The following information regarding a dependent variable (y) and an independent variable (x) is provided.
y x
4 2
3 1
4 4
6 3
8 5
SSE = 6 and SST = 16.
The least squares estimate of the slope is
To find the least squares estimate of the slope, we need to use the formula:
β₁ = Cov(x, y) / Var(x)
Here, Cov(x, y) is the covariance between the independent variable x and the dependent variable y, and Var(x) is the variance of the independent variable x.
First, let's calculate the mean of x (x̄) and y (ȳ):
x̄ = (2 + 1 + 4 + 3 + 5) / 5 = 3
ȳ = (4 + 3 + 4 + 6 + 8) / 5 = 5
Next, we calculate the deviations of x and y from their means:
xᵢ - x̄:
(2 - 3) = -1
(1 - 3) = -2
(4 - 3) = 1
(3 - 3) = 0
(5 - 3) = 2
yᵢ - ȳ:
(4 - 5) = -1
(3 - 5) = -2
(4 - 5) = -1
(6 - 5) = 1
(8 - 5) = 3
Now, we calculate the sum of the products of the deviations:
Σ[(xᵢ - x̄)(yᵢ - ȳ)] =
((-1) * (-1)) + ((-2) * (-2)) + ((1) * (-1)) + ((0) * (1)) + ((2) * (3)) = 10
Next, we calculate the variance of x (Var(x)):
Var(x) = [(xᵢ - x̄)²] / n = [(-1)² + (-2)² + (1)² + (0)² + (2)²] / 5 = 10 / 5 = 2
Now, we can calculate the least squares estimate of the slope:
β₁ = Cov(x, y) / Var(x) = Σ[(xᵢ - x̄)(yᵢ - ȳ)] / Var(x) = 10 / 2 = 5
Therefore, the least squares estimate of the slope is 5.