Question

What is a simple linear regression model?   What does the value of the linear correlation coefficient tell us?

Please type the answer as I have diffculties understanding handwritten answers. Thanks.

Answer 1

Firstly, regression is a statistical measure used to study the extent of relationship between a dependent variable and one or more independent variables.It also allows us to study the form of relationship between the variables involved.

Now in Simple linear regression model, we consider the modeling between the dependent and one independent variable. When there is only one independent variable in the linear regression model, the model is generally termed as simple linear regression model.And the term linear implies , we wish to establish a linear relationship between the two variables involved.

          The model is generally of the form :

             

where,    and are regression parameters

              y is the dependent variable and x is the independent variable

              is the error due to regression

Now before establishing a linear relationship between two sets of data (x and y), it is important to answer first that "Are the variables linearly related at all ? ".Because if in real life no linear realtionship exisits and we still fit a simple linear regression model to our data then the regression is bound to give us misleading conclusions. Here comes the role of the linear correlation coefficient.It is basically an index unit free measure whose value interprets the extent of linear relationship between the two variables under study.

It is denoted by and its value lies in between -1 and +1.

By fomula,       = Covariance(X,Y) / ( StDev(X)*StDev(Y))

If    =0, it implies that no form of linear relationship exists at all.

If   =1,   it implies perfect positive linear relationship between variable ie if x increases then y also increases proporrtionally.For ex: y=x

If   =-1, it implies perfect negative linear relationship between variable ie if x increases then y decreases proporrtionally.For ex: y= -x