1. 1 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Supplies the data to confirm a hypothesis that two variables are related Provides both a visual and statistical means to test the strength of a relationship Provides a good follow-up to cause and effect diagrams Scatter Diagram * * * * * *

2. 2 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Mathematical Model Driven Quality Decisions Y = F(x) Independent variables Inputs, and In-Process Variables Cause Problem Control Input Conditions Dependent variable (s) Output (s) Effect (s) Symptom Monitor Response YX= X1... X N

3. 3 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Scatter Plot Y X Dependent Variable (Output) Independent Variable (Inputs)

4. 4 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Scatter Diagram Example volume per day cost per day 23125 26140 29146 33160 38167 42170 50188 55195 60200

5. 5 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Scatter plot Examples Y XX X X YY Y Linear RelationshipsCurvilinear Relationships

6. 6 Prof. Indrajit Mukherjee, School of Management, IIT Bombay X Y X Y X Y X Y Scatter plot Examples Strong RelationshipsWeak Relationships (Continued)

7. 7 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Scatter plot Examples X Y X Y No Relationship (Continued)

8. 8 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Types of Correlation Positive CorrelationNegative CorrelationNo Correlation

9. 9 Prof. Indrajit Mukherjee, School of Management, IIT Bombay -3 -2 -1 0 1 2 3 0 6 5 4 3 2 1 X Y A Nonlinear Relationship for Which r = 0

10. 10 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Calculation Example Tree Height Trunk Diameter yixixiyiyi2xi2 358280122564 499441240181 27718972949 336198108936 60137803600169 21714744149 45114952025121 51126122601144 32173314214111713

11. 11 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Excel Output Excel Correlation Output Tools / data analysis / correlation…. Correlation between Tree Height and Trunk Diameter

12. 12 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Explaining Attitude Toward the City of Residence Respondent number Attitude toward the city Duration of the residence 1610 2912 38 434 51012 646 758 822 91118 1099 111017 1222

13. 13 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Simple Linear Regression Describes the linear relationship between a Predictor variable, plotted on the x-axis, and a response variable, plotted on the y-axis Predictor Independent Variable (X)

14. 14 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Types of Regression Models Regression Models Simple Multiple Linear Non- Linear Non- Linear Non- Linear Non- Linear X=1 Variable X≥2 Variables

15. 15 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Only one independent variable, x Relationship between x and y is described by a linear function Changes in y are assumed to be caused by changes in x Simple Linear Regression Model

16. 16 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Parameter Estimation Table [Volume Sales /month (Y) vs. Advertising/month (X)] 11111 21412 32946 421648 54251620 1510552637

17. 17 Prof. Indrajit Mukherjee, School of Management, IIT Bombay observation numberHydrocorban numberpurity 10.9990.01 21.0289.05 31.0591.43 41.2993.74 51.4696.73 61.3694.45 70.8787.59 81.2391.77 91.5599.42 101.493.65 111.1993.54 121.1592.52 130.9890.56 141.0189.54 151.1189.85 161.290.39 171.2693.25 181.3293.41 191.4394.98 200.9587.33 Empirical Models Table Oxygen and hydrocarbon levels

18. 18 Prof. Indrajit Mukherjee, School of Management, IIT Bombay observation number Hydrocorban numberpuritypredicted valueresidual 10.9990.0189.0690090.940991 21.0289.0589.51836-0.468136 31.0591.4391.464353-0.034353 41.2993.7493.5602790.179721 51.4696.7396.1053320.624668 61.3694.4594.608242-0.158242 70.8787.5987.2725010.317499 81.2391.7792.662025-0.892025 91.5599.4297.4527131.967287 101.493.6595.207078-1.557078 111.1993.5492.0631891.476811 121.1592.5291.6140620.905938 130.9890.5688.91931.6407 141.0189.5489.3684270.171573 151.1189.8590.865571-1.015517 161.290.3992.212898-1.822898 171.2693.2593.1111520.138848 181.3293.4194.009406-0.599406 191.4394.9895.656205-0.676205 200.9587.3388.470173-1.140173 Adequacy of the Regression Model

19. 19 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Sample Data for House Price Model House Price in Rs.1000’s ( Y)Square Feet(x) 2451400 3121600 2791700 3081875 1991100 2191550 4052350 3242450 3191425 2551700

20. 20 Prof. Indrajit Mukherjee, School of Management, IIT Bombay The Data Data on sales of breadstick baskets and margaritas for 25 weeks are shown below. Breadstick weekordersmargaritas 18601330 28501350 38001290 48501350 58801360 67801250 78151275 87801250 97501160 107101140 117401140 126751080 137201140 147301150 156451020 166501000 177301200 188701380 198901390 209101380 219401400 228301250 238401250 248151245 258001250

