The coefficient of correlation between the age and blood pressure of a sample of 8 persons was found to be +0.8, the average age is 50 years and average blood pressure is 60. Their standard deviations were 5 and 10 respectively.
a) Obtain the two regression equations
b) find the expected blood pressure of a person who is 49 years old
To obtain the two regression equations, we need to first calculate the regression coefficients.
The regression coefficient of X on Y, denoted by bXY, can be calculated using the formula:
bXY = r * (Sy / Sx)
where r is the coefficient of correlation, Sy is the standard deviation of Y, and Sx is the standard deviation of X.
Similarly, the regression coefficient of Y on X, denoted by bYX, can be calculated using the formula:
bYX = r * (Sx / Sy)
Let's calculate the regression coefficients:
Given:
Coefficient of correlation (r) = +0.8
Standard deviation of age (Sx) = 5 years
Standard deviation of blood pressure (Sy) = 10 mmHg
Using the formulas above, we can calculate the regression coefficients as follows:
bXY = 0.8 * (10 / 5) = 1.6
bYX = 0.8 * (5 / 10) = 0.4
a) The two regression equations are:
Regression equation 1: Y = a + bXY * X
Regression equation 2: X = c + bYX * Y
To find the intercept (a) and the constant term (c), we can substitute the average values of X and Y into the equations:
For regression equation 1:
X = 50 (average age)
Y = 60 (average blood pressure)
60 = a + (1.6 * 50)
60 = a + 80
a = -20
So, the regression equation 1 is: Y = -20 + 1.6 * X
For regression equation 2:
X = 50 (average age)
Y = 60 (average blood pressure)
50 = c + (0.4 * 60)
50 = c + 24
c = 26
So, the regression equation 2 is: X = 26 + 0.4 * Y
b) To find the expected blood pressure of a person who is 49 years old, we can use regression equation 1:
Y = -20 + 1.6 * X
Substituting X = 49 into the equation, we get:
Y = -20 + 1.6 * 49
Y = -20 + 78.4
Y = 58.4
Therefore, the expected blood pressure of a person who is 49 years old is approximately 58.4 mmHg.