题目

Problem

A teacher believes that those of her students with strong mathematical ability may also have enhanced short-term memory. She shows a random sample of 11 students a tray of different objects for eight seconds and then asks them to write down as many of the objects as they can remember. The results, along with their percentage score in a recent mathematics test, are given in the table below.

Student	A	B	C	D	E	F	G	H	I	J	K
No. of objects	8	11	9	15	17	6	10	14	12	13	5
% in maths test	30	62	57	80	75	43	65	51	48	55	32

(a) Calculate Spearman’s rank correlation coefficient for these data. Show your working clearly.

(5)

(b) Stating your hypotheses clearly, carry out a suitable test to assess the teacher’s belief. Use a 5% level of significance and state your critical value.

(3)

The teacher shows these results to her class and argues that spending more time trying to improve their short-term memory would improve their mathematical ability.

(1)

(Total 9 marks)

题目中文翻译

一位老师认为，她的学生中数学能力强的人也可能有更好的短期记忆力。她向一组随机抽取的 11 名学生展示一个装有不同物体的托盘 8 秒，然后要求他们写下自己能记住的物体数量。结果以及他们最近一次数学考试的百分制成绩如下表。

学生	A	B	C	D	E	F	G	H	I	J	K
记住物体数	8	11	9	15	17	6	10	14	12	13	5
数学考试百分比	30	62	57	80	75	43	65	51	48	55	32

(a) 求这些数据的 Spearman 秩相关系数。清楚写出你的工作过程。

(b) 清楚写出假设，做一个合适的检验来评估老师的看法。使用 5% 显著性水平，并写出临界值。

老师把这些结果告诉她的班级，并认为花更多时间去提高短期记忆会提高他们的数学能力。

解答

(a)

解法一

思路

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两组数据都没有并列值，可直接分别排名。为与官方评分资料一致，把最大值排为第 $1$ 。求每名学生的秩之差 $d$ ，再把 $\sum d^2$ 代入 Spearman 秩相关系数公式。

答题过程

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Ranking the largest value as 1 gives

Student	A	B	C	D	E	F	G	H	I	J	K
Objects rank	9	6	8	2	1	10	7	3	5	4	11
Mathematics rank	11	4	5	1	2	9	3	7	8	6	10
$d$	$-2$	2	3	1	$-1$	1	4	$-4$	$-3$	$-2$	1
$d^2$	4	4	9	1	1	1	16	16	9	4	1

Therefore,

\sum d^2=66.

Using $n=11$ ,

\begin{align*} r_s=&\,1-\frac{6\sum d^2}{n(n^2-1)}\\ =&\,1-\frac{6(66)}{11(11^2-1)}\\ =&\,1-\frac{396}{1320}\\ =&\,\boxed{0.7}. \end{align*}

(b)

解法一

思路

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老师认为两种能力呈正相关，所以使用单尾检验。把样本秩相关系数 $r_s=0.7$ 与 $n=11$ 、 $5\%$ 单尾检验的临界值比较，再用题目语境写结论。

答题过程

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

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

For $n=11$ at the $5\%$ significance level in a one-tailed test, the critical value is

0.5364.

Since

r_s=0.7>0.5364,

the result is significant, so $H_0$ is rejected. There is sufficient evidence to support the teacher’s belief that students with stronger mathematical ability tend to have better short-term memory.

(c)

解法一

思路

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检验支持的是两者存在正相关，而不是“改善记忆会导致数学能力提高”。相关关系不能单独证明因果关系，因此老师把相关解释成因果并不成立。

答题过程

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I do not agree. The data provide evidence of a positive correlation, but correlation does not imply causation. They do not show that improving short-term memory causes an improvement in mathematical ability.

IAL 2021 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