大量学生参加了两份数学测试卷。老师认为 Paper 1 的得分会高于 Paper 2 的得分。她随机抽取 9 名学生并比较他们的分数。分数如下表所示。

(a) 在 5% 显著性水平下，进行 Wilcoxon 配对符号秩检验，检验数据是否支持老师的看法。

(b) 说明你在进行 part (a) 中的检验时所作的一个假设。

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。

同一学生的两份试卷成绩组成一对。取第一份减第二份，按绝对差排名后计算负秩和。

Let $m_D$ be the population median difference, Paper 1 minus Paper 2. Test $H_0\colon m_D=0$ against $H_1\colon m_D>0$. The differences, Paper 1 minus Paper 2, are

Ranking their absolute values gives the corresponding signed ranks

Thus $W_-=5+3+8=16$.

For $9$ nonzero differences the lower one-sided $5\%$ critical value is $8$. Since $16>8$, we fail to reject $H_0$. There is insufficient evidence at the $5\%$ level that students tend to score higher on Paper 1 than on Paper 2.

检验对配对差值的分布形状作假设。

The population distribution of the paired differences in marks is assumed to be symmetric about its median.