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"Cement","Blast","Fly.Ash","Water","Superplasticizer","Coarse.Aggregate","Fine.Aggregate","Age","Strength" 540,0,0,162,2.5,1040,676,28,79.99 540,0,0,162,2.5,1055,676,28,61.89...

1 answer below »
"Cement","Blast","Fly.Ash","Water","Superplasticizer","Coarse.Aggregate","Fine.Aggregate","Age","Strength"
540,0,0,162,2.5,1040,676,28,79.99
540,0,0,162,2.5,1055,676,28,61.89
332.5,142.5,0,228,0,932,594,270,40.27
332.5,142.5,0,228,0,932,594,365,41.05
198.6,132.4,0,192,0,978.4,825.5,360,44.3
266,114,0,228,0,932,670,90,47.03
380,95,0,228,0,932,594,365,43.7
380,95,0,228,0,932,594,28,36.45
266,114,0,228,0,932,670,28,45.85
475,0,0,228,0,932,594,28,39.29
198.6,132.4,0,192,0,978.4,825.5,90,38.07
427.5,47.5,0,228,0,932,594,270,43.01
304,76,0,228,0,932,670,28,47.81
380,0,0,228,0,932,670,90,52.91
139.6,209.4,0,192,0,1047,806.9,90,39.36
342,38,0,228,0,932,670,365,56.14
380,95,0,228,0,932,594,90,40.56
475,0,0,228,0,932,594,180,42.62
427.5,47.5,0,228,0,932,594,180,41.84
139.6,209.4,0,192,0,1047,806.9,28,28.24
139.6,209.4,0,192,0,1047,806.9,180,44.21
380,0,0,228,0,932,670,365,52.52
380,0,0,228,0,932,670,270,53.3
380,95,0,228,0,932,594,270,41.15
342,38,0,228,0,932,670,180,52.12
427.5,47.5,0,228,0,932,594,28,37.43
475,0,0,228,0,932,594,7,38.6
304,76,0,228,0,932,670,365,55.26
198.6,132.4,0,192,0,978.4,825.5,180,41.72
475,0,0,228,0,932,594,270,42.13
190,190,0,228,0,932,670,365,53.69
237.5,237.5,0,228,0,932,594,270,38.41
237.5,237.5,0,228,0,932,594,28,30.08
332.5,142.5,0,228,0,932,594,90,37.72
475,0,0,228,0,932,594,90,42.23
237.5,237.5,0,228,0,932,594,180,36.25
342,38,0,228,0,932,670,90,50.46
427.5,47.5,0,228,0,932,594,365,43.7
237.5,237.5,0,228,0,932,594,365,39
380,0,0,228,0,932,670,180,53.1
427.5,47.5,0,228,0,932,594,90,41.54
427.5,47.5,0,228,0,932,594,7,35.08
349,0,0,192,0,1047,806.9,3,15.05
380,95,0,228,0,932,594,180,40.76
237.5,237.5,0,228,0,932,594,7,26.26
380,95,0,228,0,932,594,7,32.82
332.5,142.5,0,228,0,932,594,180,39.78
190,190,0,228,0,932,670,180,46.93
237.5,237.5,0,228,0,932,594,90,33.12
304,76,0,228,0,932,670,90,49.19
139.6,209.4,0,192,0,1047,806.9,7,14.59
475,0,0,228,0,932,594,365,41.93
198.6,132.4,0,192,0,978.4,825.5,3,9.13
304,76,0,228,0,932,670,180,50.95
332.5,142.5,0,228,0,932,594,28,33.02
304,76,0,228,0,932,670,270,54.38
266,114,0,228,0,932,670,270,51.73
310,0,0,192,0,971,850.6,3,9.87
266,114,0,228,0,932,670,180,48.7
139.6,209.4,0,192,0,1047,806.9,360,44.7
332.5,142.5,0,228,0,932,594,7,30.28
374,189.2,0,170.1,10.1,926.1,756.7,3,34.4
313.3,262.2,0,175.5,8.6,1046.9,611.8,3,28.8
425,106.3,0,153.5,16.5,852.1,887.1,3,33.4
425,106.3,0,151.4,18.6,936,803.7,3,36.3
375,93.8,0,126.6,23.4,852.1,992.6,3,29
475,118.8,0,181.1,8.9,852.1,781.5,3,37.8
469,117.2,0,137.8,32.2,852.1,840.5,3,40.2
425,106.3,0,153.5,16.5,852.1,887.1,3,33.4
388.6,97.1,0,157.9,12.1,852.1,925.7,3,28.1
531.3,0,0,141.8,28.2,852.1,893.7,3,41.3
425,106.3,0,153.5,16.5,852.1,887.1,3,33.4
401.8,94.7,0,147.4,11.4,946.8,852.1,3,41.1
362.6,189,0,164.9,11.6,944.7,755.8,3,35.3
379.5,151.2,0,153.9,15.9,1134.3,605,3,28.6
362.6,189,0,164.9,11.6,944.7,755.8,3,35.3
286.3,200.9,0,144.7,11.2,1004.6,803.7,3,24.4
362.6,189,0,164.9,11.6,944.7,755.8,3,35.3
439,177,0,186,11.1,884.9,707.9,3,39.3
389.9,189,0,145.9,22,944.7,755.8,3,40.6
362.6,189,0,164.9,11.6,944.7,755.8,3,35.3
374,189.2,0,170.1,10.1,926.1,756.7,7,46.2
425,106.3,0,153.5,16.5,852.1,887.1,7,49.2
