两个拼图有相同数量且形状相同的拼块，但上面印着不同的图案。一个拼图是海边图案，另一个是卡通图案。一位研究人员认为儿童完成卡通拼图会更快。为了检验这一观点，随机选取 10 名儿童。每名儿童完成每个拼图所用的时间（单位为秒）记录如下。

(a) 在 5% 显著性水平下进行 Wilcoxon 配对符号秩检验，检验这位研究人员的观点。

(b) 证明在 5% 显著性水平下使用配对样本符号检验，会得到与 part (a) 相反的结论。

后来发现实验的进行方式是：每名儿童都先完成海边图案拼图，然后再完成卡通图案拼图。

(c) 评价使用这个实验来检验研究人员观点的有效性。

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。

同一个儿童的两种时间组成一对。用海边减卡通，正差值对应卡通更快。

Let $m_D$ be the population median of seaside-minus-cartoon completion times. Test $H_0\colon m_D=0$ against $H_1\colon m_D>0$. In child order the differences are $21,19,-2,22,-10,4,16,27,35,29$, with signed ranks $6,5,-1,7,-3,2,4,8,10,9$. The negative rank sum is $W_-=1+3=4$.

The lower one-sided $5\%$ critical value for $10$ nonzero differences is $10$. As $4\le10$, we reject $H_0$. There is sufficient evidence at the $5\%$ level that children tend to complete the cartoon puzzle more quickly.

符号检验只数正负，不使用差值大小。按同一备择方向算至少观察到八个正差值的概率。

Eight of the ten differences are positive. Under the sign-test null hypothesis, the number $S$ of positive differences is $\operatorname{Binomial}(10,1/2)$. Its upper-tail probability is

Since this exceeds $0.05$, the sign test does not reject $H_0$. It gives the opposite decision to the signed-rank test in (a).

固定顺序把图案和先后次序混在一起。第二次做题可能受第一次练习的影响。

Every child attempted the cartoon puzzle second, so a shorter time could reflect practice with the pieces rather than the picture. The experiment cannot distinguish the picture effect from the order effect.