Question

When testing a thermocouple pyrometer, the voltage output (E in millivolt) 2 was measured against the temperature (t in 'C) of the hot junction and the following results obtained: 203 298 390 487 570 661 753 844 931 t E 6.1 9.2 12.1 14.9 18.2 21.1 24.2 26.9 30.2 E and t are connected by equation of the type E = a + bt an Find the regression line fort on E (25 Marks) a Find the regression line for E on t. b (15 Marks) c What is the value of the correlation coefficient? (20 Marks) What would the value for E be when the temperature reaches 1000°C? d (15 Marks)

Answer 1

XValues YValues 6.1 203 1200 298 9.2 12.1 390 14.9 487 1000 18.2 570 21.1 661 24.2 753 800 26.9 844 30.2 931 600 M: M: 18.1 570.7778 400 200 6 27 13 34 20 x-Independent Regression Line (y = 30.28X+ 22.63) Estimate Calculation Summary Sum of X 162.9 Sum of Y 5137 Mean X 18.1 Mean Y 570.7778 Sum of squares (SS)= 539.52 Sum of products (SP) = 16339 Regression Equation y= bX+ a b SPISSx 16339/539.52 = 30.28433 570.78 - (30.28*18.1) 22.63146 My bMx a y = 30.28433X+ 22.63146 (X-M)2 (X- M,)Y- My) Y- My X-Mx 144 4413.3333 12 -367.7778 -272.7778 -1 80.7778 -83.7778 -0.7778 90.2222 7 -8.9 2427.7222 1084.6667 -6 36 10.24 -3.2 268.0889 0.01 0.1 -0.0778 270.6667 37.2 6.1 182.2222 273.2222 1111.5556 2404.3556 8.8 146 41 12.1 360.2222 4358.6889 SP: 16339 SS 539.52 Dependent

XValues YValues 203 6.1 35 9.2 296 30 48. 14.9 18.2 21.1 570 661 25 753 24.2 26.9 30.2 844 20 931 M: 18. 15 M: 570.7778 10 5 400 1000 200 .. 600 800 .. X-Independent Regression Line ( =0.03X-0.74) Estimate Calculation Summary Sum of X- 5137 Sum of Y= 162.9 Mean X 570.7778 Mean Y 18.1 Sum of squares (SS) = 495123.5556 Sum of products (SP) = 16339 Regression Equation y bX+ a b SPISSx 16339/495123.56 0.033 a My - bMx = 18.1 - (0.03*570.78) = -0.73558 Y 0.033X-0.73558 (X-M)2 X-M (X- MxYMy) Y- My -12 135260.4938 4413.3333 -367.7778 -272.7778 -8.9 74407.716 2427.7222 1084.6667 -6 32680.6049 -180.7778 -83.7778 -0.7778 -3.2 7018.716 268.0889 0.1 0.6049 8140,0494 33204.9383 -0.0778 270.6667 1111,5556 2404.3556 90.2222 3 182.2222 6.1 273.2222 8.8 12.1 74650.3827 129760.0494 360.2222 4358.6889 sS: SP: 16339 495123.5556 Y- Dependent

XValues YValues 203 6.1 35 9.2 296 30 48. 14.9 18.2 21.1 570 661 25 753 24.2 26.9 30.2 844 20 931 M: 18. 15 M: 570.7778 10 5 400 1000 200 .. 600 800 .. X-Independent Regression Line ( =0.03X-0.74) Estimate Calculation Summary Sum of X- 5137 Sum of Y= 162.9 Mean X 570.7778 Mean Y 18.1 Sum of squares (SS) = 495123.5556 Sum of products (SP) = 16339 Regression Equation y bX+ a b SPISSx 16339/495123.56 0.033 a My - bMx = 18.1 - (0.03*570.78) = -0.73558 Y 0.033X-0.73558 (X-M)2 X-M (X- MxYMy) Y- My -12 135260.4938 4413.3333 -367.7778 -272.7778 -8.9 74407.716 2427.7222 1084.6667 -6 32680.6049 -180.7778 -83.7778 -0.7778 -3.2 7018.716 268.0889 0.1 0.6049 8140,0494 33204.9383 -0.0778 270.6667 1111,5556 2404.3556 90.2222 3 182.2222 6.1 273.2222 8.8 12.1 74650.3827 129760.0494 360.2222 4358.6889 sS: SP: 16339 495123.5556 Y- Dependent

Bstinalt ne of t on E- t 30.28 E t 22 63 4 estmaled ern ns 4 2 0.03 O74 (2) . hen t= 1000 e Co.0 O 3 X1oUD - 0.74 29.26 milliroel

X Values Y Values 6.1 9.2 203 298 14.9 18.2 21.1 570 661 26.9 844 30.2 931 XValues (X-MY-My) 4413.333 X- M Y M (X-M2 (Y My y 367.778 -12.000 144,000 135260.494 -8.900 79.210 36.000 74407,716 2427.722 1084.667 -272.778 -6.000 -180.778 32680.605 10 010 7018 .605 26 078 100 .0 778 .000 90.222 182.222 .000 8140.049 270.667 1111.556 35204.938 6.100 12.100 146.410 129760.049 360.222 4358.689 Sum 16339,000 Sum 539,520 Mx: 18.100 My: 570.778 495123,556 Result Details & Calculation Key XX Values X Values Y Values y= 162.9 Mean 18.1 (X- My)Ss, 539.52 M. Mean of X Values M Mean of Y Values X-M, & Y-M Deviation scores y Values (X-M2 & (Y- My: Deviation Squared (X-MY- M: Product of Deviation Scores 5137 Mean 570.778 Y- My2 SS= 495123.556 Xand Y Combined N 9 2X- MXY My) = 16339 R Calculation rX-M,XY-M/V(SS,SS) r 16339/((539.52) (495123.556)) 0.9997 Meta Numerics (cross-check) r 0.9997 The value of R is 0.9997. This is a strong positive correlation, which means that high X variable scores go with high Y variable scores (and vice versa)