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
4 93.6
7 87.3
9 73.5
11 67.1
13 90.9
15 55.7
19 25.5

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the linear regression line in the form f = ax + b, we first need to calculate the slope (a) and the y-intercept (b) of the line.

Step 1: Calculate the mean of x and y
Mean of x:
(4 + 7 + 9 + 11 + 13 + 15 + 19) / 7 = 78 / 7 = 11.143

Mean of y:
(93.6 + 87.3 + 73.5 + 67.1 + 90.9 + 55.7 + 25.5) / 7 ≈ 67.857

Step 2: Calculate the slope (a)
a = Σ((x - x̄)(y - ȳ)) / Σ(x - x̄)^2
where x̄ is the mean of x and ȳ is the mean of y.

Applying the formula:

a = ((4-11.143)*(93.6-67.857) + (7-11.143)*(87.3-67.857) + (9-11.143)*(73.5-67.857) + (11-11.143)*(67.1-67.857) + (13-11.143)*(90.9-67.857) + (15-11.143)*(55.7-67.857) + (19-11.143)*(25.5-67.857)) / ((4-11.143)^2 + (7-11.143)^2 + (9-11.143)^2 + (11-11.143)^2 + (13-11.143)^2 + (15-11.143)^2 + (19-11.143)^2)

a ≈ -2.867

Step 3: Calculate the y-intercept (b)
b = ȳ - a * x̄

Applying the formula:

b = 67.857 - (-2.867 * 11.143)
b ≈ 100.633

Therefore, the linear regression line in the form f = ax + b for this relationship is:
f ≈ -2.867x + 100.633

Rounded to the nearest thousandth, the slope (a) is -2.867 and the y-intercept (b) is 100.633.