To find the equation of the least squares line that best fits the data points (2, 3), (3, 2), (5, 1), (6, 0), we can use the least squares method. This involves finding the slope (b) and intercept (a) of the line represented by the equation y = a + bx.
First, let’s list our data points:
- (2, 3)
- (3, 2)
- (5, 1)
- (6, 0)
Next, we need to calculate the necessary sums:
- n = number of points = 4
- Σx = 2 + 3 + 5 + 6 = 16
- Σy = 3 + 2 + 1 + 0 = 6
- Σxy = (2*3) + (3*2) + (5*1) + (6*0) = 6 + 6 + 5 + 0 = 17
- Σx² = (2²) + (3²) + (5²) + (6²) = 4 + 9 + 25 + 36 = 74
Now, we can use the formulas for slope (b) and intercept (a) of the least squares line:
b = (nΣxy – ΣxΣy) / (nΣx² – (Σx)²)
a = (Σy – bΣx) / n
Plugging the values into the slope formula:
b = (4 * 17 – 16 * 6) / (4 * 74 – 16²)
b = (68 – 96) / (296 – 256)
b = -28 / 40 = -0.7
Now, we substitute b into the intercept formula:
a = (6 – (-0.7) * 16) / 4
a = (6 + 11.2) / 4
a = 17.2 / 4 = 4.3
Thus, the equation of the least squares line is:
y = 4.3 – 0.7x
This line minimizes the distance between itself and the data points, giving us the best possible linear approximation for the provided data.