一名驾驶教练认为，学生参加驾驶考试时的表现（通过或未通过）与他们的年龄有关。下表总结了三年内参加考试的学生中通过和未通过的人数，以及学生的年龄（单位为年）。

在 $10\%$ 显著性水平下，检验考试表现是否独立于学生年龄。

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。

按年龄组与是否通过构成列联表。先算独立假设下的期望频数，再用卡方统计量衡量偏离程度。

Let $H_0$ be that driving-test performance is independent of the student’s age group, and $H_1$ that they are associated. Under $H_0$ each expected count is its row total times its column total divided by $145$. For example, the expected number of passes in the under-20 group is $81(50)/145$. The expected counts, rounded for display, are

All six expected counts exceed $5$. Using their unrounded values in the test statistic,

There are $(2-1)(3-1)=2$ degrees of freedom. The upper $10\%$ critical value is $4.605$. Since $5.20849\ldots>4.605$, we reject $H_0$. There is sufficient evidence at the $10\%$ level that driving-test performance and age group are associated.