一所学校正在进行实验，研究儿童在炎热天气中投球距离是否会增加。在一个寒冷的日子，学校所有儿童都被要求尽可能远地投球，并测量和记录投球距离。从其中 25 名儿童组成的随机样本中，投球距离的中位数为 22.0 m。随后要求同样的 25 名儿童在炎热的日子再次投球，并测量和记录他们的投球距离。下列数据为这些距离，单位为 m。

老师声称，平均而言，在炎热天气中投球距离会更远。

在 5% 显著性水平下进行 Wilcoxon 符号秩检验，检验数据是否支持老师的说法。

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。

按题目给出的冷天中位数 $22.0$ 米作为比较基准。对热天数据减去基准，并按差值绝对值排名。

Using $22.0$ m as the reference, test $H_0\colon m=22.0$ against $H_1\colon m>22.0$, where $m$ is the population median distance on a hot day. In the order of the hot-day observations, the signed ranks of the differences from $22.0$ are

For the positive alternative, use the negative rank sum $W_-=111$. With $25$ nonzero, untied differences, the null mean is $25(26)/4=162.5$ and the variance is $25(26)(51)/24=1381.25$. With the lower-tail continuity correction, the normal statistic is

Since this exceeds the lower $5\%$ critical value $-1.645$, we fail to reject $H_0$. Using the stated reference, there is insufficient evidence at the $5\%$ level to support the teacher’s claim. The individual cold-day distances would be needed to run a signed-rank test of within-child changes.