Math Problem Statement

Given that the lines of regression of y on x and x on y are respectively y = x and 4x - y = 3, and that σx = 1, find the correlation coefficient (r) and the variance of y (σy²).

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: y=xy = x
  • Regression of x on y: 4xy=34x - y = 3

Additionally:

  • σx=1\sigma_x = 1 (standard deviation of x)

The task is to:

  1. Find the correlation coefficient rr.
  2. Find the variance of yy (σy2\sigma_y^2).

Step-by-Step Explanation of Solution:

1. The regression coefficients:

The regression equations help us calculate the regression coefficients:

  • byxb_{yx}: Regression coefficient of yy on xx
  • bxyb_{xy}: Regression coefficient of xx on yy

From the given equations:

  • y=xy = x: byx=1b_{yx} = 1
  • 4xy=34x - y = 3: Rearranging gives x=y+34x = \frac{y + 3}{4}, so bxy=14=0.25b_{xy} = \frac{1}{4} = 0.25.

2. Correlation coefficient rr:

The formula for the correlation coefficient rr is: r=±byxbxyr = \pm\sqrt{b_{yx} \cdot b_{xy}}

Substituting byx=1b_{yx} = 1 and bxy=0.25b_{xy} = 0.25: r=10.25=0.25=0.5r = \sqrt{1 \cdot 0.25} = \sqrt{0.25} = 0.5

Since both byxb_{yx} and bxyb_{xy} are positive, r=0.5r = 0.5.


3. Variance of yy:

We know the relationship between byxb_{yx}, rr, σx\sigma_x, and σy\sigma_y: byx=rσyσxb_{yx} = r \cdot \frac{\sigma_y}{\sigma_x}

Substitute byx=1b_{yx} = 1, r=0.5r = 0.5, and σx=1\sigma_x = 1: 1=0.5σy11 = 0.5 \cdot \frac{\sigma_y}{1}

Solve for σy\sigma_y: σy=2\sigma_y = 2

Thus, the variance of yy is: σy2=(σy)2=22=4\sigma_y^2 = (\sigma_y)^2 = 2^2 = 4


Final Answers:

  1. Correlation coefficient: r=0.5r = 0.5
  2. Variance of yy: σy2=4\sigma_y^2 = 4

Do you want further clarifications or details?


Related Questions:

  1. How are regression coefficients byxb_{yx} and bxyb_{xy} derived from equations?
  2. What does the correlation coefficient rr represent in a regression context?
  3. Why is rr positive in this case? When would rr be negative?
  4. How can we calculate the standard deviation of xx and yy from the regression lines?
  5. What is the significance of the regression line equations in real-world applications?

Tip:

Always ensure that the correlation coefficient rr 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)