题目

Problem

The table below shows the number of televised tournaments won and the total number of tournaments won by the top 10 ranked darts players in 2020

Player’s rank	Televised tournaments won	Total tournaments won
1	55	135
2	7	33
3	5	17
4	2	14
5	4	9
6	2	5
7	9	36
8	0	15
9	3	3
10	0	13

Michael did not want to calculate Spearman’s rank correlation coefficient between player’s rank and the rank in televised tournaments won because there would be tied ranks.

(a) Explain how Michael could have dealt with these tied ranks.

(1)

Given that the largest number of total tournaments won is ranked number 1

(b) calculate the value of Spearman’s rank correlation coefficient between player’s rank and the rank in total tournaments won.

(4)

(c) Stating your hypotheses and critical value clearly, test at the 5% level of significance, whether or not there is evidence of a positive correlation between player’s rank and the rank in total tournaments won for these darts players.

(4)

Michael does not believe that there is a positive correlation between player’s rank and the rank in total number of tournaments won.

(d) Find the largest level of significance, that is given in the tables provided, which could be used to support Michael’s claim. You must state your critical value.

(1)

(Total for Question 1 is 10 marks)

题目中文翻译

下表显示了 2020 年排名前 10 的飞镖选手赢得的电视转播赛事场数和赛事总胜场数。

选手排名	电视转播赛事胜场数	赛事总胜场数
1	55	135
2	7	33
3	5	17
4	2	14
5	4	9
6	2	5
7	9	36
8	0	15
9	3	3
10	0	13

Michael 不想计算选手排名与电视转播赛事胜场数排名之间的 Spearman 等级相关系数，因为会出现并列名次。

(a) 说明 Michael 可以如何处理这些并列名次。

已知赛事总胜场数最多的记为 1 号名次，

(b) 计算选手排名与赛事总胜场数排名之间的 Spearman 等级相关系数。

Michael 不认为选手排名与赛事总胜场数排名之间存在正相关。

(d) 求可用于支持 Michael 观点的、表中给出的最大显著性水平。你必须写出临界值。

解答

(a)

解法一

思路

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遇到并列数据时，不是放弃 Spearman 秩相关，而是把这些数据本来占据的名次取平均，并把该平均名次分配给每个并列数据。

答题过程

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Each tied value should be assigned the average of the ranks that the tied values would have occupied.

(b)

解法一

思路

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按赛事总胜场数由大到小排名，再与题目给出的选手排名逐项作差。求出 $\sum d^2$ 后代入 Spearman 秩相关系数公式。

答题过程

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The ranks and squared differences are:

Player’s rank	Total-tournaments rank	$d$	$d^2$
1	1	0	0
2	3	$-1$	1
3	4	$-1$	1
4	6	$-2$	4
5	8	$-3$	9
6	9	$-3$	9
7	2	5	25
8	5	3	9
9	10	$-1$	1
10	7	3	9

Therefore,

\sum d^2=68.

With $n=10$ ,

\begin{align*} r_s =&\,1-\frac{6\sum d^2}{n(n^2-1)}\\ =&\,1-\frac{6(68)}{10(10^2-1)}\\ =&\,0.58787\ldots\\ =&\,\boxed{0.588}. \end{align*}

(c)

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 (b) 的 $r_s$ 与 $n=10$ 、 $5\%$ 单尾临界值比较。

答题过程

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Let $\rho$ be the population rank correlation coefficient between player rank and total-tournaments rank.

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

For $n=10$ at the $5\%$ significance level in a one-tailed test, the critical value is

r_s=0.5636.

Since

0.5879\ldots>0.5636,

the observed value lies in the critical region, so $H_0$ is rejected. There is sufficient evidence at the $5\%$ significance level of a positive correlation between player rank and total-tournaments rank for these darts players.

(d)

解法一

思路

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若要支持 Michael“不存在正相关证据”的观点，需要选取一个使样本系数不落入临界域的显著性水平。由表中逐级检查临界值， $2.5\%$ 单尾临界值为 0.6485，而 $0.5879<0.6485$ ；更大的 $5\%$ 水平却会拒绝原假设。

答题过程

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At the $2.5\%$ significance level for a one-tailed test with $n=10$ , the critical value is

r_s=0.6485.

Since

0.5879\ldots<0.6485,

the result is not significant at this level. As the result is significant at the next larger tabulated level, $5\%$ , the largest tabulated significance level that supports Michael’s claim is

\boxed{2.5\%}.

IAL 2022 June S3 Q1

题目

解答

(a)

解法一

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(b)

解法一

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(c)

解法一

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(d)

解法一

思路

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