The age (X) and blood pressure (Y) of 9 patients were measured and the results are as follows:
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
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.
To calculate the slope of the regression line, we need to find the correlation coefficient (r) between the age (X) and blood pressure (Y). The formula for calculating the slope (m) is given by:
m = r * (std Dev of Y) / (std Dev of X)
First, let's find the mean of both X and Y:
Mean of X (meanX) = (20 + 25 + 30 + 40 + 46 + 50 + 57 + 60 + 70) / 9 = 44.2222
Mean of Y (meanY) = (120 + 125 + 126 + 128 + 130 + 135 + 138 + 140 + 145) / 9 = 131.8889
Next, let's calculate the sum of squares of each variable (X and Y):
SS(X) = (20 - 44.2222)^2 + (25 - 44.2222)^2 + (30 - 44.2222)^2 + (40 - 44.2222)^2 + (46 - 44.2222)^2 + (50 - 44.2222)^2 + (57 - 44.2222)^2 + (60 - 44.2222)^2 + (70 - 44.2222)^2 = 805.3333
SS(Y) = (120 - 131.8889)^2 + (125 - 131.8889)^2 + (126 - 131.8889)^2 + (128 - 131.8889)^2 + (130 - 131.8889)^2 + (135 - 131.8889)^2 + (138 - 131.8889)^2 + (140 - 131.8889)^2 + (145 - 131.8889)^2 = 693.5556
Now, let's calculate the cross-product sum (SP):
SP = (20 - 44.2222) * (120 - 131.8889) + (25 - 44.2222) * (125 - 131.8889) + (30 - 44.2222) * (126 - 131.8889) + (40 - 44.2222) * (128 - 131.8889) + (46 - 44.2222) * (130 - 131.8889) + (50 - 44.2222) * (135 - 131.8889) + (57 - 44.2222) * (138 - 131.8889) + (60 - 44.2222) * (140 - 131.8889) + (70 - 44.2222) * (145 - 131.8889) = 819.4444
Next, let's calculate the standard deviation of X (stdDevX) and Y (stdDevY):
stdDevX = sqrt(SS(X) / (n - 1)) = sqrt(805.3333 / 8) = 9.3346
stdDevY = sqrt(SS(Y) / (n - 1)) = sqrt(693.5556 / 8) = 8.4853
Now, let's calculate the correlation coefficient (r):
r = SP / sqrt(SS(X) * SS(Y)) = 819.4444 / sqrt(805.3333 * 693.5556) = 0.4714
Finally, let's calculate the slope (m):
m = r * (stdDevY) / (stdDevX) = 0.4714 * 8.4853 / 9.3346 = 0.4282
Therefore, the slope of the regression line is approximately 0.4282.
To calculate the slope of the regression line, we need to use the formula:
slope = ( Σ(XY) - ( ΣX * ΣY ) / n * Σ(X^2) - ( ΣX )^2 )
Where:
- Σ(XY) represents the sum of the products of each X and Y value.
- ΣX represents the sum of all X values.
- ΣY represents the sum of all Y values.
- n represents the number of data points.
- Σ(X^2) represents the sum of the squares of each X value.
Let's calculate the slope step by step using the given data:
1. Calculate the sum of the products of each X and Y value (Σ(XY)):
Σ(XY) = (20 * 120) + (25 * 125) + (30 * 126) + (40 * 128) + (46 * 130) + (50 * 135) + (57 * 138) + (60 * 140) + (70 * 145)
= 2400 + 3125 + 3780 + 5120 + 5980 + 6750 + 7866 + 8400 + 10150
= 52971
2. Calculate the sum of all X values (ΣX):
ΣX = 20 + 25 + 30 + 40 + 46 + 50 + 57 + 60 + 70
= 398
3. Calculate the sum of all Y values (ΣY):
ΣY = 120 + 125 + 126 + 128 + 130 + 135 + 138 + 140 + 145
= 1137
4. Calculate the sum of the squares of each X value (Σ(X^2)):
Σ(X^2) = (20^2) + (25^2) + (30^2) + (40^2) + (46^2) + (50^2) + (57^2) + (60^2) + (70^2)
= 400 + 625 + 900 + 1600 + 2116 + 2500 + 3249 + 3600 + 4900
= 20390
5. Calculate the slope:
slope = ( Σ(XY) - ( ΣX * ΣY ) ) / ( n * Σ(X^2) - ( ΣX )^2 )
= ( 52971 - ( 398 * 1137 ) ) / ( 9 * 20390 - 398^2 )
= ( 52971 - 452286 ) / ( 183510 - 158404 )
= ( -399315 ) / ( 25106 )
≈ -15.914
Therefore, the slope of the regression line is approximately -15.914.