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
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To obtain the regression equations, we can use the formula:
b = r * (Sy / Sx)
where:
- b is the slope of the regression line
- r is the coefficient of correlation
- Sy is the standard deviation of the dependent variable (blood pressure)
- Sx is the standard deviation of the independent variable (age)
Step 1: Calculate the slope for each regression line.
For the regression line of blood pressure as the dependent variable:
b1 = r * (Sy / Sx)
= 0.8 * (10 / 5)
= 1.6
For the regression line of age as the dependent variable:
b2 = r * (Sx / Sy)
= 0.8 * (5 / 10)
= 0.4
Step 2: Calculate the intercept for each regression line.
For the regression line of blood pressure as the dependent variable:
Using the formula: a1 = Y̅ - b1 * X̅
where:
- a1 is the intercept
- Y̅ is the average blood pressure
- X̅ is the average age
a1 = 60 - (1.6 * 50)
= 60 - 80
= -20
For the regression line of age as the dependent variable:
Using the formula: a2 = Y̅ - b2 * X̅
where:
- a2 is the intercept
- Y̅ is the average age
- X̅ is the average blood pressure
a2 = 50 - (0.4 * 60)
= 50 - 24
= 26
So the two regression equations are:
1) Blood pressure (Y) as the dependent variable:
Y = -20 + 1.6 * X
2) Age (X) as the dependent variable:
X = 26 + 0.4 * Y
Now, we can proceed to find the expected blood pressure of a person who is 49 years old.
b) To find the expected blood pressure of a person who is 49 years old, we can use the regression equation for blood pressure (Y) as the dependent variable:
Y = -20 + 1.6 * X
Substitute X with 49:
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 58.4.