Least-Squares Regression

The lest-squares regression line based on sample data is

b1 is the slope of the line, or the change in y for every unit change in x, and is calculated as

where b0 is the y-intercept and is calculated as

Example: Rising Hills Manufacturing Inc. wishes to study the relationship between the number of workers, X, and the number of tables, Y, produced in its Redwood Falls plant. It has obtained a random samples of 10 hours of production. The following (x, y) combinations of points were obtained:
       (12, 20)             (30, 60)     (15, 27)        (24, 50)   (14, 21)
       (18, 30)             (28, 61)     (26, 54)        (19, 32)   (27, 57)

Compute the covariance and correlation coefficient. Discuss briefly the relationship between the number of workers and the number of tables produced per hour.

Solution: The computations are set out in the Table bellow.

From the covariance we see that the direction of the relationship is positive, the high correlation of 0.989 also indicate that the same data points are very close to some increasing straight line.

We can also use a statistical software package such as Minitab or spreadsheet such as Excel to obtain the same regression coefficients and regression line.