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

4 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Scatter Diagram Example volume per day cost per day

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

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

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

9 Prof. Indrajit Mukherjee, School of Management, IIT Bombay X Y A Nonlinear Relationship for Which r = 0

10 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Calculation Example Tree Height Trunk Diameter yixixiyiyi2xi

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 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

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 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 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 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Parameter Estimation Table [Volume Sales /month (Y) vs. Advertising/month (X)]

17 Prof. Indrajit Mukherjee, School of Management, IIT Bombay observation numberHydrocorban numberpurity Empirical Models Table Oxygen and hydrocarbon levels

18 Prof. Indrajit Mukherjee, School of Management, IIT Bombay observation number Hydrocorban numberpuritypredicted valueresidual Adequacy of the Regression Model

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)

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

21 Prof. Indrajit Mukherjee, School of Management, IIT Bombay YearIncome(X)Retail sales(Y) Check This Regression Analysis in Excel

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 Multiple Linear Regression Models Example

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 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 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 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 regression Error or residual total

27 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Observation numberPull strengthwire lengthDie heightObservation numberPull strengthwire lengthDie height Multiple Linear Regression Models Example

28 Prof. Indrajit Mukherjee, School of Management, IIT Bombay observationTemp(X)Feed rate(X2)Viscosity(Y) Assignment (Contd) Table:-

29 Prof. Indrajit Mukherjee, School of Management, IIT Bombay yearrevenuenumber of officesProfit margin()

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 Prof. Indrajit Mukherjee, School of Management, IIT Bombay observation numer Surface finishRPM Type of cutting tool observation numer Surface finishRPM Type of cutting tool Multiple Regression Modeling What to Do in Such Cases?-Check Book