13-Correlation-Regression

BSTA 200 3. Correlation and Regression 3 Prof. Joshua Emmanuel LEAST SQUARES PRINCIPLE : The regression equation is obtained by minimizing the sum of the squares of the vertical distance between the actual y values and the predicted values of y. The scatter diagram on the right (with regression line) shows the relationship between wait time (before seeing a doctor) and satisfaction rating at a hospital. Regression Equation : An equation that expresses the linear relationship between two variables . General Form of Linear Regression Equation: 𝒚𝒚 � = 𝒃𝒃 𝟎𝟎 + 𝒃𝒃 𝟏𝟏 𝒙𝒙 or 𝒚𝒚 � = 𝒂𝒂 + 𝒃𝒃𝒙𝒙 𝒚𝒚 � = 𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑 𝒗𝒗𝒂𝒂𝒗𝒗𝒗𝒗𝒑𝒑 𝒐𝒐𝒐𝒐 𝒑𝒑𝒕𝒕𝒑𝒑 𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒅𝒅𝒑𝒑𝒑𝒑𝒅𝒅𝒑𝒑 𝒗𝒗𝒂𝒂𝒑𝒑𝒑𝒑𝒂𝒂𝒃𝒃𝒗𝒗𝒑𝒑 𝒃𝒃 𝒐𝒐𝒑𝒑 𝒃𝒃 𝟏𝟏 is the slope 𝒂𝒂 𝒐𝒐𝒑𝒑 𝒃𝒃 𝟎𝟎 is the y-intercept 𝒃𝒃 = 𝑛𝑛 ( ∑𝑥𝑥𝑥𝑥 ) − ∑𝑥𝑥∑𝑥𝑥 𝑛𝑛∑𝑥𝑥 2 − ( ∑𝑥𝑥 ) 2 𝒂𝒂 = ∑𝑥𝑥 𝑛𝑛 − 𝑏𝑏 ∑𝑥𝑥 𝑛𝑛 Slope 𝒃𝒃 𝒐𝒐𝒑𝒑 𝒃𝒃 𝟏𝟏 : the expected change in the value of y for a unit increase in x . The slope is interpreted as the change in the Y variable associated with a unit change in the X variable. y-intercept, 𝒂𝒂 𝒐𝒐𝒑𝒑 𝒃𝒃 𝟎𝟎 : the point where the regression line crosses the Y axis. The Y intercept is the predicted value of Y for an X value of zero. 0 20 40 60 80 100 0 10 20 30 40 50 60 70 SATISTIFACTION RATING WAIT TIME SATSIFACTION RATING