Exercise 12.1: Concept and Foundations of Regression
Comprehensive guide and practice questions for CBSE Class 11 Applied Mathematics curriculum (2026-27).
Comprehensive Concepts & Principles: Foundations of Regression Analysis
First introduced into biological statistics by Sir Francis Galton in 1877 during his seminal studies on hereditary stature ("stepping back" or "regressing" toward the average height of the population), Regression Analysis has evolved into the foremost mathematical tool for economic forecasting and causal modeling. While correlation assesses whether two variables move together, regression quantifies the precise mathematical functional relationship to predict the unknown value of a dependent variable from known values of an independent variable.
Correlation vs. Regression: Core Analytical Distinctions:
1. Objective: Correlation measures the strength and direction of mutual co-variation. Regression enables estimation and prediction of one variable given another.
2. Symmetry: Correlation is completely symmetric ($r_{xy} = r_{yx}$). Regression is strictly asymmetric: the regression line of $Y$ on $X$ is distinct from the regression line of $X$ on $Y$.
3. Dimensionality & Units: Correlation coefficient $r$ is a pure dimensionless number without units ($-1 \le r \le 1$). Regression coefficients ($b_{yx}, b_{xy}$) are dimensional rates of change carrying physical units (e.g., ₹ of sales per ₹ of advertising).
4. Causality & Dependency: In correlation, neither variable is treated as dependent or independent. In regression, one variable is designated as the independent predictor ($X$) and the other as the dependent response ($Y$).
The Principle of Least Squares & The Two Regression Lines
A fundamental question in bivariate statistics is: Why are there two distinct lines of regression?
• Line of Regression of $Y$ on $X$:
- Purpose: Predicts $Y$ for a known value of $X$.
- Mathematical Criterion: Minimizes the sum of squares of vertical deviations: $\sum (y_i - \hat{y}_i)^2 = \text{minimum}$.
• Line of Regression of $X$ on $Y$:
- Purpose: Predicts $X$ for a known value of $Y$.
- Mathematical Criterion: Minimizes the sum of squares of horizontal deviations: $\sum (x_i - \hat{x}_i)^2 = \text{minimum}$.
• Point of Intersection: The two regression lines ALWAYS intersect at the point of means $(\bar{X}, \bar{Y})$.
• Angular Separation:
- If $r = \pm 1$: The two regression lines coincide into a single line (zero angle).
- If $r = 0$: The two regression lines are perpendicular to each other ($90^\circ$ angle).
Worked Benchmark Examples
Example 1 (Intersection Point & Means Determination):
The equations of two lines of regression obtained from a market survey are:
$$2X + 3Y - 18 = 0 \quad \text{and} \quad 4X + Y - 16 = 0$$
Find the arithmetic mean of $X$ ($\bar{X}$) and the arithmetic mean of $Y$ ($\bar{Y}$).
Show Step-by-Step Solution
• Property: The two regression lines always pass through the mean point $(\bar{X}, \bar{Y})$. Therefore, $(\bar{X}, \bar{Y})$ must satisfy both equations simultaneously:
(1) $2\bar{X} + 3\bar{Y} = 18$
(2) $4\bar{X} + \bar{Y} = 16$
• Multiply equation (2) by 3: $12\bar{X} + 3\bar{Y} = 48$.
• Subtract equation (1):
$$(12\bar{X} + 3\bar{Y}) - (2\bar{X} + 3\bar{Y}) = 48 - 18 \implies 10\bar{X} = 30 \implies \mathbf{\bar{X} = 3}$$
• Substitute $\bar{X} = 3$ into equation (2):
$$4(3) + \bar{Y} = 16 \implies 12 + \bar{Y} = 16 \implies \mathbf{\bar{Y} = 4}$$ Answer: $\bar{X} = 3$ and $\bar{Y} = 4$.
Example 2 (Appropriate Line Selection for Prediction):
A corporate analyst establishes the regression line of sales ($Y$) on advertising expenditure ($X$) as $Y = 4X + 25$, and the regression line of $X$ on $Y$ as $X = 0.16Y - 2$.
(a) If advertising budget is set to ₹10 Lakhs, which line should be used to forecast sales, and what is the predicted sales figure?
(b) If the board mandates a target sales revenue of ₹125 Lakhs, which line should be used to estimate required advertising, and what is the estimate?
