The age (X) and blood pressure (Y) of 9 patients were measured and the results are as follows:

Question

Patient Age (X) Blood pressure (Y)
1 20 120
2 25 125
3 30 126
4 40 128
5 46 130
6 50 135
7 57 138
8 60 140
9 70 145


Calculate the slope of the regression line.
526.8889
131.8889
111.0440
0.4714
44.2222

Answer 1

To calculate the slope of the regression line, we need to use the formula:

slope = Σ((X - X̄)(Y - Ȳ)) / Σ((X - X̄)^2)

where X is the age, Y is the blood pressure, X̄ is the mean of the age, and Ȳ is the mean of the blood pressure.

Calculating the mean of the age:
X̄ = (20 + 25 + 30 + 40 + 46 + 50 + 57 + 60 + 70) / 9
X̄ = 418 / 9
X̄ ≈ 46.4444

Calculating the mean of the blood pressure:
Ȳ = (120 + 125 + 126 + 128 + 130 + 135 + 138 + 140 + 145) / 9
Ȳ = 1157 / 9
Ȳ ≈ 128.5556

Calculating the sums:
Σ((X - X̄)(Y - Ȳ)) = (20 - 46.4444)(120 - 128.5556)
+ (25 - 46.4444)(125 - 128.5556)
+ (30 - 46.4444)(126 - 128.5556)
+ (40 - 46.4444)(128 - 128.5556)
+ (46 - 46.4444)(130 - 128.5556)
+ (50 - 46.4444)(135 - 128.5556)
+ (57 - 46.4444)(138 - 128.5556)
+ (60 - 46.4444)(140 - 128.5556)
+ (70 - 46.4444)(145 - 128.5556)
Σ((X - X̄)(Y - Ȳ)) ≈ -131.8889

Σ((X - X̄)^2) = (20 - 46.4444)^2
+ (25 - 46.4444)^2
+ (30 - 46.4444)^2
+ (40 - 46.4444)^2
+ (46 - 46.4444)^2
+ (50 - 46.4444)^2
+ (57 - 46.4444)^2
+ (60 - 46.4444)^2
+ (70 - 46.4444)^2
Σ((X - X̄)^2) ≈ 2220.2222

Substituting these values into the slope formula:

slope = Σ((X - X̄)(Y - Ȳ)) / Σ((X - X̄)^2)
slope ≈ -131.8889 / 2220.2222
slope ≈ -0.0594

Therefore, the slope of the regression line is approximately -0.0594.