对某种机器随机抽取 11 台，测量它们产生的声音水平。结果以适当单位表示如下。

(a) 使用 Wilcoxon 符号秩检验，检验这种机器产生的平均声音水平是否大于 8.00。使用 5% 显著性水平。

(b) 说明在这种情况下 Wilcoxon 符号秩检验为什么可能比 $t$ 检验更合适。

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。

用每个声级减去 $8.00$，按绝对差排名并恢复符号。若总体位置较高，负秩和应偏小。

Let $m$ be the population median sound level. Test $H_0\colon m=8.00$ against $H_1\colon m>8.00$. The differences and signed ranks in machine order are

The negative rank sum is $W_-=10+2+8+4=24$. The lower one-sided $5\%$ critical value for $11$ nonzero differences is $13$. Since $24>13$, we fail to reject $H_0$. There is insufficient evidence at the $5\%$ level that the typical sound level exceeds $8.00$.

样本较小，题目也没有给出声级总体正态的条件。比较两种检验对总体形状的要求。

A one-sample $t$-test with this small sample requires a normally distributed population. The signed-rank test can be used when the population is symmetric without requiring it to be normal.