis that there is an association between driving test centre and result.

Under $H_0$, each expected frequency is calculated using

For example,

The complete table of expected frequencies, rounded to 2 decimal places, is

Now calculate each contribution $\frac{(O-E)^2}{E}$:

Therefore,

The degrees of freedom are

At the $5\%$ significance level, the critical value is

so the critical region is $\chi^2\geq5.991$.

Since

the result is significant and $H_0$ is rejected. There is evidence, at the $5\%$ significance level, of an association between driving test centre and driving test result.

The official alternative calculation gives the same statistic:

进一步调查应聚焦于对检验统计量贡献最大的中心。把同一中心“通过”和“未通过”两格的贡献相加，C 中心的总贡献约为 $5.23$，远高于 A、B，说明 C 中心的观察频数与独立性假设下的期望频数差异最大。

The contributions to the test statistic from each centre are

Therefore, the researcher should investigate $\boxed{\text{centre C}}$, because it makes the largest contribution to the $\chi^2$ test statistic. In particular, its observed pass frequency, 68, is much lower than its expected frequency, approximately 83.55.