What is the difference between the regression equation\(\hat y = {b_0} + {b_1}x\)and the regression equation\(y = {\beta _0} + {\beta _1}x\)?

The two regression equations are provided as,

\(\hat y = {b_0} + {b_1}x\)

\(y = {\beta _0} + {\beta _1}x\)

The regression equation is given as,

\(\hat y = {b_0} + {b_1}x\)

Where,

\(\hat y\)is the predicted value of the dependent variable,\({b_0}\)is the y-intercept for the regression equation,\({b_1}\)is the slope of the regression equation and x is the independent variable.

For, the regression equation,

\(y = {\beta _0} + {\beta _1}x\)

Where,

y is the dependent variable,\({\beta _0}\)is thepopulation parameter for the y-intercept for the regression equation,\({\beta _1}\)is thepopulation slope parameter for the independent variable x.

From the provided regression equations, it can be observed that \(\hat y = {b_0} + {b_1}x\) is the regression equation based on sample statistics while \(y = {\beta _0} + {\beta _1}x\) is the regression equation for the population based on parameters.

Also, the resultant values provide thepredicted value of y \(\left( {\hat y} \right)\)and the observed value of y (y).