题目

Problem

Secondary schools in a region conduct ability testing at the start of Year 7 and the start of Year 8. Each year a regional education officer randomly selects 240 Year 7 students and 240 Year 8 students from across the region. The results for last year are summarised in the table below.

	Mean score	Variance of scores
Year 7	101	38
Year 8	103	42

The regional education officer claims that there is no difference between the mean scores of these two year groups.

(a) Test the regional education officer’s claim at the 1% significance level. You should state your hypotheses, test statistic and critical value clearly.

(7)

(b) Explain the significance of the Central Limit Theorem in part (a).

(1)

(Total for Question 2 is 8 marks)

题目中文翻译

某地区的中学在 7 年级开始和 8 年级开始时进行能力测试。每年，一名地区教育官员随机从全区抽取 240 名 7 年级学生和 240 名 8 年级学生。去年的结果汇总如下表。

	平均分	分数方差
7 年级	101	38
8 年级	103	42

该地区教育官员声称，这两个年级组的平均分没有差异。

(a) 在 1% 显著性水平下检验该说法。请清楚写出原假设、检验统计量和临界值。

(b) 解释 (a) 小题中中心极限定理的意义。

解答

(a)

解法一

思路

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教育官员声称两个总体均值没有差异，所以把均值相等写作原假设；题目要检验该说法，使用双尾备择假设。两个独立样本容量均为 $240$ ，可用样本方差估计总体方差，并以标准正态分布检验样本均值之差。

答题过程

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Let $\mu_7$ and $\mu_8$ be the population mean scores for Year 7 and Year 8 students respectively.

\begin{aligned} H_0:&\ \mu_7=\mu_8,\\ H_1:&\ \mu_7\ne\mu_8. \end{aligned}

Under $H_0$ , the estimated standard error is

\begin{align*} \operatorname{SE} =&\,\sqrt{\frac{38}{240}+\frac{42}{240}}\\ =&\,0.57735\ldots \end{align*}

The test statistic is

\begin{align*} Z =&\,\frac{103-101}{0.57735\ldots}\\ =&\,3.4641\ldots \end{align*}

For a two-tailed test at the $1\%$ significance level, the critical values are

z=\pm2.5758.

Since

3.4641\ldots>2.5758,

the test statistic lies in the critical region, so $H_0$ is rejected. There is sufficient evidence at the $1\%$ significance level to suggest that the mean scores of the two year groups are different. Therefore, the regional education officer’s claim is not supported.

(b)

解法一

思路

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这里要说明中心极限定理为何使 (a) 的正态检验合理。重点是样本均值，而不是总体均值：样本量很大时，即使原总体分布未知，两个样本均值也都近似服从正态分布。

答题过程

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Since both samples are large, the Central Limit Theorem implies that the distributions of the two sample means are approximately normal, even if the underlying score distributions are not normal. This justifies the normal test used in part (a).

IAL 2022 Jan S3 Q2

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程