A physics instructor investigates the relationship between the time, t t\,t hours, spent by 8 students on an independent research project and their final ranking, rrr, in a national competition. The data are recorded in the table below, with rank 1 representing the highest level of achievement.
| Student | Research Time (ttt) | Time Rank (xxx) | Competition Rank (rrr) |
|---|---|---|---|
| A | 12 | 7 | 6 |
| B | 45 | 2 | 3 |
| C | 30 | 5 | |
| D | 18 | 7 | |
| E | 55 | 1 | 1 |
| F | 8 | 8 | |
| G | 22 | 4 | |
| H | 38 | 3 | 2 |
Complete the ranks for the research time (assigning rank 1 to the longest time) and calculate the Spearman's rank correlation coefficient for this data.
Stating your hypotheses clearly, test at the 5% level of significance whether there is a positive correlation between the time spent on research and the competition performance. (The critical value for n=8n=8n=8 at the 5% significance level is 0.6429).