The temperature of heat treatment, TTT in ∘C^\circ\text{C}∘C, and the resulting tensile strength, SSS in MPa, of a random sample of 8 experimental alloy specimens are shown in the table below.
| Specimen | P | Q | R | S | T | U | V | W |
|---|---|---|---|---|---|---|---|---|
| Temperature (TTT) | 200 | 250 | 300 | 350 | 400 | 450 | 500 | 550 |
| Strength (SSS) | 420 | 410 | 460 | 450 | 490 | 480 | 530 | 510 |
Given that STT=105 000S_{TT} = 105\,000STT=105000, SSS=12 287.5S_{SS} = 12\,287.5SSS=12287.5 and STS=33 250S_{TS} = 33\,250STS=33250, calculate, to 3 decimal places, the product moment correlation coefficient (PMCC) between heat treatment temperature and tensile strength.
Using a 5% significance level, test whether there is evidence of a positive correlation between treatment temperature and tensile strength. State your hypotheses clearly.
An apprentice, Jonas, inspects the 8 specimens and ranks them by 'expected durability' based on surface finish. His ranking from least durable to most durable is:
Q,R,P,T,S,V,W,U Q, R, P, T, S, V, W, U Q,R,P,T,S,V,W,UFind, to 3 decimal places, Spearman's rank correlation coefficient between Jonas's durability ranks and the actual tensile strength rankings.
Use your value from part (c) to test whether Jonas has a significant ability to rank the specimens by strength at a 5% significance level. State your hypotheses clearly.
