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CIE 9231 2023 Nov Paper 41 Q1

A Level / CIE / FS

CIE 9231 2023 Nov Paper 41 Paper · Question 1

题目

Problem

Maya is an athlete who competes in 1500-metre races. Last summer her practice run times had mean 4.22 minutes. Over the winter she has done some intense training to try to improve her times. A random sample of 10 of her practice run times, xx minutes, this summer are summarised as follows.

∑x=42.05∑x2=176.83\sum x = 42.05 \qquad \sum x^2 = 176.83

Maya’s new practice run times are normally distributed. She believes that on average her times have improved as a result of her training.

Test, at the 5% significance level, whether Maya’s belief is supported by the data.

[6]
题目中文翻译

Maya 是一名参加 1500 米比赛的运动员。去年夏天，她的训练跑用时均值为 4.22 分钟。冬天期间，她进行了高强度训练，希望提高成绩。今年夏天，她随机抽取了 10 次训练跑用时，记为 xx 分钟，汇总如下。

∑x=42.05∑x2=176.83\sum x = 42.05 \qquad \sum x^2 = 176.83

Maya 新的训练跑用时服从正态分布。她认为，经过训练后，她的平均用时有所改善。

在 5% 显著性水平下检验数据是否支持 Maya 的看法。

(6)
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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用新的样本均值与去年的基准比较。跑得更快对应平均时间更短，因此做左尾单样本 tt 检验。

答题过程

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Let μ\mu be the population mean of Maya’s new practice times. Test H0 ⁣:μ=4.22H_0\colon\mu=4.22 against H1 ⁣:μ<4.22H_1\colon\mu<4.22. From the sample,

 xˉ=42.05/10=4.205, s2=176.83−42.052/109=0.00108333….\begin{align*} &\,\bar x=42.05/10=4.205,\\ &\,s^2=\frac{176.83-42.05^2/10}{9}=0.00108333\ldots. \end{align*}

The statistic is

t=4.205−4.22s2/10=−1.441….t=\frac{4.205-4.22}{\sqrt{s^2/10}}=-1.441\ldots.

For 99 degrees of freedom, the lower 5%5\% critical value is −1.833-1.833. Since −1.441…>−1.833-1.441\ldots>-1.833, we fail to reject H0H_0. There is insufficient evidence at the 5%5\% level that Maya’s population mean practice time has decreased.