Show Step-by-Step Solution
• (a) Predicting $Y$ given $X = 10$:
Use the regression line of $Y$ on $X$:
$$\hat{Y} = 4(10) + 25 = 40 + 25 = \mathbf{₹65 \text{ Lakhs}}$$
• (b) Predicting $X$ given $Y = 125$:
Use the regression line of $X$ on $Y$ (never invert the $Y$ on $X$ line!):
$$\hat{X} = 0.16(125) - 2 = 20 - 2 = \mathbf{₹18 \text{ Lakhs}}$$
CHECK YOUR PROGRESS 12.1
Graded textbook exercises on Regression Foundations, Mean Intersections, and Predictive Selection from CBSE Class 11 Applied Mathematics.
Problem 1Mean Determination Drill
The two regression lines between variables $X$ and $Y$ are given by:
$$3X + 2Y - 26 = 0 \quad \text{and} \quad 6X + Y - 31 = 0$$
Find the mean of $X$ and the mean of $Y$.
Show Step-by-Step Solution
• Since regression lines intersect at $(\bar{X}, \bar{Y})$:
(1) $3\bar{X} + 2\bar{Y} = 26$
(2) $6\bar{X} + \bar{Y} = 31$
• From (2), $\bar{Y} = 31 - 6\bar{X}$. Substitute into (1):
$$3\bar{X} + 2(31 - 6\bar{X}) = 26 \implies 3\bar{X} + 62 - 12\bar{X} = 26$$
$$-9\bar{X} = 26 - 62 = -36 \implies \mathbf{\bar{X} = 4}$$
• $\bar{Y} = 31 - 6(4) = 31 - 24 = \mathbf{7}$. Answer: Mean of $X = 4$, Mean of $Y = 7$.
Problem 2Line Selection Rule
Explain why we cannot simply solve the regression line of $Y$ on $X$ for $X$ to estimate $X$ from $Y$.
Show Step-by-Step Solution
• The regression line of $Y$ on $X$ is derived by minimizing the sum of squares of vertical errors $\sum (y - \hat{y})^2$ under the assumption that $X$ is error-free.
• When estimating $X$ from $Y$, the objective is to minimize horizontal errors $\sum (x - \hat{x})^2$ treating $Y$ as given.
• Mathematically, the line minimizing vertical errors is distinct from the line minimizing horizontal errors (unless $r = \pm 1$). Inverting the $Y$ on $X$ equation produces a mathematically suboptimal, biased estimate of $X$.
Problem 3Angle Between Regression Lines
What is the geometric relationship between the two regression lines when:
(a) $r = +1$
(b) $r = -1$
(c) $r = 0$?
Show Step-by-Step Solution
• (a) When $r = +1$, the two lines coincide into a single upward-sloping line.
• (b) When $r = -1$, the two lines coincide into a single downward-sloping line.
• (c) When $r = 0$, the two lines are perpendicular ($90^\circ$): the regression line of $Y$ on $X$ is horizontal ($Y = \bar{Y}$), and the regression line of $X$ on $Y$ is vertical ($X = \bar{X}$).
Problem 4Origin of the Word 'Regression'
Briefly explain the historical context of the word 'regression' as coined by Sir Francis Galton in his biological studies.
Show Step-by-Step Solution
• Sir Francis Galton observed that although tall parents tended to have tall children, the average height of their sons was closer to the general population mean than the parents' height (and vice versa for short parents).
• He called this tendency to revert back to the population average a "regression toward mediocrity". Over time, the mathematical statistical technique developed for this prediction retained the name 'regression'.
Problem 5Least Squares Formulation
State the mathematical least-squares minimization condition used to derive the regression line $\hat{Y} = a + bX$.
Show Step-by-Step Solution
• The criterion requires minimizing the sum of squared residuals ($S$):
$$S = \sum_{i=1}^n e_i^2 = \sum_{i=1}^n (y_i - \hat{y}_i)^2 = \sum_{i=1}^n [y_i - (a + bx_i)]^2 = \text{Minimum}$$
• Setting partial derivatives $\frac{\partial S}{\partial a} = 0$ and \frac{\partial S}{\partial b} = 0$ yields the two normal equations of linear regression.
