一名科学家正在研究记忆能力是否取决于年龄。随机抽取 150 名不同年龄组的学生。每名学生观看一组 20 个物品 30 秒，然后被要求列出他们能记住的尽可能多的物品。根据正确记住的物品数量，将学生评为 A 或 B：正确记住 16 个或更多为 grade A，少于 16 个为 grade B。结果如下表所示。

(a) 在 2.5% 显著性水平下进行 $\chi^2$ 检验，检验 grade 是否与学生年龄相互独立。

科学家转而决定使用三个 grade：正确记住 16 个或更多为 grade A，正确记住 10 到 15 个为 grade B，少于 10 个为 grade C。结果如下表所示。

对于这第二组数据，检验统计量计算为 10.91。

(b) 在 2.5% 显著性水平下，完成针对第二组数据的 $\chi^2$ 检验。

(c) 说明你会更倾向于使用 part (a) 还是 part (b) 的结果来研究记忆能力是否取决于年龄，并给出理由。

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。

用两行三列的列联表计算期望频数及卡方统计量。用对应自由度的临界值判断关联。

Test $H_0$ that grade is independent of age group against $H_1$ that it is associated with age group. The expected counts, calculated as row total times column total divided by $150$, are

All exceed $5$. The statistic is

There are $(2-1)(3-1)=2$ degrees of freedom. The upper $2.5\%$ critical value is $7.378$. Since $8.40396\ldots>7.378$, we reject $H_0$. There is sufficient evidence at the $2.5\%$ level that grade and age group are associated under this grading scheme.

现在有三行三列，使用新的自由度。将题给统计量与新的临界值比较。

With the same independence hypotheses and the three grades, there are $(3-1)(3-1)=4$ degrees of freedom. The upper $2.5\%$ critical value is $11.143$. Since $10.91<11.143$, we fail to reject $H_0$. Under this grading scheme there is insufficient evidence at the $2.5\%$ level that grade and age group are associated.

如果关注整体记忆表现，比较两个分组方式各自保留了哪些成绩信息。

I would use the result from (b) for overall recall performance. Its extra category distinguishes students who remembered fewer than $10$ objects from those who remembered $10$ to $15$. Part (a) pools these students into a single group.