375,93.8,0,126.6,23.4,852.1,992.6,7,45.7
475,118.8,0,181.1,8.9,852.1,781.5,7,55.6
469,117.2,0,137.8,32.2,852.1,840.5,7,54.9
425,106.3,0,153.5,16.5,852.1,887.1,7,49.2
388.6,97.1,0,157.9,12.1,852.1,925.7,7,34.9
531.3,0,0,141.8,28.2,852.1,893.7,7,46.9
425,106.3,0,153.5,16.5,852.1,887.1,7,49.2
318.8,212.5,0,155.7,14.3,852.1,880.4,7,33.4
401.8,94.7,0,147.4,11.4,946.8,852.1,7,54.1
362.6,189,0,164.9,11.6,944.7,755.8,7,55.9
379.5,151.2,0,153.9,15.9,1134.3,605,7,47.1
362.6,189,0,164.9,11.6,944.7,755.8,7,55.9
286.3,200.9,0,144.7,11.2,1004.6,803.7,7,38
362.6,189,0,164.9,11.6,944.7,755.8,7,55.9
439,177,0,186,11.1,884.9,707.9,7,56.1
389.9,189,0,145.9,22,944.7,755.8,7,59.09
337.9,189,0,174.9,9.5,944.7,755.8,7,35.1
374,189.2,0,170.1,10.1,926.1,756.7,28,61.09
313.3,262.2,0,175.5,8.6,1046.9,611.8,28,59.8
425,106.3,0,153.5,16.5,852.1,887.1,28,60.29
375,93.8,0,126.6,23.4,852.1,992.6,28,56.7
475,118.8,0,181.1,8.9,852.1,781.5,28,68.3
469,117.2,0,137.8,32.2,852.1,840.5,28,66.9
425,106.3,0,153.5,16.5,852.1,887.1,28,60.29
388.6,97.1,0,157.9,12.1,852.1,925.7,28,50.7
531.3,0,0,141.8,28.2,852.1,893.7,28,56.4
318.8,212.5,0,155.7,14.3,852.1,880.4,28,55.5
401.8,94.7,0,147.4,11.4,946.8,852.1,28,68.5
362.6,189,0,164.9,11.6,944.7,755.8,28,71.3
323.7,282.8,0,183.8,10.3,942.7,659.9,28,74.7
379.5,151.2,0,153.9,15.9,1134.3,605,28,52.2
362.6,189,0,164.9,11.6,944.7,755.8,28,71.3
286.3,200.9,0,144.7,11.2,1004.6,803.7,28,67.7
389.9,189,0,145.9,22,944.7,755.8,28,74.5
362.6,189,0,164.9,11.6,944.7,755.8,28,71.3
337.9,189,0,174.9,9.5,944.7,755.8,28,49.9
374,189.2,0,170.1,10.1,926.1,756.7,56,63.4
313.3,262.2,0,175.5,8.6,1046.9,611.8,56,64.9
425,106.3,0,153.5,16.5,852.1,887.1,56,64.3
425,106.3,0,151.4,18.6,936,803.7,56,64.9
375,93.8,0,126.6,23.4,852.1,992.6,56,60.2
475,118.8,0,181.1,8.9,852.1,781.5,56,72.3
469,117.2,0,137.8,32.2,852.1,840.5,56,69.3
425,106.3,0,153.5,16.5,852.1,887.1,56,64.3
388.6,97.1,0,157.9,12.1,852.1,925.7,56,55.2
531.3,0,0,141.8,28.2,852.1,893.7,56,58.8
425,106.3,0,153.5,16.5,852.1,887.1,56,64.3
318.8,212.5,0,155.7,14.3,852.1,880.4,56,66.1
401.8,94.7,0,147.4,11.4,946.8,852.1,56,73.7
362.6,189,0,164.9,11.6,944.7,755.8,56,77.3
323.7,282.8,0,183.8,10.3,942.7,659.9,56,80.2
362.6,189,0,164.9,11.6,944.7,755.8,56,77.3
286.3,200.9,0,144.7,11.2,1004.6,803.7,56,72.99
362.6,189,0,164.9,11.6,944.7,755.8,56,77.3
337.9,189,0,174.9,9.5,944.7,755.8,56,59.89
313.3,262.2,0,175.5,8.6,1046.9,611.8,91,66.6
425,106.3,0,153.5,16.5,852.1,887.1,91,65.2
425,106.3,0,151.4,18.6,936,803.7,91,66.7
375,93.8,0,126.6,23.4,852.1,992.6,91,62.5
475,118.8,0,181.1,8.9,852.1,781.5,91,74.19
469,117.2,0,137.8,32.2,852.1,840.5,91,70.7
425,106.3,0,153.5,16.5,852.1,887.1,91,65.2
388.6,97.1,0,157.9,12.1,852.1,925.7,91,57.6
531.3,0,0,141.8,28.2,852.1,893.7,91,59.2
425,106.3,0,153.5,16.5
concretetest-44elf3tg.csv concretetrain-rsah1xns.csv
Answered Same Day Oct 01, 2021

Solution

Pooja answered on Oct 03 2021
138 Votes
Analysis:
a)
The regression equation is given by Strength = -34.5 + 0.12521 Cement + 0.1114 Blast + 0.0982 Fly.Ash - 0.1302 Water + 0.240 Superplasticizer + 0.0197 Coarse.Aggregate + 0.0251 Fine.Aggregate + 0.10812 Age
Null hypothesis, the model is not significant. Versus alternative hypothesis that the model is significant. With F=166.81, p<5%, the null hypothesis is rejected at 5% level of significance and conclusion is that the model is significant.
