Regression Analysis — Previous Year Questions

📊 Topic Overview

Metric	Value
Papers Appeared	18 of 27 (66.7%)
Priority	🔴 VERY HIGH
Section Distribution	Sec A: ~12
Most Repeated	"Compute regression equation for X and Y based on the following data"
Probability	Very High — Section A computation in ~2 of every 3 papers

🎯 Must-Know Concepts

1. Regression is used for prediction — predicting one variable from another
2. Two regression equations: Y on X (predicting Y from X) and X on Y (predicting X from Y)
3. Regression coefficient (b) = slope of the regression line
4. Relationship between correlation and regression
5. Linear vs multiple regression
6. Assumptions underlying regression

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🔥 Core Computation Procedure

Regression Equations (Section A)

Two regression equations are needed for full marks:

Regression of Y on X (predicting Y from X): $Y = a_{yx} + b_{yx}X$

Where: $b_{yx} = \frac{n\Sigma XY - \Sigma X \cdot \Sigma Y}{n\Sigma X^2 - (\Sigma X)^2}$

$a_{yx} = \bar{Y} - b_{yx}\bar{X}$

Regression of X on Y (predicting X from Y): $X = a_{xy} + b_{xy}Y$

Where: $b_{xy} = \frac{n\Sigma XY - \Sigma X \cdot \Sigma Y}{n\Sigma Y^2 - (\Sigma Y)^2}$

$a_{xy} = \bar{X} - b_{xy}\bar{Y}$

Computation Table

N	X	Y	X²	Y²	XY
1
2
...
Σ	ΣX	ΣY	ΣX²	ΣY²	ΣXY

Step 1: Calculate ΣX, ΣY, ΣX², ΣY², ΣXY, and n
Step 2: Compute X̄ and Ȳ
Step 3: Compute b_yx using the formula
Step 4: Compute a_yx = Ȳ − b_yx × X̄
Step 5: Write equation: Ŷ = a_yx + b_yx × X
Step 6: Repeat for X on Y regression
Step 7: Verify — point (X̄, Ȳ) must lie on both regression lines

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📝 Conceptual Questions

Relationship between Correlation and Regression (Section B — ~250 words)

Correlation and regression are closely related but serve different purposes:

• Correlation measures the degree and direction of the linear relationship between two variables. It is symmetrical — r(X,Y) = r(Y,X).
• Regression uses the relationship to predict one variable from the other. It is asymmetrical — there are two distinct regression equations.

The regression coefficients b_yx and b_xy are related to the correlation coefficient: $r = \sqrt{b_{yx} \times b_{xy}}$

This means: if you multiply the two regression coefficients and take the square root, you get Pearson's r. Also: if r = 0, the regression lines are perpendicular to the axes and prediction is impossible; if r = ±1, both regression lines coincide.

Assumptions of linear regression:

1. Linear relationship between variables
2. Normality of residuals
3. Homoscedasticity (equal variance of residuals)
4. Independence of observations

Linear vs Multiple Regression:

• Linear (simple) regression: One predictor variable; one criterion variable
• Multiple regression: Two or more predictor variables; one criterion variable. Better prediction accuracy but more complex computation.

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🧠 Memory Aid

Regression Steps (TSBE):

• Table: compute ΣX, ΣY, ΣX², ΣY², ΣXY
• Slope b: apply formula
• B-intercept a: a = Ȳ − bX̄
• Equation: write Ŷ = a + bX

r connects b_yx and b_xy: $r = \sqrt{b_{yx} \times b_{xy}}$

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📚 Related Content

• Linear Regression: Introduction and Prediction
• Bivariate Regression: OLS and Computation
• Multiple Regression

✅ Self-Assessment

1. Write the formula for regression coefficient b_yx.
2. What is the relationship between the two regression coefficients and r?
3. If X̄ = 5, Ȳ = 7, and b_yx = 0.8, what is a_yx?
4. Why are there two regression equations?
5. What is the purpose of the regression equation?

💡 Exam Tips

• Always compute both regression equations for full marks in Section A.
• The computation table with ΣX, ΣY, ΣX², ΣY², ΣXY is essential — set it up first.
• Verify your computation: r² = b_yx × b_xy.
• Questions often ask you to also predict a value using the equation — substitute and solve.
• Know the formula r = √(b_yx × b_xy) — it frequently appears as a short note or within conceptual answers.