题目

Problem

The table shows the time, in seconds, of the fastest qualifying lap for 10 different Formula One racing drivers and their finishing position in the actual race.

Driver	A	B	C	D	E	F	G	H	I	J
Fastest qualifying lap	62.94	63.92	63.63	62.95	63.97	63.87	64.31	64.64	65.18	64.21
Finishing position	1	2	3	4	5	6	7	8	9	10

(a) Calculate the value of 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 fastest qualifying lap time and finishing position for these Formula One racing drivers.

(4)

(Total 8 marks)

题目中文翻译

下表给出了 10 位不同的一级方程式赛车手最快排位赛圈速的时间（单位：秒），以及他们在正式比赛中的完赛名次。

车手	A	B	C	D	E	F	G	H	I	J
最快排位赛圈速	62.94	63.92	63.63	62.95	63.97	63.87	64.31	64.64	65.18	64.21
完赛名次	1	2	3	4	5	6	7	8	9	10

(a) 计算这些数据的斯皮尔曼等级相关系数。

(b) 清楚写出假设，在 1% 显著性水平下检验：对于这些一级方程式赛车手，最快排位赛圈速与完赛名次之间是否有正相关证据。

解答

(a)

解法一

思路

展开

完赛名次已经是 $1$ 至 $10$ 。把最快排位赛圈速按时间由短到长排名，使最快时间为第 $1$ 名；再逐项求秩差 $d$ 和 $d^2$ ，代入 Spearman 秩相关系数公式。

答题过程

展开

Ranking the fastest qualifying lap times from shortest to longest gives:

Driver	A	B	C	D	E	F	G	H	I	J
Lap-time rank	1	5	3	2	6	4	8	9	10	7
Finishing rank	1	2	3	4	5	6	7	8	9	10
$d$	0	3	0	$-2$	1	$-2$	1	1	1	$-3$
$d^2$	0	9	0	4	1	4	1	1	1	9

Therefore,

\sum d^2=30.

With $n=10$ ,

\begin{align*} r_s =&\,1-\frac{6\sum d^2}{n(n^2-1)}\\ =&\,1-\frac{6(30)}{10(10^2-1)}\\ =&\,0.818181\ldots\\ \approx&\,\boxed{0.818}. \end{align*}

(b)

解法一

思路

展开

题目检验是否有正相关证据，因此采用右尾 Spearman 秩相关检验。把 (a) 的 $r_s$ 与 $n=10$ 、 $1\%$ 单尾临界值比较，并在结论中明确提到最快圈速与完赛名次。

答题过程

展开

Let $\rho_s$ be the population rank correlation coefficient between fastest qualifying lap time and finishing position.

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

For $n=10$ at the $1\%$ significance level in a one-tailed test, the critical value is

r_s=0.7455.

Since

0.818181\ldots>0.7455,

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 fastest qualifying lap time and finishing position for these Formula One racing drivers.

IAL 2022 Jan S3 UNUSED 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