BUS606: Operations and Supply Chain Management

To test to see if the regression equation really helps, see how much of the error that would be made using the mean of all of the $y$'s to predict is eliminated by using the regression equation to predict. By testing to see if the regression helps predict, you are testing to see if there is a functional relationship in the population.

Imagine that you have found the mean price of the apartments in our sample, and for each apartment, you have made the simple prediction that price of apartment will be equal to the sample mean, $\bar{y}$. This is not a very sophisticated prediction technique, but remember that the sample mean is an unbiased estimator of population mean, so on average you will be right. For each apartment, you could compute your error by finding the difference between your prediction (the sample mean, $\bar{y}$) and the actual price of an apartment.

As an alternative way to predict the price, you can have a computer find the intercept,$\alpha,$, and slope, $\beta$, of the sample regression line. Now, you can make another prediction of how much each apartment in the sample may be worth by computing:

$\hat{y}=\alpha+\beta(\text { distance })$

Once again, you can find the error made for each apartment by finding the difference between the price of apartments predicted using the regression equation $\hat{y}$, and the observed price, $\bar{y}$. Finally, find how much using the regression improves your prediction by finding the difference between the price predicted using the mean, $\bar{y}$, and the price predicted using regression, $\hat{y}$. Notice that the measures of these differences could be positive or negative numbers, but that error or improvement implies a positive distance.