Math Problem Statement
Solution
From the provided image, let me break down the problem and solution step by step:
Problem:
We are given the lines of regression:
- Regression of y on x:
- Regression of x on y:
Additionally:
- (standard deviation of x)
The task is to:
- Find the correlation coefficient .
- Find the variance of ().
Step-by-Step Explanation of Solution:
1. The regression coefficients:
The regression equations help us calculate the regression coefficients:
- : Regression coefficient of on
- : Regression coefficient of on
From the given equations:
- :
- : Rearranging gives , so .
2. Correlation coefficient :
The formula for the correlation coefficient is:
Substituting and :
Since both and are positive, .
3. Variance of :
We know the relationship between , , , and :
Substitute , , and :
Solve for :
Thus, the variance of is:
Final Answers:
- Correlation coefficient:
- Variance of :
Do you want further clarifications or details?
Related Questions:
- How are regression coefficients and derived from equations?
- What does the correlation coefficient represent in a regression context?
- Why is positive in this case? When would be negative?
- How can we calculate the standard deviation of and from the regression lines?
- What is the significance of the regression line equations in real-world applications?
Tip:
Always ensure that the correlation coefficient lies between -1 and 1. Its sign depends on the direction of the relationship between the variables.
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Math Problem Analysis
Mathematical Concepts
Regression Analysis
Correlation Coefficient
Variance
Standard Deviation
Formulas
r = ±√(b_yx * b_xy)
b_yx = r * (σ_y / σ_x)
b_xy = r * (σ_x / σ_y)
Theorems
Regression coefficient relationship with correlation coefficient
Variance and standard deviation relationship
Suitable Grade Level
Undergraduate or Advanced High School (Grade 11-12)