一家工厂生产金属圆片。经理声称这些圆片直径的中位数为 22.0 mm。该工厂生产的 12 个圆片的随机样本直径如下，单位为 mm。

(a) 在 10% 显著性水平下进行 Wilcoxon 符号秩检验，检验是否存在反对经理声称的证据。

(b) 说明使该检验有效所需要的一个假设。

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。

用直径减去 $22.0$，对绝对差排名后恢复符号。题目问是否有任何反证，因此做双侧检验。

Let $m$ be the population median diameter. Test $H_0\colon m=22.0$ against $H_1\colon m\ne22.0$. In observation order, the differences are

The corresponding signed ranks are

The positive and negative rank sums are $54$ and $24$, giving $T=24$. The two-sided $10\%$ critical value with $12$ observations is $17$. Since $24>17$, we fail to reject $H_0$. There is insufficient evidence at the $10\%$ level that the population median diameter differs from $22.0$ mm.

符号秩检验不仅用差值的正负，也用大小的排序，因此需要总体对称。

The population distribution of disc diameters is assumed to be symmetric about its median.