Question

Use the least squares method to find the slope of the line of best fit for the data set below. Round to three decimal places.

x 67 28 26 47 86 32 15 70
y 46 34 33 38 46 37 35 44(1 point)
Responses

−0.202
negative 0.202

0.202
0.202

−4.430
negative 4.430

4.430

Answer 1

To find the slope of the line of best fit using the least squares method, we first need to calculate the mean of x (x̄) and the mean of y (ȳ).

Σx = 67 + 28 + 26 + 47 + 86 + 32 + 15 + 70 = 361
Σy = 46 + 34 + 33 + 38 + 46 + 37 + 35 + 44 = 313

x̄ = Σx / n = 361 / 8 = 45.125
ȳ = Σy / n = 313 / 8 = 39.125

Next, we calculate the sums of squares for x (SSx), y (SSy), and the sum of products (SP).

SSx = Σ(xi - x̄)² = (67-45.125)² + (28-45.125)² + (26-45.125)² + (47-45.125)² + (86-45.125)² + (32-45.125)² + (15-45.125)² + (70-45.125)² = 3442.875
SSy = Σ(yi - ȳ)² = (46-39.125)² + (34-39.125)² + (33-39.125)² + (38-39.125)² + (46-39.125)² + (37-39.125)² + (35-39.125)² + (44-39.125)² = 171.875
SP = Σ(xi - x̄)(yi - ȳ) = (67-45.125)(46-39.125) + (28-45.125)(34-39.125) + (26-45.125)(33-39.125) + (47-45.125)(38-39.125) + (86-45.125)(46-39.125) + (32-45.125)(37-39.125) + (15-45.125)(35-39.125) + (70-45.125)(44-39.125) = 742.875

The slope (b) of the line of best fit is calculated as:

b = SP / SSx = 742.875 / 3442.875 ≈ 0.215

Therefore, the slope of the line of best fit for the data set is 0.215 (rounded to three decimal places).