21. 21 Prof. Indrajit Mukherjee, School of Management, IIT Bombay YearIncome(X)Retail sales(Y) 190985492 291385540 390945305 492825507 592295418 693475320 795255538 897565692 9102825871 10106626157 11110196342 12113075907 13114326124 14114496186 15116976224 16118716496 17120186718 18125236921 19120536471 20120886394 21122156555 22124946755 Check This Regression Analysis in Excel

22. 22 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Observation number Pull strength wire length Die height Observation number Pull strength wire length Die height 19.95540585600400205360 224.4581101521.654205 331.75111201617.894400 43510550176920600 525.0282951810.31585 616.8642001934.9310540 714.3823752046.5915250 89.62522144.8815290 924.3591002254.1216510 1027.583002356.6317590 1117.0844122422.136100 1237114002521.155400 1341.9512500 Multiple Linear Regression Models Example

23. 23 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Multiple Linear Regression Models Least Squares Estimation of the Parameters The method of least squares may be used to estimate the regression Coefficient, in the multiple regression model, equation suppose that n>k Observations are available, and let xij denote the ith observation or level of variable xj, the observations are (x i1,x i2,…,x ik,y i ), i=1,2,...,n and n>k It is customary to present the data for multiple regression in a table such as table. Table data for multiple regression yx1x1 x2x2 …xkxk Y1Y1 x 11 x 12 …x 1k Y2Y2 x 21 x 22 …x 2k … ynyn x n1 x n2 …x nk

24. 24 Prof. Indrajit Mukherjee, School of Management, IIT Bombay yx1x1 x2x2 …xkxk Y1Y1 x 11 x 12 …x 1k Y2Y2 x 21 x 22 …x 2k … ynyn x n1 x n2 …x nk Table data for multiple regression

25. 25 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Hypothesis Tests in Multiple Linear Regression Test for Significance of Regression Source of variation Sum of squaresDegrees of freedom Mean squareF0F0 regressionSS R kMS R MS R /MS E Error or residualSS E n-pMS E totalSS T n-1

26. 26 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Source of variation Sum of squares Degrees of freedom Mean squareF0F0 regressionSS R kMS R MS R /MS E Error or residual SS E n-pMS E totalSS T n-1 Hypothesis Tests in Multiple Linear Regression Test for Significance of Regression Source of variation Sum of squares Degrees of freedom Mean squareF0F0 regression5990.771222995.3856572.17 Error or residual 115.1735 225.2352 total 6105.9447 24

27. 27 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Observation numberPull strengthwire lengthDie heightObservation numberPull strengthwire lengthDie height 19.958.381.571411.6612.26-0.60 224.4525.60-1.151521.6515.815.84 331.7533.95-2.201617.8918.25-0.36 43596.60-1.60176964.674.33 525.0227.91-2.891810.312.34-2.04 616.8615.751.111934.9336.47-1.54 714.3812.451.932046.5946.56-0.03 89.68.401.202144.8847.06-2.18 924.3528.21-3.862254.1252.561.56 1027.527.98-0.482356.6356.310.32 1117.0818.40-1.322422.1319.982.15 123737.46-0.462521.1521.000.15 1341.9541.460.49 Multiple Linear Regression Models Example

28. 28 Prof. Indrajit Mukherjee, School of Management, IIT Bombay observationTemp(X)Feed rate(X2)Viscosity(Y) 18082256 29392340 3100102426 482122293 590112330 69982368 78182250 896102409 994122364 109112379 11397132440 1295112364 1310082404 1485122317 158692309 1687122328 Assignment (Contd) Table:-

29. 29 Prof. Indrajit Mukherjee, School of Management, IIT Bombay yearrevenuenumber of officesProfit margin() 13.9272980.75 23.6168550.71 33.3266360.66 43.0765060.61 53.0664500.7 63.1164020.72 73.2163680.77 83.2663400.74 93.4263490.9 103.4263520.82 113.4563610.75 123.5863690.77 133.6665460.78 143.7866720.84 153.8268900.79 163.9771150.7 174.0773270.68 184.2575460.72 194.4179310.55 204.4980970.63 214.784680.56 224.5897170.41 234.6989910.51 244.7191790.47 254.7893180.32

30. 30 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Neural Networks Neural Network: A collection of neurons which are interconnected. The output of one connects to several others with different strength connections. – Initially, neural networks have no knowledge. (All information is learned from experience using the network.) Neuron 1 Neuron 2 Output from Neuron2 Output from Neuron 1 Input 2 Input 3 Input 1

31. 31 Prof. Indrajit Mukherjee, School of Management, IIT Bombay observation numer Surface finishRPM Type of cutting tool observation numer Surface finishRPM Type of cutting tool 145.442253021133.5224416 242.032003021231.23212416 350.012503021337.52248416 448.752453021437.13260416 547.922353021534.7243416 647.792373021633.92238416 752.262653021732.13224416 850.522593021835.47251416 945.582213021933.49232416 1044.782183022032.29216416 Multiple Regression Modeling What to Do in Such Cases?-Check Book