一家大型公司声称其雇员工资的中位数为 $32\,500$ 美元。随机选取的 15 名雇员的工资（美元）如下。

$18\,750 \quad 30\,500 \quad 125\,000 \quad 42\,500 \quad 25\,000$

$26\,000 \quad 52\,500 \quad 23\,000 \quad 27\,500 \quad 19\,500$

$25\,500 \quad 33\,000 \quad 30\,000 \quad 21\,500 \quad 29\,000$

(a) 解释为什么在这种情况下，Wilcoxon 符号秩检验可能不适合用于检验该公司的声称。

(b) 在 10% 显著性水平下进行符号检验，以调查该公司的声称。

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。

符号秩检验要求总体关于其中位数对称。观察工资中明显偏大的值及长尾方向。

The very large salary of $125\,000$ suggests a long upper tail. The salary distribution may be right-skewed, so the symmetry assumption of the Wilcoxon signed-rank test may be inappropriate.

只数在公司声称的中位数之上和之下的工资。因为要检验是否不同，计算双侧概率。

Let $m$ be the population median salary. Test $H_0\colon m=32500$ against $H_1\colon m\ne32500$. Four salaries exceed $32500$ and eleven are below it, with no ties. Under $H_0$, the number $S$ above $32500$ has distribution $\operatorname{Binomial}(15,1/2)$. The exact two-sided $p$-value is

Since this exceeds $0.10$, we fail to reject $H_0$. There is insufficient evidence at the $10\%$ level that the company’s median salary differs from $32500$.

也可以用正态分布近似符号个数的二项分布。检验为双侧，计算下尾时要作连续性修正。

Test $H_0\colon m=32500$ against $H_1\colon m\ne32500$, where $m$ is the population median salary. There are four salaries above $32500$ and no ties, so under $H_0$, $S\sim\operatorname{Binomial}(15,1/2)$.

Approximate $S$ by a normal variable with mean $15/2=7.5$ and variance $15(1/2)(1/2)=3.75$. For the lower-tail event $S\le4$, the continuity-corrected statistic is

The critical values for a two-sided $10\%$ test are $\pm1.645$. Since $-1.54919\ldots>-1.645$, we do not reject $H_0$. There is insufficient evidence at the $10\%$ level that the population median salary differs from $32500$.