科技公司 $A$ 的经理声称，他的员工每年收入高于科技公司 $B$ 的员工。下表给出了从公司 $A$ 随机抽取的 12 名员工，以及从公司 $B$ 独立随机抽取的 12 名员工的年收入，单位为百美元。

(a) 在 5% 显著性水平下进行 Wilcoxon 秩和检验，检验数据是否支持该经理的说法。

(b) 如果收入服从正态分布，说明配对样本 $t$ 检验是否适合用于检验该经理的说法。

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。

合并两公司的工资从小到大排名。若 $A$ 工资较高，则 $B$ 的秩和偏小，用 $B$ 的秩和做左尾近似。

Let $m_A,m_B$ be the population median earnings under the common-shape location model. Test $H_0\colon m_A=m_B$ against $H_1\colon m_A>m_B$. In the order given, the ranks for company A are

which sum to $173$. All ranks sum to $24(25)/2=300$, so company B has rank sum $W_B=127$. Its null mean is $12(25)/2=150$ and variance is $12(12)(25)/12=300$. The continuity-corrected lower-tail statistic is

Since $z>-1.645$, we fail to reject $H_0$ at the one-sided $5\%$ level. There is insufficient evidence that employees at company A have higher typical annual earnings than those at company B.

配对检验要求每个观察都能与另一组中的对应观察形成有意义的一对。

A paired $t$-test is inappropriate. The observations are independent samples of different employees with no matched pairs, so there is no within-pair difference to analyse.