An engineering team is testing the flight endurance of 8 different commercial drones. The table below shows the initial battery charge level, xxx percentage, and the final endurance rank (where rank 1 is the drone that stayed in the air the longest) for each drone during a standardised test flight.
| Drone | A | B | C | D | E | F | G | H |
|---|---|---|---|---|---|---|---|---|
| Initial Charge (xxx %) | 98.5 | 94.2 | 96.8 | 92.0 | 88.5 | 90.2 | 85.4 | 82.0 |
| Endurance Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
Given that the highest initial charge level is ranked number 1:
Calculate the Spearman's rank correlation coefficient for these data.
Stating your hypotheses clearly, test at the 5% level of significance whether there is evidence of a positive correlation between the rank of the initial battery charge and the endurance rank.
An analyst suggests using the actual flight durations, yyy minutes, instead of ranks. For the 8 drones, it is given that:
Sxx=220.36,Syy=150.0,Sxy=170.87 S_{xx} = 220.36, \quad S_{yy} = 150.0, \quad S_{xy} = 170.87 Sxx=220.36,Syy=150.0,Sxy=170.87Calculate the product moment correlation coefficient (PMCC) between the initial battery charge and the flight duration. Give your answer to 3 decimal places.
Use your PMCC value to test at the 5% level of significance whether there is evidence of a positive correlation between the initial battery charge and the flight duration.