2.2.2 Scatter diagrams and correlation

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Question 77

A materials engineer is investigating the relationship between the cooling rate, $R$ ( $∘C/min^{\circ}\text{C/min}$ ), and the ultimate tensile strength, $S$ ( $MPa\text{MPa}$ ), of a newly developed alloy. Data is collected from 8 different sample batches and is summarised in the table below.

\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}

[You may assume that $∑R=360\sum R = 360$ , $∑S=3962\sum S = 3962$ , $∑R2=20400\sum R^2 = 20400$ , $∑RS=168330\sum RS = 168330$ and $S_{SS} = 23911.5$ ]

Calculate $S_{RS}$ and $S_{RR}$ . Give your answers to 3 significant figures.

[3]

Calculate the product moment correlation coefficient for this data.

[2]

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.

[2]

Find the equation of the regression line of $S$ on $R$ giving your answer in the form $S = a + b R$ .

[3]

Interpret the value of $b$ .

[1]

Estimate the decrease in tensile strength when moving from a cooling rate of 25 $∘C/min^{\circ}\text{C/min}$ to a cooling rate of 75 $∘C/min^{\circ}\text{C/min}$ .

[2]

2.2.2 Scatter diagrams and correlation

Question 77

\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}

[You may assume that $∑R=360\sum R = 360$ , $∑S=3962\sum S = 3962$ , $∑R2=20400\sum R^2 = 20400$ , $∑RS=168330\sum RS = 168330$ and $S_{SS} = 23911.5$ ]

Calculate $S_{RS}$ and $S_{RR}$ . Give your answers to 3 significant figures.

[3]

Calculate the product moment correlation coefficient for this data.

[2]

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.

[2]

Find the equation of the regression line of $S$ on $R$ giving your answer in the form $S = a + b R$ .

[3]

Interpret the value of $b$ .

[1]

Estimate the decrease in tensile strength when moving from a cooling rate of 25 $∘C/min^{\circ}\text{C/min}$ to a cooling rate of 75 $∘C/min^{\circ}\text{C/min}$ .

[2]

2.2.2 Scatter diagrams and correlation

Question 77

2.2.2 Scatter diagrams and correlation Questions

2.2.2 Scatter diagrams and correlation

Question 77

2.2.2 Scatter diagrams and correlation Questions