Lab6Yuvi
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ECOR 1010 – INTRODUCTION TO ENGINEERING
ASSIGNMENT #6
Bivariate Data Regression TO Marking TA:
Name: Grael Miller Email: graelmiller@cmail.carleton.ca
Lab Section: L7
Room: ME 2256 – Carleton University
Number of Figures and Tables (Including handwritten ones): 9
Last Date and Time of Revision: 20 November 2018
INTRODUCTION
In today’s time a Metallurgical Engineer must deal with more than just one data provided, known to be bivariate data. This lab will introduce the issues with the steel carbon content and the strength of it. This lab will dive further into the investigation of many graded steels its carbon content, and strength. We have plotted a diagram and figures out many equations to the problem. One of which was to linearize all the data points and analyse a trend to it. Same task was carried for both polynomial function one with a power of 6 and the other with a power of 3. This helped us determine
Yuvrajsinh Zala
the new prediction of the strength based on our equation. This lab will talk further about which equation should be used
for better Strength predictions. MATERIALS AND METHODS
We have conducted line of best fit on table () This will help us determine the strength of the steel when compared to it’s
original strength. This process is also called the regression of the data where we have taken the means of both the carbon content and compared it the sheer strength of the steel. The same process was executed for the data or polynomial of 6 and 3. Additionally we also calculated the interpolation and extrapolation predicted steel strength for a better report. RESULTS & DISCUSSION Tables 2 and 4 have the results stated. When we compare the predicated the strength values of the steel it can be found
that the all the values have great difference. This can provide us with the answer of the which steel and what grade type would provide us with the best carbon content. We used the y = 326.623215614041000x + 281.923594630017000, y = -1451.674147346980000x
3
+ 1913.433841886300000x
2
- 291.645226527383000x + 324.703716891106000, and y = -427275.091254234000000x
6
+ 1315861.040483760000000x
5
- 1582271.812993400000000x
4
+ 940923.544962666000000x
3
- 288750.586099818000000x
2
+ 43412.608121276500000x - 2173.281249976480000. All these three formulas are made though using the regression formula. When we analyze the graph, it shows that when moving up polynomial order the steel seems to get stronger, however there is always a curve at the very end of the graph. This can mean that the after some grade of steel it starts to lose its strength. This can result into the strength of the steel not being accurately presented. This is the reason why we tested the Extrapolation, and interpolation values is so we can determine the values of the strength of the steel and through which we can use to compare the linear and polynomial equation. This is done so we can see how our answers would become. Overall these values are all calculated by using the linear and polynomial regression. These calculations can be referred attached at the end of the report. CONCLUSIONS
The whole point of the lab is to conclude if the new predicted strength values of the steel are going to correlate with the
values of 0.33(wt%) and the value of 1.00(wt%). However, when we analyse all the graph the together, we can determine the table 5 best accurately represents the predicted the strength values. This is since when linearization the data we can see the data points not getting tampered. The linear graph provides a line of best fit, through which we can determine a proper average/ mean. This means that when the carbon content increases the stronger the steel will become. APPENDICES- FIGURES AND TABLES
