The best predicted value for the response variable is the sample mean. The reason is as follows:
Assuming the true model relating the response and the predictor is the following:
Y = β0 + β1X + error
The graph of the above equation is a straight line (also called linear regression line) with a slope β1 and intercept β0, with the error term determining the scatter of the points around the straight line.
No linear correlation between X and Y means β1 =0. Thus, under this assumption, our model is reduced to
Y = β0 + error
Thus the regression line is constant, with the error term determining the scatter of the points around the straight line.
Given a sample data from the above model, if you determine the best fitting line (same as best fitting β0) based on the data (using the least squares method), you can show the best fitting β0 is the sample mean. Thus, Yhat (the predicted Y)= Ybar (sample mean). Thus, at every value of x, the Yhat (the predicted response or y) is the same, which is Ybar (or the sample mean).
