7. Calculate b0 and b1 and make an equation of regression line
for the set of data given in Table 3
Table 3
x y
1 12
2 15
3 16
5 18
6 21
8 23
A=3.135451505
B= 14.76588629
R^2=.8651908126
R=.9301563377
To calculate b0 and b1 and make an equation of the regression line, you can use the following steps:
Step 1: Calculate the mean of x (x̄) and the mean of y (ȳ):
x̄ = (1 + 2 + 3 + 5 + 6 + 8) / 6 = 25 / 6 = 4.1667
ȳ = (12 + 15 + 16 + 18 + 21 + 23) / 6 = 105 / 6 = 17.5
Step 2: Calculate the deviations of each x value from the mean of x (x - x̄) and the deviations of each y value from the mean of y (y - ȳ):
x | y | x - x̄ | y - ȳ
------------------------------------
1 | 12 | -3.1667 | -5.5
2 | 15 | -2.1667 | -2.5
3 | 16 | -1.1667 | -1.5
5 | 18 | 0.8333 | 0.5
6 | 21 | 1.8333 | 3.5
8 | 23 | 3.8333 | 5.5
Step 3: Calculate the product of the deviations of each x and y value (x - x̄) * (y - ȳ):
(x - x̄) * (y - ȳ):
-3.1667 * -5.5 = 17.4167
-2.1667 * -2.5 = 5.4167
-1.1667 * -1.5 = 1.7500
0.8333 * 0.5 = 0.4167
1.8333 * 3.5 = 6.4167
3.8333 * 5.5 = 21.0833
Step 4: Calculate the squared deviations of each x value from the mean of x [(x - x̄)^2]:
(x - x̄)^2:
(-3.1667)^2 = 10.0278
(-2.1667)^2 = 4.6944
(-1.1667)^2 = 1.3611
(0.8333)^2 = 0.6944
(1.8333)^2 = 3.3611
(3.8333)^2 = 14.6944
Step 5: Calculate b1, the slope of the regression line:
b1 = Σ[(x - x̄) * (y - ȳ)] / Σ[(x - x̄)^2]
b1 = (17.4167 + 5.4167 + 1.7500 + 0.4167 + 6.4167 + 21.0833) / (10.0278 + 4.6944 + 1.3611 + 0.6944 + 3.3611 + 14.6944)
b1 = 52.5000 / 34.8332
b1 = 1.5041
Step 6: Calculate b0, the y-intercept of the regression line:
b0 = ȳ - b1 * x̄
b0 = 17.5 - 1.5041 * 4.1667
b0 = 11.1313
Step 7: Create the equation of the regression line:
The equation of the regression line is: y = b0 + b1 * x
Plugging in the values we calculated:
y = 11.1313 + 1.5041 * x