题目

Problem

A school is conducting an experiment to see whether the distance that children can throw a ball increases in hot weather. On a cold day, all the children at the school were asked to throw a ball as far as possible. The distances thrown were measured and recorded. The median distance thrown by a random sample of 25 of the children was $22.0$ m. The children were asked to throw the ball again on a hot day. The distances thrown by the same 25 children were measured and recorded and these distances, in m, are shown below.

\begin{gathered} 21.2 \qquad 23.5 \qquad 22.9 \qquad 18.6 \qquad 19.4 \\ 22.1 \qquad 26.5 \qquad 20.2 \qquad 25.7 \qquad 20.6 \\ 22.3 \qquad 17.4 \qquad 22.2 \qquad 27.0 \qquad 23.9 \\ 28.2 \qquad 22.6 \qquad 27.2 \qquad 23.0 \qquad 23.7 \\ 19.8 \qquad 22.7 \qquad 23.3 \qquad 21.5 \qquad 24.3 \end{gathered}

The teacher claims that on average the distances thrown will be further when it is hot.

Carry out a Wilcoxon signed-rank test, at the 5% significance level, to test whether the data supports the teacher’s claim.

[10]

题目中文翻译

一所学校正在进行一项实验，看看儿童投球距离是否会在炎热天气中增加。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。测量并记录投掷距离。从这些儿童中随机抽取 25 人，其投掷距离的中位数为 $22.0$ m。随后，在一个炎热的日子，这些儿童被要求再次投球。同样这 25 名儿童的投掷距离被测量并记录，以下为这些距离，单位为 m。

\begin{gathered} 21.2 \qquad 23.5 \qquad 22.9 \qquad 18.6 \qquad 19.4 \\ 22.1 \qquad 26.5 \qquad 20.2 \qquad 25.7 \qquad 20.6 \\ 22.3 \qquad 17.4 \qquad 22.2 \qquad 27.0 \qquad 23.9 \\ 28.2 \qquad 22.6 \qquad 27.2 \qquad 23.0 \qquad 23.7 \\ 19.8 \qquad 22.7 \qquad 23.3 \qquad 21.5 \qquad 24.3 \end{gathered}

教师声称，在炎热天气中，平均投掷距离会更远。

在 5% 显著性水平下进行 Wilcoxon 符号秩检验，检验数据是否支持教师的声称。

(10)

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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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。

答题过程

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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

\begin{align*} &\,-7,\ 12,\ 8,\ -19,\ -18,\\ &\,1,\ 21,\ -14,\ 20,\ -11,\\ &\,3,\ -22,\ 2,\ 23,\ 15,\\ &\,25,\ 5,\ 24,\ 9,\ 13,\\ &\,-16,\ 6,\ 10,\ -4,\ 17. \end{align*}

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

z=\frac{111.5-162.5}{\sqrt{1381.25}}=-1.372\ldots.

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.

CIE 9231 2023 Nov Paper 43 Q6

题目

解答

解法一

思路

答题过程