题目

Problem

A sports college keeps records of the times taken by students to run one lap of a running track. The population median time taken is 51.0 seconds. After a month of intensive training, a random sample of 22 new students run one lap of the track, giving times, in seconds, as follows.

\begin{gathered} 51.3 \qquad 52.0 \qquad 53.4 \qquad 49.2 \\ 49.3 \qquad 51.1 \qquad 52.2 \qquad 47.2 \\ 53.0 \qquad 48.5 \qquad 49.4 \qquad 50.3 \\ 50.8 \qquad 51.6 \qquad 49.1 \qquad 52.3 \\ 51.8 \qquad 52.4 \qquad 47.9 \qquad 48.9 \\ 50.6 \qquad 51.9 \end{gathered}

It is claimed that the intensive training has led to a decrease in the median time taken to run one lap of the track.

Carry out a Wilcoxon signed-rank test, at the 5% significance level, to test whether there is sufficient evidence to support the claim.

[9]

题目中文翻译

一所体育学院记录学生跑完一圈跑道所用的时间。总体中位数时间为 51.0 秒。经过一个月的强化训练后，随机抽取 22 名新学生跑一圈跑道，得到如下时间，单位为秒。

\begin{gathered} 51.3 \qquad 52.0 \qquad 53.4 \qquad 49.2 \\ 49.3 \qquad 51.1 \qquad 52.2 \qquad 47.2 \\ 53.0 \qquad 48.5 \qquad 49.4 \qquad 50.3 \\ 50.8 \qquad 51.6 \qquad 49.1 \qquad 52.3 \\ 51.8 \qquad 52.4 \qquad 47.9 \qquad 48.9 \\ 50.6 \qquad 51.9 \end{gathered}

有人声称强化训练使跑完一圈跑道所用时间的中位数降低。

在 5% 显著性水平下进行 Wilcoxon 符号秩检验，检验是否有足够证据支持该说法。

(9)

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

解法一

思路

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以 $51.0$ 秒为基准计算差值，再按绝对值排名。成绩变快对应负差值，因此用正秩和较小的方向做左尾检验。

答题过程

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Let $m$ be the population median time after training. Test $H_0\colon m=51.0$ against $H_1\colon m<51.0$ . Subtract $51.0$ from each time. There are no zero differences or tied absolute differences. Ranking their absolute values and restoring their signs gives, in the order of the data,

\begin{align*} &\,3,\ 9,\ 19,\ -15,\ -14,\ 1,\ 10,\ -22,\\ &\,17,\ -20,\ -13,\ -6,\ -2,\ 5,\ -16,\ 11,\\ &\,7,\ 12,\ -21,\ -18,\ -4,\ 8. \end{align*}

The sum of the positive ranks is $W_+=102$ . For $n=22$ the null mean and variance of $W_+$ are

\begin{align*} &\,\frac{22(23)}4=126.5,\\ &\,\frac{22(23)(45)}{24}=948.75. \end{align*}

Applying the normal approximation with a lower-tail continuity correction,

z=\frac{102.5-126.5}{\sqrt{948.75}}=-0.779175605\ldots.

The lower $5\%$ critical value is $-1.645$ . Since $-0.779\ldots>-1.645$ , we fail to reject $H_0$ . There is insufficient evidence at the $5\%$ level that the population median lap time after training is below $51.0$ seconds.

CIE 9231 2024 Nov Paper 42 Q6

题目

解答

解法一

思路

答题过程