Topic 8 Correlation and Regression Analysis Business Statistics Topic 8 Correlation and Regression Analysis

Business Statistics:Topic 8 Learning Objectives By the end of this topic you will be able to: Measure the strength of association using correlation analysis Use Linear Regression models Set up the linear regression equation Make predictions using the regression equation Test goodness of fit of the regression model Business Statistics:Topic 8

Business Statistics:Topic 8 Correlation analysis A statistical technique used to measure the strength of the relationship (correlation) between two variables. Business Statistics:Topic 8

Business Statistics:Topic 8 Examples Is there any relationship between height & weight? Is there any relationship between housing loan interest rates and the number of housing loans? Business Statistics:Topic 8

Correlation coefficient The index which defines the strength of the association between two given variables Population correlation coefficient  (Rho) Sample correlation coefficient r is used to estimate  Business Statistics:Topic 8

Choosing Variables for Assessing Dependent Variable is the one whose value is to be predicted - usually denoted by ‘Y’. Independent Variable is the one whose value is used to make the prediction- usually denoted by ‘X’. Assumption: The two variables are bivariate normally distributed. Business Statistics:Topic 8

Business Statistics:Topic 8 Scatter Plot A scatter diagram is a display in which the ordered pairs of measurements are plotted on a coordinate axes. The chart portrays the relationship between the two variables of interest. Plot of all (Xi, Yi) pairs Suggests how the the two variables are related Business Statistics:Topic 8

Business Statistics:Topic 8 Example A well known company wants to assess: How are ‘advertising cost’ and ‘sales’ related? What is the dependent variable? ‘Sales’ chosen as ‘Y’ for the analysis Advertising cost chosen as ‘X’ for the analysis Business Statistics:Topic 8

Business Statistics:Topic 8 What Do you Notice? Advertising and sales move together (positive) The relationship appears to be linear Business Statistics:Topic 8

Business Statistics:Topic 8 Scatter Plots Positive Linear Relationship Negative Linear Relationship Non- Linear Relationship No Relationship Business Statistics:Topic 8

Sample Correlation Coefficient ‘r’ Business Statistics:Topic 8

Properties of the correlation coefficient r always lies between -1 and +1. r = 1, the two variables have perfect positive correlation. On a scatter plot, this means that the points all lie on a straight line with positive slope. r = -1, the two variables have perfect negative correlation. On a scatter plot, this means that the points all lie on a straight line with negative slope Business Statistics:Topic 8

Business Statistics:Topic 8 Properties Closer to +1 strong positive linear relation Closer to –1 strong negative linear relation Closer to 0 weak linear relationship Business Statistics:Topic 8

Scatter plot & ‘r’ values Business Statistics:Topic 8

Population Correlation Coefficient  Is used to measure the strength of relationship between any two variables. The sample correlation coefficient ‘r’ is an estimate of  Business Statistics:Topic 8

Testing for Correlation Hypothesis H0:  = 0 (There is no correlation) H1:   0 (There is correlation) Rule Two sided ‘t’ test with  = .05 and degrees of freedom n-2: t,n-2 Test Statistic: Business Statistics:Topic 8

Example: WOM Insurance company An insurance company manager is concerned with the health of female adults, since the company is prepared to give a reduced premium rate to those who have a certain level of fitness. In particular, he would like to investigate how their height is related to their weight, with a view to possibly using these measurements as a fitness criterion. He selects a random sample of 12 adult females and measures both their height and weight. The results are shown in the following table: Business Statistics:Topic 8

Example: Assessing Height & Weight Since the value of ‘r’ is close to 1 (.82), there is a strong positive linear relationship between ‘height’ and ‘weight’. r= 0.82 is an estimate of  Business Statistics:Topic 8

Testing Correlation: Height & Weight Test Statistic Critical Value using  = .05 and degrees of freedom n-2 is 2.228. Decision: Reject H0 Conclusion: There is evidence that there is a correlation between Height & Weight Business Statistics:Topic 8

Business Statistics:Topic 8 Regression Analysis Regression Analysis is used to establish a mathematical equation between two related variables. Predict variable (Y) based on the independent variable(X) Explain the effect of the independent variable (X) on the dependent variable (Y) Business Statistics:Topic 8

Sample Linear Regression Model Y is the dependent variable X is the independent variable is the sample Y-intercept is the sample slope coefficient e is the residual or error Business Statistics:Topic 8

Sample Linear Regression Function: SRF Is used for predicting the Y value for a given X Business Statistics:Topic 8

Sample Regression coefficients is the value of Y when the value of X is zero. measures the change in the value of Y as a result of a one-unit change in X. Business Statistics:Topic 8

Sample Regression Coefficients Business Statistics:Topic 8

Example: WOM Insurance Company = b0 + b1 x = = -106 + 1.05X Business Statistics:Topic 8

Business Statistics:Topic 8 Slope The slope b1 means that for each one unit of increase in X, the estimated Y value increases by 1.05 units In this example, if the height increases by 1 cm, then the estimated weight increases by 1.05 kg. Business Statistics:Topic 8

Coefficient of Determination Measures the percentage of variation in Y that is explained by the independent variable X Business Statistics:Topic 8

Example Height & Weight r2 = 0.67 67% of the variation in the weight of people can be explained by the variability in the height of people Also if r2 is close to 1, then the linear model fits the data well. Business Statistics:Topic 8

Business Statistics:Topic 8 Estimation Using SRF To estimate the weight if a person is 170 cm tall, Put x=170, in the SRF r2 = 0.67 As r2 is close to 1, the linear model fits in the data well. Hence the estimation is good = b0 + b1 x = = -106 + 1.05X = -106 + 1.05*170  72 kg Business Statistics:Topic 8

Estimated Population Linear Regression Equation Y is the dependent variable X is the independent variable is the population Y-intercept is the population slope coefficient is a point estimate of is a point estimate of Business Statistics:Topic 8

Estimated Regression Equation Height & Weight Business Statistics:Topic 8

Business Statistics:Topic 8 Testing the Slope Hypothesis H0: 1 = 0 (no linear dependence) H1: 1  0 (linear dependence) Rule Two sided ‘t’ test with  = .05 and degrees of freedom n-2: t,n-2 Test Statistic: Business Statistics:Topic 8

Testing the slope: Height & Weight Model Test statistic Critical value with  = .05 and degrees of freedom n-2=10 is 2.228 Decision: Reject H0 Conclusion: Evidence that height & weight have a linear relationship Business Statistics:Topic 8

Business Statistics:Topic 8 Summary In this topic you have discussed: Assessing the strength of relationship between variables The Linear Regression model Estimation using the linear regression equation Testing the estimation model Business Statistics:Topic 8