A materials engineer is investigating the relationship between the cooling rate, RRR (∘C/min^{\circ}\text{C/min}∘C/min), and the ultimate tensile strength, SSS (MPa\text{MPa}MPa), of a newly developed alloy. Data is collected from 8 different sample batches and is summarised in the table below.
R1020304050607080S582545535502495458435410 \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline R & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 \\ \hline S & 582 & 545 & 535 & 502 & 495 & 458 & 435 & 410 \\ \hline \end{array} RS1058220545305354050250495604587043580410[You may assume that ∑R=360\sum R = 360∑R=360, ∑S=3962\sum S = 3962∑S=3962, ∑R2=20400\sum R^2 = 20400∑R2=20400, ∑RS=168330\sum RS = 168330∑RS=168330 and SSS=23911.5S_{SS} = 23911.5SSS=23911.5]
Calculate SRSS_{RS}SRS and SRRS_{RR}SRR. Give your answers to 3 significant figures.
Calculate the product moment correlation coefficient for this data.
State whether or not your value supports the use of a linear regression equation to predict tensile strength based on cooling rates. Give a reason for your answer.
Find the equation of the regression line of SSS on RRR giving your answer in the form S=a+bRS = a + bRS=a+bR.
Interpret the value of bbb.
Estimate the decrease in tensile strength when moving from a cooling rate of 25 ∘C/min^{\circ}\text{C/min}∘C/min to a cooling rate of 75 ∘C/min^{\circ}\text{C/min}∘C/min.
