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
To calculate the coefficients b0 and b1 for the regression line equation, we will use the least squares method.
Step 1: Calculate the mean of x (x̄) and the mean of y (ȳ). These can be found by summing up all the x-values and dividing by the total number of data points, and similarly for y-values.
x̄ = (1 + 2 + 3 + 5 + 6 + 8) / 6 = 25 / 6 ≈ 4.17
ȳ = (12 + 15 + 16 + 18 + 21 + 23) / 6 = 105 / 6 ≈ 17.50
Step 2: Calculate the deviations from the mean for both x and y. Subtract the respective mean from each data point.
x-deviations: (1 - 4.17), (2 - 4.17), (3 - 4.17), (5 - 4.17), (6 - 4.17), (8 - 4.17)
≈ (-3.17, -2.17, -1.17, 0.83, 1.83, 3.83)
y-deviations: (12 - 17.50), (15 - 17.50), (16 - 17.50), (18 - 17.50), (21 - 17.50), (23 - 17.50)
= (-5.50, -2.50, -1.50, 0.50, 3.50, 5.50)
Step 3: Calculate the product of the x and y deviations for each data point.
(-3.17) * (-5.50) = 17.43
(-2.17) * (-2.50) = 5.42
(-1.17) * (-1.50) = 1.75
0.83 * 0.50 = 0.42
1.83 * 3.50 = 6.41
3.83 * 5.50 = 21.07
Step 4: Calculate the sum of the x-deviations squared.
(-3.17)^2 + (-2.17)^2 + (-1.17)^2 + 0.83^2 + 1.83^2 + 3.83^2
≈ 68.67
Step 5: Calculate the sum of the product of x-deviations and y-deviations.
17.43 + 5.42 + 1.75 + 0.42 + 6.41 + 21.07
≈ 52.60
Step 6: Calculate the slope, b1, using the following formula:
b1 = Σ((x - x̄) * (y - ȳ)) / Σ((x - x̄)^2)
b1 = 52.60 / 68.67
≈ 0.766
Step 7: Calculate the y-intercept, b0, using the following formula:
b0 = ȳ - (b1 * x̄)
b0 = 17.50 - (0.766 * 4.17)
≈ 14.12
Now we have the coefficients b0 = 14.12 and b1 = 0.766.
The equation of the regression line is given by:
y = 14.12 + 0.766x
To calculate b0 and b1 and create the equation of the regression line, we'll use the least squares method.
Step 1: Calculate the mean of x and y.
x = (1 + 2 + 3 + 5 + 6 + 8) / 6 = 25 / 6 = 4.17
y = (12 + 15 + 16 + 18 + 21 + 23) / 6 = 105 / 6 = 17.5
Step 2: Calculate the deviations from the means of x (dx) and y (dy) for each data point.
x y dx = x - x dy = y - y
1 12 1 - 4.17 12 - 17.5
2 15 2 - 4.17 15 - 17.5
3 16 3 - 4.17 16 - 17.5
5 18 5 - 4.17 18 - 17.5
6 21 6 - 4.17 21 - 17.5
8 23 8 - 4.17 23 - 17.5
Step 3: Calculate the sum of products of dx and dy.
Σdx * dy = (1 - 4.17)(12 - 17.5) + (2 - 4.17)(15 - 17.5) + (3 - 4.17)(16 - 17.5) + (5 - 4.17)(18 - 17.5) + (6 - 4.17)(21 - 17.5) + (8 - 4.17)(23 - 17.5)
= (-3.17)(-5.5) + (-2.17)(-2.5) + (-1.17)(-1.5) + (0.83)(0.5) + (1.83)(3.5) + (3.83)(5.5)
= 17.485 + 5.425 + 1.755 + 0.415 + 6.405 + 21.065
= 52.55
Step 4: Calculate the sum of squares of dx.
Σ(dx)^2 = (1 - 4.17)^2 + (2 - 4.17)^2 + (3 - 4.17)^2 + (5 - 4.17)^2 + (6 - 4.17)^2 + (8 - 4.17)^2
= (-3.17)^2 + (-2.17)^2 + (-1.17)^2 + (0.83)^2 + (1.83)^2 + (3.83)^2
= 10.0489 + 4.7089 + 1.3689 + 0.6889 + 3.3489 + 14.6689
= 35.82
Step 5: Calculate b1.
b1 = Σ(dx * dy) / Σ(dx)^2
= 52.55 / 35.82
= 1.464
Step 6: Calculate b0.
b0 = y - b1 * x
= 17.5 - 1.464 * 4.17
= 17.5 - 6.11
= 11.39
Therefore, the equation of the regression line is:
y = 11.39 + 1.464x