随机抽取 13 家科技公司，并记录它们在 2018 年和 2022 年的员工人数。

一名研究者声称，2018 年到 2022 年间，科技公司的员工人数中位数有所增加。

(a) 在 5% 显著性水平下进行 Wilcoxon 配对符号秩检验，检验这些数据是否支持该声称。

研究者注意到，公司 G 的数据记录有误。事实上，2018 年的员工人数是 32，2022 年的员工人数是 35。

(b) 结合数值说明，解释 (a) 中检验的结论是否保持不变。

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。

按公司配对，用 2022 年员工数减去 2018 年员工数。将绝对差排名后，计算负秩和以检验增加。

Let $m_D$ be the population median of the within-company changes, defined as $2022$ minus $2018$. Test $H_0\colon m_D=0$ against $H_1\colon m_D>0$. For A through G the changes and signed ranks, respectively, are

For H through M they are

The negative rank sum is $W_-=5+3+7+8=23$. The lower one-sided $5\%$ critical value for $13$ pairs is $21$. Since $23>21$, we fail to reject $H_0$. There is insufficient evidence at the $5\%$ level of a positive median change in employee numbers.

更正只使 G 的差值从负变正，绝对差不变。把 G 的秩从负秩和移走，再与临界值比较。

For G the change becomes $35-32=3$ instead of $-3$. Its absolute rank is still $3$. The negative rank sum becomes $23-3=20$. Since $20\le21$, the corrected data lead to rejection of $H_0$, so the conclusion changes. There is now sufficient evidence at the $5\%$ level of a positive median change.