题目

Problem

A professor claims that undergraduates studying History have a typing speed of more than 15 words per minute faster than undergraduates studying Maths.

A sample is taken of 38 undergraduates studying History and 45 undergraduates studying Maths. The typing speed, x words per minute, of each undergraduate is recorded. The results are summarised in the table below.

	$n$	$\overline x$	$s^2$
Undergraduates studying History	38	56.3	27.2
Undergraduates studying Maths	45	39.8	18.5

(a) Use a suitable test, at the 5% level of significance, to investigate the professor’s claim. State clearly your hypotheses, test statistic and critical value.

(7)

(b) State two assumptions you have made in carrying out the test in part (a).

(2)

(Total for Question 5 is 9 marks)

题目中文翻译

一位教授声称，学习历史的本科生打字速度比学习数学的本科生每分钟快 15 个词以上。

抽取了 38 名学习历史的本科生和 45 名学习数学的本科生作为样本。每位本科生的打字速度 x（每分钟词数）都被记录下来。结果汇总如下表。

	$n$	$\overline x$	$s^2$
学习历史的本科生	38	56.3	27.2
学习数学的本科生	45	39.8	18.5

(a) 在 5% 显著性水平下，使用适当的检验来调查教授的说法。请清楚写出你的原假设、检验统计量和临界值。

(b) 写出在完成 (a) 小题检验时所作的两个假设。

解答

(a)

解法一

思路

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教授声称历史系总体平均打字速度比数学系高出超过 15，因此进行右尾两样本均值差检验。以两组样本方差估计总体方差，计算 $z$ 统计量后与 $5\%$ 单尾临界值比较。

答题过程

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Let $\mu_H$ and $\mu_M$ be the population mean typing speeds of History and Maths undergraduates respectively. The hypotheses are

\begin{aligned} H_0:&\ \mu_H-\mu_M=15,\\ H_1:&\ \mu_H-\mu_M>15. \end{aligned}

The estimated standard error is

\begin{align*} \operatorname{SE}(\overline X_H-\overline X_M) =&\,\sqrt{\frac{27.2}{38}+\frac{18.5}{45}}\\ =&\,1.0615\ldots. \end{align*}

Therefore, the test statistic is

\begin{align*} z =&\,\frac{(56.3-39.8)-15} {\sqrt{\frac{27.2}{38}+\frac{18.5}{45}}}\\ =&\,1.4130\ldots\\ =&\,1.41. \end{align*}

For a one-tailed test at the $5\%$ significance level, the critical value is $1.6449$ , giving the rejection region

z\geqslant1.6449.

Since $1.41<1.6449$ , $H_0$ is not rejected. There is insufficient evidence at the $5\%$ significance level to support the professor’s claim that History undergraduates type more than 15 words per minute faster than Maths undergraduates.

(b)

解法一

思路

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检验中以两组样本方差代替相应总体方差，并依赖两组样本均值的正态近似，因此分别写出这两个针对历史系和数学系样本的假设。

答题过程

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The assumptions are:

For both History and Maths undergraduates, the sample variance is approximately equal to the corresponding population variance.
Both sample sizes are sufficiently large for the central limit theorem to apply, or the typing speeds in both populations are normally distributed.

IAL 2024 Jan S3 Q5

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程