题目

Problem

Members of a cycling club use the same circular route every week. Throughout the year, the members change between completing the route in a clockwise direction one week and completing the route in an anticlockwise direction the following week.

The club captain believes that the anticlockwise route takes longer to complete on average than the clockwise route. To test this belief, the club captain selects a random sample of $10$ members. Each of the $10$ club members gives their personal best times, in minutes, for completing the route in both directions over the previous year. These times are as follows.

club member	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$	$I$	$J$
clockwise	127.1	129.1	139.3	151.4	142.5	123.6	153.4	129.7	140.9	137.4
anticlockwise	131.0	132.1	147.3	150.0	144.3	127.3	153.5	129.9	140.6	140.3

(a) Explain the advantage of using paired samples to investigate the club captain’s belief.

(1)

(b) Carry out a Wilcoxon matched-pairs signed-rank test at the $5\%$ significance level to test the club captain’s belief.

(6)

(1)

题目中文翻译

某自行车俱乐部的成员每周骑行同一条环形路线。一年中，成员一周沿顺时针方向完成路线，下一周则沿逆时针方向完成路线，如此交替。

俱乐部队长认为，沿逆时针方向完成路线所需的时间平均而言长于沿顺时针方向所需的时间。为检验这一看法，队长随机抽取 $10$ 位成员。这 $10$ 位成员分别提供过去一年中沿两个方向完成路线的个人最佳时间，单位为分钟，结果如下。

俱乐部成员	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$	$I$	$J$
顺时针	127.1	129.1	139.3	151.4	142.5	123.6	153.4	129.7	140.9	137.4
逆时针	131.0	132.1	147.3	150.0	144.3	127.3	153.5	129.9	140.6	140.3

(a) 解释使用配对样本考察队长看法的优点。

(1)

(b) 在 $5\%$ 的显著性水平下进行 Wilcoxon 配对符号秩检验，检验队长的看法。

(6)

(1)

原卷通用作答要求

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。

解答

(a)

解法一

思路

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比较同一个人的两个方向，可以消除不同骑手本身速度的差异。

答题过程

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Pairing compares the two directions within the same rider. This controls for differences in general cycling speed between riders, so the paired changes more directly reflect the direction of travel.

(b)

解法一

思路

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每位骑手用逆时针减顺时针时间。按绝对差排名后，用负秩和检验逆时针时间偏长。

答题过程

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Let $m_D$ be the population median of anticlockwise-minus-clockwise times. Test $H_0\colon m_D=0$ against $H_1\colon m_D>0$ . In rider order the differences are

\begin{align*} &\,3.9,\ 3.0,\ 8.0,\ -1.4,\ 1.8,\\ &\,3.7,\ 0.1,\ 0.2,\ -0.3,\ 2.9. \end{align*}

Their corresponding signed ranks are

\begin{align*} &\,9,\ 7,\ 10,\ -4,\ 5,\\ &\,8,\ 1,\ 2,\ -3,\ 6. \end{align*}

The negative rank sum is $W_-=4+3=7$ .

The lower one-sided $5\%$ critical value for $10$ nonzero differences is $10$ . Since $7\le10$ , we reject $H_0$ . There is sufficient evidence at the $5\%$ level that the anticlockwise direction tends to take longer.

(c)

解法一

思路

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符号秩检验对配对时间差的分布形状作假设。

答题过程

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The population distribution of the paired time differences is assumed to be symmetric about its median.

CIE 9231 2026 June Paper 44 Q4

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程

(c)

解法一

思路

答题过程