Linear Regression – Straight Line Problems

 

Linear Regression – Straight Line Problems

Linear regression is used to find the best-fitting straight line that shows the relationship between two variables.

The general equation of a straight line:

$$Y = a + bX$$

Where:

• a = Intercept

• b = Slope (regression coefficient)

• Y = Dependent variable

• X = Independent variable

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1️⃣ Example 1: Find Regression Equation (Y on X)

🔹 Question

Find the regression equation of Y on X for the following data:

X	1	2	3	4	5
Y	2	4	5	4	5

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🔹 Step 1: Formula for Y on X

$$Y = a + bX$$ $$b = \frac{n\sum XY - \sum X \sum Y}{n\sum X^2 - (\sum X)^2}$$ $$a = \bar{Y} - b\bar{X}$$

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🔹 Step 2: Prepare Table

X	Y	X²	XY
1	2	1	2
2	4	4	8
3	5	9	15
4	4	16	16
5	5	25	25

Now compute totals:

$$\sum X = 15$$ $$\sum Y = 20$$ $$\sum X^2 = 55$$ $$\sum XY = 66$$ $$n = 5$$

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🔹 Step 3: Calculate b

$$b = \frac{5(66) - (15)(20)}{5(55) - (15)^2}$$ $$= \frac{330 - 300}{275 - 225}$$ $$= \frac{30}{50}$$ $$b = 0.6$$

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🔹 Step 4: Calculate a

$$\bar{X} = 15/5 = 3$$ $$\bar{Y} = 20/5 = 4$$ $$a = 4 - (0.6)(3)$$ $$a = 4 - 1.8$$ $$a = 2.2$$

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✅ Regression Equation

$$Y = 2.2 + 0.6X$$

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🔹 Step 5: Prediction Example

If X = 6, then:

$$Y = 2.2 + 0.6(6)$$ $$Y = 2.2 + 3.6$$ $$Y = 5.8$$

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2️⃣ Example 2: Find Both Regression Lines

🔹 Given:

X	10	20	30	40
Y	15	25	35	45

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Observation

Notice:

$$Y = X + 5$$

So:

Regression of Y on X:

$$Y = X + 5$$

Regression of X on Y:

$$X = Y - 5$$

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📌 Important Points for Exams

1. There are two regression lines:

   • Y on X

   • X on Y

2. Both lines pass through:

$$(\bar{X}, \bar{Y})$$

3. If correlation (r) = ±1 → both regression lines coincide.

4. Sign of regression coefficient is same as correlation coefficient.