一所大型学校为某个年级的所有学生安排了一次英语考试，由两位教师批改试卷。为检查两位教师评分的一致性，随机抽取 $10$ 份试卷，由每位教师独立评分。所给分数如下表。

(a) 给出一个理由，说明在这种情况下，Wilcoxon 配对符号秩检验可能比配对样本 $t$ 检验更合适。

(b) 在 $5\%$ 的显著性水平下进行 Wilcoxon 配对符号秩检验，考察两位教师所给分数在平均水平上是否存在差异。

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。

同一份答卷的两位老师评分是配对的。检查差值总体是否有理由假设正态。

With only ten paired observations and no stated normality for the population of mark differences, the normality assumption of a paired $t$-test may be unsuitable. A signed-rank test can be used under a symmetry assumption for those differences.

取第二位老师减第一位老师的分数。对绝对差排名，恢复符号后取较小秩和做双侧检验。

Let $m_D$ be the population median of Teacher 2 minus Teacher 1 marks for the same script. Test $H_0\colon m_D=0$ against $H_1\colon m_D\ne0$. The differences, in student order, are

The corresponding signed ranks are

Their negative and positive rank sums are $10$ and $45$, so the statistic is $T=10$.

For $10$ nonzero differences the two-sided $5\%$ critical value is $8$. Since $10>8$, we fail to reject $H_0$. There is insufficient evidence at the $5\%$ level of a systematic difference between the marks awarded by the two teachers.