In theory, the P value for the constant could be used to determine whether the constant could be removed from the model. In practice, we do not usually do that. There are two reasons for this. Most studies are performed with the independent variable far removed from 0. While a straight line may be appropriate for the range of data values studied, the relationship may not be a straight line all the way down to values of 0 for the predictor.
Nothing is simpler than a constant.
So if a change of Y with X is to be place in a model, the constant should be included, too. It tells the story of how the regression equation accounts for variablity in the response variable. The total amount of variability in the response is the Total Sum of Squares , . If a prediction had to be made without any other information, the best that could be done, in a certain sense, is to predict every value to be equal to the sample mean. It is sometimes called the Error Sum of Squares.