题目
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.
(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.
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.
(c) Explain whether or not you agree with the teacher’s argument.
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。
(c) 说明你是否同意老师的这个说法。