题目

Problem

The table shows the annual tea consumption, t (kg/person), and population, p (millions), for a random sample of 7 European countries.

Country	A	B	C	D	E	F	G
Annual tea consumption, t (kg/person)	0.27	0.15	0.42	0.06	1.94	0.78	0.44
Population, p (millions)	5.4	5.8	9	10.2	67.9	17.1	8.7

(You may use $S_{tt}=2.486$ , $S_{pp}=3026.234$ and $S_{pt}=83.634$ .)

Angela suggests using the product moment correlation coefficient to calculate the correlation between annual tea consumption and population.

(a) Use Angela’s suggestion to test, at the 5% level of significance, whether or not there is evidence of any correlation between annual tea consumption and population. State your hypotheses clearly and the critical value used.

(5)

Johan suggests using Spearman’s rank correlation coefficient to calculate the correlation between the rank of annual tea consumption and the rank of population.

(b) Calculate Spearman’s rank correlation coefficient between the rank of annual tea consumption and the rank of population.

(4)

(c) Use Johan’s suggestion to test, at the 5% level of significance, whether or not there is evidence of a positive correlation between annual tea consumption and population. State your hypotheses clearly and the critical value used.

(3)

(Total for Question 3 is 12 marks)

题目中文翻译

下表给出了 7 个欧洲国家的随机样本中，年茶消费量 t（kg/人）与人口 p（百万）。

国家	A	B	C	D	E	F	G
年茶消费量 t（kg/人）	0.27	0.15	0.42	0.06	1.94	0.78	0.44
人口 p（百万）	5.4	5.8	9	10.2	67.9	17.1	8.7

（你可以使用 $S_{tt}=2.486$ 、 $S_{pp}=3026.234$ 和 $S_{pt}=83.634$ 。）

Angela 建议使用积差相关系数来计算年茶消费量与人口之间的相关性。

(a) 按照 Angela 的建议，在 5% 显著性水平下检验年茶消费量与人口之间是否有相关性的证据。请清楚写出你的原假设和所用的临界值。

Johan 建议使用 Spearman 等级相关系数来计算年茶消费量排名与人口排名之间的相关性。

(b) 计算年茶消费量排名与人口排名之间的 Spearman 等级相关系数。

解答

(a)

解法一

思路

展开

“是否有相关”对应双尾检验。先用题给的三个修正平方和与乘积和计算样本积差相关系数 $r$ ，再将 $|r|$ 与 $n=7$ 、 $5\%$ 双尾检验的临界值比较。

答题过程

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Let $\rho$ be the population product moment correlation coefficient. The hypotheses are

\begin{aligned} H_0:&\ \rho=0,\\ H_1:&\ \rho\neq0. \end{aligned}

The sample product moment correlation coefficient is

\begin{align*} r =&\,\frac{S_{pt}}{\sqrt{S_{pp}S_{tt}}}\\ =&\,\frac{83.634} {\sqrt{(3026.234)(2.486)}}\\ =&\,0.9642\ldots\\ =&\,0.964\quad\text{(3 s.f.)}. \end{align*}

For $n=7$ , the critical value for a two-tailed test at the $5\%$ significance level is $0.7545$ . Since

|0.964|>0.7545,

$H_0$ is rejected. There is evidence at the $5\%$ significance level of a correlation between annual tea consumption and population.

(b)

解法一

思路

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对茶消费量和人口都按由大到小排秩，逐个国家计算秩差 $d$ 。没有并列数据，所以可直接使用标准 Spearman 公式。

答题过程

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Using rank 1 for the largest value,

Country	A	B	C	D	E	F	G
Tea rank	5	6	4	7	1	2	3
Population rank	7	6	4	3	1	2	5
$d$	$-2$	0	0	4	0	0	$-2$
$d^2$	4	0	0	16	0	0	4

Therefore,

\sum d^2=4+0+0+16+0+0+4=24.

Hence

\begin{align*} r_s =&\,1-\frac{6\sum d^2}{n(n^2-1)}\\ =&\,1-\frac{6(24)}{7(7^2-1)}\\ =&\,\frac47\\ =&\,\boxed{0.571}\quad\text{(3 s.f.)}. \end{align*}

(c)

解法一

思路

展开

Johan 检验的是正等级相关，因此使用单尾假设。把 (b) 的 $r_s$ 与 $n=7$ 、 $5\%$ 单尾检验临界值比较。

答题过程

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Let $\rho_s$ be the population Spearman rank correlation coefficient. The hypotheses are

\begin{aligned} H_0:&\ \rho_s=0,\\ H_1:&\ \rho_s>0. \end{aligned}

For $n=7$ , the critical value for a one-tailed test at the $5\%$ significance level is $0.7143$ . Since

0.571<0.7143,

$H_0$ is not rejected. There is insufficient evidence at the $5\%$ significance level of a positive correlation between annual tea consumption and population.

IAL 2024 Jan S3 Q3

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程

(c)

解法一

思路

答题过程