The owner of an intracity moving company typically has his most experienced manager predict the total number of labor hours that will be required to complete an upcoming move. This approach has proved useful in the past, but he would like to be able to develop a more accurate method of predicting labor hours by using the amount of cubic feet moved. In a preliminary effort to provide a more accurate method, he has collected data for 36 moves in which the origin and destination were within the borough of Manhattan in New York City, and the travel time was an insignificant portion of the hours worked. See Excel spreadsheet.
a. Construct a scatter diagram.
b. Assuming a linear relationship, use the least-squares method to find the regression coefficients bo and b1.
c. Interpret the meaning of the slope b1 in this problem.
d. Predict the mean labor hours for moving 500 cubic feet.
Solutions: a. Using EXCEL we have the following scatter diagram
We notice a reasonable degree of linear relationship on this sample. To conclude more precisely the level of linear relationship, we must compute , the coefficient which represents the degree of linear association. Using EXCEL again we get
which gives us a strong linear association between moving time and cubic feet moved.
b. Now, we are looking for a linear model of the form
where represents the cubic feet moved, represents the moving time, and represents the predicted cubic feet moved. Using least-squares method we get
Using EXCEL we obtain
We find that
Our model is therefore
c. The interpretation of is the following:
corresponds to an estimate of the real population parameter which appears in assumed linear relationship between the moving time and the cubic feet moved. In this case, , which means that in average, for each extra cubic foot of moved material, we have extra 0.05008 hours of labor hours for moving, in average.
d. In order to predict the mean labor hours for moving 500 cubic feet, we need to use the predicted linear model:
and evaluate at . The predicted amount of hours will be
Therefore, the mean labor hours for moving 500 cubic feet are 22.67034 hours.