题目

Problem

The table below shows the distance travelled by car and the amount of commission earned by each of 8 salespersons in 2015.

Salesperson	Distance travelled (in 1000s of km)	Commission earned (in $1000s)
A	20.4	17.7
B	22.2	24.1
C	29.9	20.3
D	37.8	28.3
E	25.5	34.9
F	30.2	29.3
G	35.3	23.6
H	16.5	26.8

(a) Find Spearman’s rank correlation coefficient for these data.

(4)

(b) Stating your hypotheses clearly, test, at the 5% level of significance, whether or not there is evidence of a positive correlation between the distance travelled by car and the amount of commission earned.

(4)

(Total for Question 1 is 8 marks)

题目中文翻译

下表给出了 2015 年 8 名销售人员的汽车行驶距离和所得佣金。

销售人员	行驶距离（千公里）	所得佣金（千美元）
A	20.4	17.7
B	22.2	24.1
C	29.9	20.3
D	37.8	28.3
E	25.5	34.9
F	30.2	29.3
G	35.3	23.6
H	16.5	26.8

(a) 求这些数据的 Spearman 等级相关系数。

(b) 清楚写出假设，并在 $5\%$ 的显著性水平下检验：是否有证据表明汽车行驶距离与所得佣金之间存在正相关。

解答

(a)

解法一

思路

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分别把行驶距离和佣金按同一方向排名，再计算每名销售人员两项排名之差 $d$ 。官方评分资料允许把最大值或最小值定为第 1 名；只要两列采用相同方向， $d$ 的符号可能改变，但 $\sum d^2$ 和最终的 Spearman 等级相关系数不变。

答题过程

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Assigning rank 1 to the largest value gives

Salesperson	A	B	C	D	E	F	G	H
Rank for distance	7	6	4	1	5	3	2	8
Rank for commission	8	5	7	3	1	2	6	4
$d$	$-1$	$1$	$-3$	$-2$	$4$	$1$	$-4$	$4$
$d^2$	1	1	9	4	16	1	16	16

Therefore,

\sum d^2=1+1+9+4+16+1+16+16=64.

With $n=8$ ,

\begin{align*} r_s=&\,1-\frac{6\sum d^2}{n(n^2-1)}\\ =&\,1-\frac{6(64)}{8(8^2-1)}\\ =&\,\frac{5}{21}\\ =&\,\boxed{0.238}\quad\text{(3 s.f.)}. \end{align*}

Equivalently, assigning rank 1 to the smallest value gives distance ranks

2,\ 3,\ 5,\ 8,\ 4,\ 6,\ 7,\ 1

and commission ranks

1,\ 4,\ 2,\ 6,\ 8,\ 7,\ 3,\ 5.

This reverses every rank difference, so $\sum d^2=64$ and the same value of $r_s$ is obtained.

(b)

解法一

思路

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题目要求检验“正相关”，所以使用单尾检验。把算得的 $r_s$ 与 $n=8$ 、 $5\%$ 单尾检验的临界值 $0.6429$ 比较；由于统计量没有进入临界域，不能拒绝原假设。

答题过程

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Let $\rho_s$ denote the population Spearman rank correlation coefficient.

H_0:\rho_s=0, \qquad H_1:\rho_s>0.

For a one-tailed test with $n=8$ at the $5\%$ significance level, the critical value is

r_s=0.6429,

so the critical region is $r_s\geq0.6429$ .

Since

0.238<0.6429,

the result is not significant and $H_0$ is not rejected. There is insufficient evidence, at the $5\%$ significance level, of a positive correlation between distance travelled and commission earned.

IAL 2026 Jan S3 A Q1

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程