题目
The table shows the season’s best times, x seconds, for the 8 athletes who took part in the 200 m final in the 2021 Tokyo Olympics. It also shows their finishing position in the race.
| Athlete | A | B | C | D | E | F | G | H | | --- | --- | --- | --- | --- | --- | --- | --- | | Season’s best time | 19.89 | 19.83 | 19.74 | 19.84 | 19.91 | 19.99 | 20.13 | 20.10 | | Finishing position | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
Given that the fastest season’s best time is ranked number 1
(a) calculate the value of the Spearman’s rank correlation coefficient for these data.
(b) Stating your hypotheses clearly, test, at the 1% level of significance, whether or not there is evidence of a positive correlation between the rank of the season’s best time and the finishing position for these athletes.
Chris suggests that it would be better to use the actual finishing time, y seconds, of these athletes rather than their finishing position. Given that
(c) calculate the product moment correlation coefficient between the season’s best time and the finishing time for these athletes. Give your answer correct to 3 decimal places.
(d) Use your value of the product moment correlation coefficient to test, at the 1% level of significance, whether or not there is evidence of a positive correlation between the season’s best time and the finishing time for these athletes.
题目中文翻译
表格给出了参加 2021 年东京奥运会 200 米决赛的 8 名运动员的赛季最佳成绩 （秒），以及他们在比赛中的名次。
| 运动员 | A | B | C | D | E | F | G | H |
|---|---|---|---|---|---|---|---|---|
| 赛季最佳成绩 | 19.89 | 19.83 | 19.74 | 19.84 | 19.91 | 19.99 | 20.13 | 20.10 |
| 完赛名次 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
已知最快的赛季最佳成绩记为 1 号名次，
(a) 计算这些数据的 Spearman 等级相关系数。
(b) 清楚写出原假设，在 1% 显著性水平下检验是否有证据表明赛季最佳成绩的名次与这些运动员的完赛名次呈正相关。
Chris 认为，使用这些运动员的实际完赛时间 （秒）而不是他们的完赛名次会更好。 已知
(c) 计算赛季最佳成绩与完赛时间之间的积差相关系数，答案保留 3 位小数。
(d) 利用你在 (c) 小题求得的积差相关系数，在 1% 显著性水平下检验赛季最佳成绩与完赛时间之间是否存在正相关证据。