Use the coordinates to write a prediction equation.
(9,17)
(4,10)
(3,10)
(1,7)
(10,19)
(6,12)
(12,20)
(8,15)
To write a prediction equation, we can use the method of linear regression. Linear regression helps us find the line that best fits the given set of points. In this case, we are given a set of coordinates:
(9, 17)
(4, 10)
(3, 10)
(1, 7)
(10, 19)
(6, 12)
(12, 20)
(8, 15)
To start, we need to calculate the slope (m) and the y-intercept (b) of the line that best fits these points. The equation for a line is given by y = mx + b.
1. Calculate the average of the x-values and the average of the y-values:
Average of x-values = (9 + 4 + 3 + 1 + 10 + 6 + 12 + 8) / 8 = 6.5
Average of y-values = (17 + 10 + 10 + 7 + 19 + 12 + 20 + 15) / 8 = 13.5
2. Calculate the differences between each x-value and the average of x-values, and the differences between each y-value and the average of y-values:
(9 - 6.5, 17 - 13.5) = (2.5, 3.5)
(4 - 6.5, 10 - 13.5) = (-2.5, -3.5)
(3 - 6.5, 10 - 13.5) = (-3.5, -3.5)
(1 - 6.5, 7 - 13.5) = (-5.5, -6.5)
(10 - 6.5, 19 - 13.5) = (3.5, 5.5)
(6 - 6.5, 12 - 13.5) = (-0.5, -1.5)
(12 - 6.5, 20 - 13.5) = (5.5, 6.5)
(8 - 6.5, 15 - 13.5) = (1.5, 1.5)
3. Calculate the sum of the products of these differences:
Sum[(x - x̄) * (y - ȳ)] = 2.5 * 3.5 + (-2.5) * (-3.5) + (-3.5) * (-3.5) + (-5.5) * (-6.5) + 3.5 * 5.5 + (-0.5) * (-1.5) + 5.5 * 6.5 + 1.5 * 1.5
= 8.75 + 8.75 + 12.25 + 35.75 + 19.25 + 0.75 + 35.75 + 2.25
= 123.75
4. Calculate the sum of the squares of the differences between the x-values and the average of x-values:
Sum[(x - x̄)^2] = (2.5)^2 + (-2.5)^2 + (-3.5)^2 + (-5.5)^2 + (3.5)^2 + (-0.5)^2 + (5.5)^2 + (1.5)^2
= 6.25 + 6.25 + 12.25 + 30.25 + 12.25 + 0.25 + 30.25 + 2.25
= 100.75
5. Calculate the slope:
m = Sum[(x - x̄) * (y - ȳ)] / Sum[(x - x̄)^2]
= 123.75 / 100.75
≈ 1.228
6. Calculate the y-intercept:
b = ȳ - m * x̄
= 13.5 - 1.228 * 6.5
≈ 5.67
The prediction equation is:
y = 1.228x + 5.67