题目

Problem

The table shows the season’s best times, x seconds, for the 8 athletes who took part in the 200 m final in the 2021 Tokyo Olympics. It also shows their finishing position in the race.

Athlete	A	B	C	D	E	F	G	H
Season’s best time	19.89	19.83	19.74	19.84	19.91	19.99	20.13	20.10
Finishing position	1	2	3	4	5	6	7	8

Given that the fastest season’s best time is ranked number 1

(a) calculate the value of the Spearman’s rank correlation coefficient for these data.

(4)

(b) Stating your hypotheses clearly, test, at the 1% level of significance, whether or not there is evidence of a positive correlation between the rank of the season’s best time and the finishing position for these athletes.

(4)

Chris suggests that it would be better to use the actual finishing time, y seconds, of these athletes rather than their finishing position. Given that

S_{xx}=0.1286875 \qquad S_{yy}=0.55275 \qquad S_{xy}=0.225175

(c) calculate the product moment correlation coefficient between the season’s best time and the finishing time for these athletes. Give your answer correct to 3 decimal places.

(2)

(d) Use your value of the product moment correlation coefficient to test, at the 1% level of significance, whether or not there is evidence of a positive correlation between the season’s best time and the finishing time for these athletes.

(2)

(Total for Question 2 is 12 marks)

题目中文翻译

表格给出了参加 2021 年东京奥运会 200 米决赛的 8 名运动员的赛季最佳成绩 $x$ （秒），以及他们在比赛中的名次。

运动员	A	B	C	D	E	F	G	H
赛季最佳成绩	19.89	19.83	19.74	19.84	19.91	19.99	20.13	20.10
完赛名次	1	2	3	4	5	6	7	8

已知最快的赛季最佳成绩记为 1 号名次，

(a) 计算这些数据的 Spearman 等级相关系数。

(b) 清楚写出原假设，在 1% 显著性水平下检验是否有证据表明赛季最佳成绩的名次与这些运动员的完赛名次呈正相关。

Chris 认为，使用这些运动员的实际完赛时间 $y$ （秒）而不是他们的完赛名次会更好。已知

S_{xx}=0.1286875 \qquad S_{yy}=0.55275 \qquad S_{xy}=0.225175

(d) 利用你在 (c) 小题求得的积差相关系数，在 1% 显著性水平下检验赛季最佳成绩与完赛时间之间是否存在正相关证据。

解答

(a)

解法一

思路

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先按时间由小到大给赛季最佳成绩排名，再与完赛名次逐项作差。利用 $\sum d^2$ 代入 Spearman 秩相关系数公式。

答题过程

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The ranks and squared differences are:

Athlete	SBT rank	Finishing-position rank	$d$	$d^2$
A	4	1	3	9
B	2	2	0	0
C	1	3	$-2$	4
D	3	4	$-1$	1
E	5	5	0	0
F	6	6	0	0
G	8	7	1	1
H	7	8	$-1$	1

Therefore,

\sum d^2=16.

With $n=8$ ,

\begin{align*} r_s =&\,1-\frac{6\sum d^2}{n(n^2-1)}\\ =&\,1-\frac{6(16)}{8(8^2-1)}\\ =&\,0.80952\ldots\\ =&\,\boxed{0.810}. \end{align*}

(b)

解法一

思路

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题目检验正相关，因此使用右尾 Spearman 秩相关检验。将 (a) 的 $r_s$ 与 $n=8$ 、 $1\%$ 单尾临界值比较。

答题过程

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Let $\rho$ be the population rank correlation coefficient between the rank of season’s best time and finishing position.

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

For $n=8$ at the $1\%$ significance level in a one-tailed test, the critical value is

r_s=0.8333.

Since

0.8095\ldots<0.8333,

the observed value does not lie in the critical region. Therefore, $H_0$ is not rejected. There is insufficient evidence at the $1\%$ significance level of a positive correlation between the rank of season’s best time and finishing position for these athletes.

(c)

解法一

思路

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积差相关系数由 $S_{xy}/\sqrt{S_{xx}S_{yy}}$ 计算。题目已经给出三个修正平方和与乘积和，直接代入并按要求保留 3 位小数。

答题过程

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The product moment correlation coefficient is

\begin{align*} r =&\,\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}\\ =&\,\frac{0.225175} {\sqrt{0.1286875(0.55275)}}\\ =&\,0.84428\ldots\\ =&\,\boxed{0.844}. \end{align*}

(d)

解法一

思路

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继续进行右尾检验，但这次检验的是总体积差相关系数。将 (c) 的 $r$ 与 $n=8$ 、 $1\%$ 单尾 PMCC 临界值比较。

答题过程

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Let $\rho$ be the population product moment correlation coefficient between season’s best time and finishing time.

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

For $n=8$ at the $1\%$ significance level in a one-tailed test, the critical value is

r=0.7887.

Since

0.844>0.7887,

the observed value lies in the critical region, so $H_0$ is rejected. There is sufficient evidence at the $1\%$ significance level of a positive correlation between season’s best time and finishing time for these athletes.

IAL 2023 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

(c)

解法一

思路

答题过程

(d)

解法一

思路

答题过程