Y x1 x2
57 220 125
43 216 147
73 250 118
37 241 160
64 305 128
48 258 149
56 354 145
50 321 150
39 370 140
43 375 115
69 385 155
60 385 152
Calculate the fitted line regression
To calculate the fitted line regression, you can use the least squares method to find the equation of the line that best fits the data.
First, calculate the mean of x1, x2, and Y:
mean_x1 = (220+216+250+241+305+258+354+321+370+375+385+385) / 12 = 313.42
mean_x2 = (125+147+118+160+128+149+145+150+140+115+155+152) / 12 = 140.17
mean_Y = (57+43+73+37+64+48+56+50+39+43+69+60) / 12 = 51.75
Next, calculate the deviations of x1, x2, and Y from their mean:
dev_x1 = (220-313.42, 216-313.42, 250-313.42, 241-313.42, 305-313.42, 258-313.42, 354-313.42, 321-313.42, 370-313.42, 375-313.42, 385-313.42, 385-313.42) = (-93.42, -97.42, -63.42, -72.42, -8.42, -55.42, 40.58, 7.58, 56.58, 61.58, 71.58, 71.58)
dev_x2 = (125-140.17, 147-140.17, 118-140.17, 160-140.17, 128-140.17, 149-140.17, 145-140.17, 150-140.17, 140-140.17, 115-140.17, 155-140.17, 152-140.17) = (-15.17, 6.83, -22.17, 19.83, -12.17, 8.83, 4.83, 9.83, -0.17, -25.17, 14.83, 11.83)
dev_Y = (57-51.75, 43-51.75, 73-51.75, 37-51.75, 64-51.75, 48-51.75, 56-51.75, 50-51.75, 39-51.75, 43-51.75, 69-51.75, 60-51.75) = (5.25, -8.75, 21.25, -14.75, 12.25, -3.75, 4.25, -1.75, -12.75, -8.75, 17.25, 8.25)
Then, calculate the products of the deviations:
prod_x1x2 = (-93.42*-15.17, -97.42*6.83, -63.42*-22.17, -72.42*19.83, -8.42*-12.17, -55.42*8.83, 40.58*4.83, 7.58*9.83, 56.58*-0.17, 61.58*-25.17, 71.58*14.83, 71.58*11.83) = (1414.9514, -666.0986, 1406.6994, -1436.1786, 102.9514, -489.0046, 196.1414, 74.4794, -9.6014, -1543.5426, 1060.5474, 844.4994)
prod_x1Y = (-93.42*5.25, -97.42*-8.75, -63.42*21.25, -72.42*-14.75, -8.42*12.25, -55.42*-3.75, 40.58*4.25, 7.58*-1.75, 56.58*-12.75, 61.58*-8.75, 71.58*17.25, 71.58*8.25) = (-489.855, 851.285, -1347.515, 1069.145, -102.955, 207.575, 172.9995, -13.265, -722.505, -540.295, 1229.635, 591.81)
Next, calculate the sum of squared deviations of x1, x2, and Y:
sum_sqdev_x1 = (-93.42)^2 + (-97.42)^2 + (-63.42)^2 + (-72.42)^2 + (-8.42)^2 + (-55.42)^2 + (40.58)^2 + (7.58)^2 + (56.58)^2 + (61.58)^2 + (71.58)^2 + (71.58)^2 = 27908.7492
sum_sqdev_x2 = (-15.17)^2 + (6.83)^2 + (-22.17)^2 + (19.83)^2 + (-12.17)^2 + (8.83)^2 + (4.83)^2 + (9.83)^2 + (-0.17)^2 + (-25.17)^2 + (14.83)^2 + (11.83)^2 = 3674.4398
sum_sqdev_Y = (5.25)^2 + (-8.75)^2 + (21.25)^2 + (-14.75)^2 + (12.25)^2 + (-3.75)^2 + (4.25)^2 + (-1.75)^2 + (-12.75)^2 + (-8.75)^2 + (17.25)^2 + (8.25)^2 = 1105.75
Finally, calculate the slope (b) and y-intercept (a) of the fitted line regression:
b = sum(prod_x1x2) / sum_sqdev_x1 = (1414.9514 + -666.0986 + 1406.6994 + -1436.1786 + 102.9514 + -489.0046 + 196.1414 + 74.4794 + -9.6014 + -1543.5426 + 1060.5474 + 844.4994) / 27908.7492 = -0.345
a = mean_Y - b * mean_x1 = 51.75 - (-0.345 * 313.42) = 163.74
Therefore, the fitted line regression equation is Y = -0.345 * x1 + 163.74.