Question

Given that the linear relationship between x and y is statistically significant, find the linear regression line in the form f = ax + b for this relationship. Round an and b to the nearest thousandth.

X У
5 98.5
8 86.2
10 71.3
12 66.8
15 90.8
17 54.2
20 26.3

Answer 1

[AI-generated answer — Bot GPT 3.5]

First, we need to calculate the values for a and b in the linear regression line equation f = ax + b.

Step 1: Calculate the means of X and Y
Mean of X (x̄) = (5 + 8 + 10 + 12 + 15 + 17 + 20) / 7 = 11.2857
Mean of Y (ȳ) = (98.5 + 86.2 + 71.3 + 66.8 + 90.8 + 54.2 + 26.3) / 7 = 70.85

Step 2: Calculate the values needed for a
Σ(x - x̄)(y - ȳ) = (5-11.2857)(98.5-70.85) + (8-11.2857)(86.2-70.85) + (10-11.2857)(71.3-70.85) + (12-11.2857)(66.8-70.85) + (15-11.2857)(90.8-70.85) + (17-11.2857)(54.2-70.85) + (20-11.2857)(26.3-70.85)
Σ(x - x̄)(y - ȳ) = -14.0657 * 27.65 + -3.2857 * 15.35 + -1.2857 * 0.45 + 0.7143 * -4.05 + 3.7143 * 19.95 + 5.7143 * -16.65 + 8.7143 * -44.55
Σ(x - x̄)(y - ȳ) = -388.5702 - 50.5099 - 0.5786 - 2.8900 + 74.1148 - 94.9781 - 387.0810
Σ(x - x̄)(y - ȳ) = -850.4930

Σ(x - x̄)^2 = (5-11.2857)^2 + (8-11.2857)^2 + (10-11.2857)^2 + (12-11.2857)^2 + (15-11.2857)^2 + (17-11.2857)^2 + (20-11.2857)^2
Σ(x - x̄)^2 = 39.6331 + 10.7685 + 1.6746 + 0.5104 + 13.4352 + 32.5131 + 81.1558
Σ(x - x̄)^2 = 179.6707

a = Σ(x - x̄)(y - ȳ) / Σ(x - x̄)^2
a = -850.4930 / 179.6707
a = -4.735

Step 3: Calculate the value for b
b = ȳ - a * x̄
b = 70.85 - (-4.735 * 11.2857)
b = 70.85 + 53.4208
b = 124.271

Therefore, the linear regression line equation is f = -4.735x + 124.271 rounded to the nearest thousandth.