The coefficient of determination is 61.8%. There is 61.8% variation in strength which is explained by all independent variables involved in this model namely Cement, Blast, Fly.Ash, Water, Superplasticizer, Coarse.Aggregate, Fine.Aggregate, and Age.
The adjusted R square is 60.8%. This implies there is 60.8% variation in strength which is explained by significant independent variables in the model.
This model tells us that for a 1 unit increase in cement, the strength is increased by 0.125. 1 unit increase in blast, the strength is increased by 0.1114. For 1 unit increase in fly ash, the strength is increased by 0.09. For 1 unit increase in water, the strength is decreased by 0.13. For 1 unit increase in Superplasticizer, the strength is increased by 0.24. For 1 unit increase in Coarse.Aggregate, the strength is increased by 0.0197. For 1 unit increase in Fine.Aggregate, the strength is increased by 0.0251. For 1 unit increase in age, the strength is increased by 0.108. 
With t=1.87, p-value greater than 5%, I can say that the insignificant variable is Coarse.Aggregate. 
The standard e
or is 10.3 which is considered low indicating that the predictions based on this model available.
)
Using the method of backward selection, the final regression equation is given by Strength = 26.95 + 0.10766 Cement + 0.09041 Blast + 0.07441 Fly.Ash - 0.2114 Water + 0.1931 Superplasticizer + 0.10736 Age
This model tells us that for a 1 unit increase in cement, the strength is increased by 0.10766. 1 unit increase in blast, the strength is increased by 0.09. For 1 unit increase in fly ash, the strength is increased by 0.07. For 1 unit increase in water, the strength is decreased by 0.2114. For 1 unit increase in Superplasticizer, the strength is increased by 0.1931. For 1 unit increase in age, the strength is increased by 0.107. All the values (coefficient of all independent variables) are significant with p-value less than 5%.
c)
The value of R squared and adjusted R Square for the initial model is 61.9% and 61.54% respectively. The value of R squared and adjusted R Square for the model using backward elimination is 61.7% and 61.42%. The number of independent variables in the initial model is 9 whereas the model obtained using backward elimination has only 7 independent variables. Even though with the same value of adjusted R square in both the models, the model using backward elimination is prefe
ed.  this is because it has low number of independent variables which saves time and cost.
d)
From the co
elation Matrix, it is evident that none of the independent variables involved in the backward elimination method strongly co
elated with each other. And there is no problem of multicollinearity in the data.
e)
from the histogram of the residuals, it is evident that the distribution of e
ors is approximately normal. Hence the assumption of normality of e
or residuals is followed.
From the residential plot all the points are randomly plotted indicating that the variance of e
ors is constant. Hence the assumption of equality of variances also satisfied.
There is no problem of multicollinearity in the data, as the value of variance inflation factor for independent variables in the model is less than 10.
References
Fox, J. (2015). Applied regression analysis and generalized linear models. Sage Publications.
Wilcox, R. R. (2011). Introduction to robust estimation and hypothesis testing. Academic press.
Tartakovsky, A., Nikiforov, I., & Basseville, M. (2014). Sequential analysis: Hypothesis testing and changepoint detection. Chapman and Hall/CRC.
Output:
Regression model
Regression Analysis: Strength versus Cement, Blast, Fly.Ash, Water, Superplastic, ...
Analysis of Variance
Source DF Adj SS Adj MS F-Value P-Value
Regression 8 142779 17847.4 166.81 0.000
Cement 1 18469 18469.4 172.62 0.000
Blast 1 10308 10307.6 96.34 0.000
Fly.Ash 1 5161 5161.3 48.24 0.000
Water 1 876 876.3 8.19 0.004
Superplasticizer 1 582 582.3 5.44 0.020
Coarse.Aggregate 1 373 373.4 3.49 0.062
...
SOLUTION.PDF
SOLUTION.PDF

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