一所大型学院的一位教师给所有学生出了一道数学谜题。随机抽取的 $24$ 名学生完成这道谜题所需时间的中位数为 $18.0$ 分钟。随后，学生接受了解谜练习。两周后，学生又做了一道与第一道类型相同的数学谜题。这同一随机样本中的 $24$ 名学生完成新谜题所需的时间如下，单位为分钟。

教师声称，这些练习并未改变完成这类谜题所需的平均时间。

在 $10\%$ 的显著性水平下进行 Wilcoxon 符号秩检验，检验是否有充分证据拒绝教师的声称。

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。

以题给的 $18.0$ 分钟为比较基准。对练习后时间的差值排名，用较小秩和做双侧检验。

Using $18.0$ minutes as the reference, test $H_0\colon m=18.0$ against $H_1\colon m\ne18.0$, where $m$ is the population median time after practice. The differences from $18.0$, in data order, are

Ranking their absolute values gives the signed ranks, in the same order,

The negative rank sum is $110$ and the positive rank sum is $190$. Under $H_0$ the sum of positive ranks has mean $24(25)/4=150$ and variance $24(25)(49)/24=1225$. For the smaller observed rank sum, use the lower-tail continuity correction, so

By symmetry, the lower critical value for the two-sided $10\%$ test is $-1.645$. Since $z>-1.645$, we fail to reject $H_0$. Using the stated benchmark, there is insufficient evidence at the $10\%$ level to reject the teacher’s claim. A test of individual before-and-after changes would require the original times for each student.