2
Yuvrajsinh Zala
Table – 1 Steel Grade
Carbon Content
(Weight %)
Strength
(MPa)
1015
1020
1022
1030
1040
1050
1060
1080
1095
0.15
0.20
0.22
0.30
0.40
0.50
0.60
0.80
0.95
315
330
358
345
415
525
483
585
527
Table - 2
0.15
315
-0.30778
-116.444
0.09472716
13559.3086
35.83901235
330.917077
10105.75161
253.3533393
0.2
330
-0.25778
-101.444
0.066449383
10290.9753
26.15012346
347.2482378
7089.001221
297.5017056
0.22
358
-0.23778
-73.4444
0.056538272
5394.08642
17.46345679
353.7807021
6031.65688
17.80247506
0.3
345
-0.15778
-86.4444
0.024893827
7472.64198
13.63901235
379.9105593
2655.741317
1218.747152
0.4
415
-0.05778
-16.4444
0.003338272
270.419753
0.950123457
412.5728809
356.1359115
5.890907244
0.5
525
0.042222
93.55556
0.001782716
8752.64198
3.950123457
445.2352024
190.185006
6362.42293
0.6
483
0.142222
51.55556
0.02022716
2657.97531
7.332345679
477.897524
2157.8886
26.03526135
0.8
585
0.342222
153.5556
0.117116049
23579.3086
52.55012346
543.2221671
12494.25929
1745.38732
0.95
527
0.492222
95.55556
0.242282716
9130.8642
47.0345679
592.2156495
25847.38036
4253.080935
Carbon Content (weight %)
Strength (MPa)
x
x
Table - 3
Number of data
9
Slope
326.623215614041
0.627355556
Intercept
281.923594630017
81108.222
Correlation Coefficient
0.908388198
204.9088889
Coefficient of Determination
0.825169118
0.457777778
431.444444
0.28003472
100.6902566
TSS
81108.2222
SSR
66928.0002
SSE
14180.22203
Table - 4
3
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Yuvrajsinh Zala
Carbon Content
(%)
Measured
Strength
(MPa)
Predicted
Strength
(Linear)
Predicted
Strength
(Polynomial
Degree=3)
Predicted
Strength
(Polynomial
Degree=6)
0.15
315
330.91707
7
319.1098
311.3699
0.2
330
347.24823
78
331.2986
348.7003
0.22
358
353.78070
21
337.6945
344.0534
0.3
345
379.91055
93
370.224
337.5413
0.4
415
412.57288
09
421.2879
428.915
0.5
525
445.23520
24
475.7803
513.3153
0.6
483
477.89752
4
524.9911
487.5392
0.8
585
543.22216
71
572.728
584.4948
0.95
527
592.21564
95
529.8857
527.0709
R
2
value for the regression
0.8251691
18
0.93278806944
978
0.988114084347
108
Interpolation:
0.33
-
389.70925
58
384.6649
355.2781
Extrapolation:
1.00
-
608.54681
02
494.8182
-273.578
Table - 5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
100
200
300
400
500
600
700
f(x) = 326.62 x + 281.92
R² = 0.83
Yuvrajsinh Zala - Strength as a function of Carbon Content Carbon Content (Weight%) Strength (MPa)
4
Yuvrajsinh Zala
Table - 6
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
100
200
300
400
500
600
700
f(x) = − 427275.09 x⁶ + 1315861.02 x⁵ − 1582271.79 x⁴ + 940923.53 x³ − 288750.58 x² + 43412.61 x − 2173.28
R² = 0.99
Yuvrajsinh Zala - Strenght as function of carbon content Table - 7
5
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0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
100
200
300
400
500
600
700
f(x) = − 1451.67 x³ + 1913.43 x² − 291.65 x + 324.7
R² = 0.93
Yuvrajsinh Zala - Strength as a function of Carbon Content Carbon Content (Weignt%)
Strength (MPa)
Regression
Statistics
Multiple R
0.908388
2
R Square
0.825169
1
Adjusted R Square
0.800193
3
Standard Error
45.00828
8
Observatio
ns
9
ANOVA
df
SS
MS
F
Significanc
e F
Regression
1
66928.000
2
66928.000
2
33.038692
94
0.0006999
03
Residual
7
14180.222
03
2025.7460
04
Total
8
81108.222
22
Coefficien
ts
Standard
Error
t Stat
P-value
Lower 95%
Upper 95%
Lower
95.0%
Upper
95.0%
6
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Intercept
281.9235
9
30.029297
46
9.3882847
24
3.23842E-
05
210.91558
96
352.93159
97
210.91558
96
352.93159
97
X Variable 1
326.6232
2
56.824499
77
5.7479294
48
0.0006999
03
192.25462
54
460.99180
59
192.25462
54
460.99180
59
Table - 7
Yuvrajsinh Zala
8