Regression Coefficients – Problems, Properties & Solutions

curvilinear Regression

🔹 1️⃣ Meaning of Regression Coefficients

In simple linear regression:

$$Y = a + b_{yx}X$$

• $b_{yx}$ = Regression coefficient of Y on X

• $b_{xy}$ = Regression coefficient of X on Y

They measure the rate of change of one variable with respect to the other.

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🔑 Important Properties of Regression Coefficients

1. Same Sign Property
   Both $b_{yx}$ and $b_{xy}$ have the same sign as correlation (r).

2. Product Rule

   $$b_{yx} \cdot b_{xy} = r^2$$

3. Correlation Relation

   $$r = \pm \sqrt{b_{yx} \cdot b_{xy}}$$

4. Independent of Change of Origin
   Regression coefficients do not change if origin changes.

5. Not Independent of Scale
   They change if scale changes.

6. If one regression coefficient is greater than 1, the other must be less than 1.

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🔢 Problem 1: Find Regression Coefficients

🔹 Given:

• $\bar{X} = 20$

• $\bar{Y} = 30$

• $\sigma_X = 4$

• $\sigma_Y = 5$

• $r = 0.6$

Find:

• $b_{yx}$

• $b_{xy}$

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✅ Solution

Formula:

$$b_{yx} = r \frac{\sigma_Y}{\sigma_X}$$ $$b_{xy} = r \frac{\sigma_X}{\sigma_Y}$$

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Step 1: Calculate $b_{yx}$

$$b_{yx} = 0.6 \times \frac{5}{4}$$ $$b_{yx} = 0.6 \times 1.25$$ $$b_{yx} = 0.75$$

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Step 2: Calculate $b_{xy}$

$$b_{xy} = 0.6 \times \frac{4}{5}$$ $$b_{xy} = 0.6 \times 0.8$$ $$b_{xy} = 0.48$$

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✅ Check Property

$$b_{yx} \cdot b_{xy} = 0.75 \times 0.48$$ $$= 0.36$$ $$r^2 = (0.6)^2 = 0.36$$

✔ Property Verified

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🔢 Problem 2: Find Correlation from Regression Coefficients

🔹 Given:

$$b_{yx} = 0.8$$ $$b_{xy} = 0.5$$

Find r.

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✅ Solution

$$r = \pm \sqrt{b_{yx} \cdot b_{xy}}$$ $$r = \sqrt{0.8 \times 0.5}$$ $$r = \sqrt{0.4}$$ $$r = 0.632$$

Since both coefficients are positive,

$$r = +0.632$$

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🔢 Problem 3: Find Regression Equations

🔹 Given:

Regression lines:

$$Y = 2X + 5$$ $$X = 0.5Y + 1$$

Find:

• Regression coefficients

• Correlation coefficient

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✅ Step 1: Identify Coefficients

From:

$$Y = 2X + 5$$ $$b_{yx} = 2$$

From:

$$X = 0.5Y + 1$$ $$b_{xy} = 0.5$$

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✅ Step 2: Find r

$$r = \pm \sqrt{2 \times 0.5}$$ $$r = \sqrt{1}$$ $$r = 1$$

Since both slopes are positive,

$$r = +1$$

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✅ Interpretation

Perfect positive correlation.
Both regression lines coincide.

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🔢 Problem 4: If One Coefficient > 1

Given:

$$b_{yx} = 1.2$$

Find possible value of $b_{xy}$.

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Using Property:

$$b_{yx} \cdot b_{xy} = r^2 \le 1$$ $$1.2 \times b_{xy} \le 1$$ $$b_{xy} \le 0.833$$

So the other coefficient must be less than 1.

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📌 Quick Summary Table

Concept	Formula
$b_{yx}$	$r \frac{\sigma_Y}{\sigma_X}$
$b_{xy}$	$r \frac{\sigma_X}{\sigma_Y}$
Product rule	$b_{yx} b_{xy} = r^2$
Correlation	$r = \pm \sqrt{b_{yx} b_{xy}}$