Comprehensive Practice Worksheet: Regression Foundations
Graded problems matching CBSE Class 11 Applied Mathematics examination blueprint.
Q1: At which coordinate point in the Cartesian plane do the two regression lines always intersect?
Show Answer & Scheme
Answer [1 Mark]: At the point of means, $(\bar{X}, \bar{Y})$.
Q2: State whether the regression relationship between $X$ and $Y$ is symmetric or asymmetric.
Show Answer & Scheme
Answer [1 Mark]:Asymmetric, because the regression line of $Y$ on $X$ is distinct from the regression line of $X$ on $Y$.
SECTION B: 2-Mark Short Answer Problems
Q3: State two fundamental differences between Correlation and Regression.
Show Answer & Scheme
Differences [2 Marks]:
(1) Purpose: Correlation measures the strength and direction of linear association; Regression models the functional relationship to forecast or predict values.
(2) Units: Correlation coefficient $r$ is dimensionless; Regression coefficients carry physical units ($ ext{units of } Y / ext{units of } X$).
Q4: If two regression lines are given by $X + 2Y - 5 = 0$ and $2X + 3Y - 8 = 0$, determine $\bar{X}$ and $\bar{Y}$.
Q5: A retail commercial chain determines the following regression equations connecting weekly advertising expenditure ($X$ in ₹ Thousands) and weekly sales revenue ($Y$ in ₹ Lakhs):
• Regression of $Y$ on $X$: $Y = 0.5X + 12$
• Regression of $X$ on $Y$: $X = 0.8Y - 4$
(i) Find the average weekly advertising expenditure $\bar{X}$ and the average weekly sales revenue $\bar{Y}$.
(ii) If the store plans to spend ₹20,000 on advertising next week ($X = 20$), predict the expected sales revenue.
(iii) If the store targets a sales revenue of ₹25 Lakhs ($Y = 25$), estimate the required advertising expenditure.
Part (ii) Predicting $Y$ from $X=20$ [1.25 Marks]:
• Use line of $Y$ on $X$:
$$\hat{Y} = 0.5(20) + 12 = 10 + 12 = \mathbf{₹22 \text{ Lakhs}}$$
Part (iii) Predicting $X$ from $Y=25$ [1.25 Marks]:
• Use line of $X$ on $Y$:
$$\hat{X} = 0.8(25) - 4 = 20 - 4 = \mathbf{₹16 \text{ Thousands}} \quad (₹16,000)$$
High-Yield Quick Revision: Regression Foundations
Summary of core concepts, comparison tables, and geometric properties for Chapter 12.1.
Correlation vs. Regression Master Table
Criteria
Correlation ($r$)
Regression ($b_{yx}, b_{xy}$)
Purpose
Measures degree of co-association
Estimates unknown values of dependent variable
Symmetry
Symmetric: $r_{xy} = r_{yx}$
Asymmetric: Line of $Y$ on $X \neq$ Line of $X$ on $Y$
Units
Pure number, dimensionless
Expressed in physical units ($ ext{units of } Y / ext{units of } X$)
Mathematical Bound
$-1 \le r \le +1$
Coefficients can exceed 1 (provided $b_{yx} \cdot b_{xy} \le 1$)
Essential Exam Rules
Intersection Point: If you are given two regression equations and asked to find the arithmetic means $\bar{X}$ and $\bar{Y}$, simply solve the two equations simultaneously.
Line Selection Protocol: To predict $Y$, use the line of $Y$ on $X$. To predict $X$, use the line of $X$ on $Y$. Never algebraicly re-arrange one line to predict the other!
10-Question Diagnostic Board Mock Examination: Foundations of Regression
Standard timed test: 4 Objective MCQs / Assertion-Reason with instant automatic grading, plus 6 multi-mark subjective problems with CBSE marking rubrics.
PART 1: Objective Section (4 Marks)
Q1: The two lines of regression intersect at the point:
Solution: (B) The two regression lines always pass through the mean point $(\bar{X}, \bar{Y})$.
Q2: If the correlation between two variables is zero ($r = 0$), the angle between the two regression lines is:
Solution: (C) When $r = 0$, regression lines are $Y = \bar{Y}$ and $X = \bar{X}$, which are mutually perpendicular ($90^\circ$).
Q3: Which of the following is TRUE regarding correlation and regression?
Solution: (C) Correlation is symmetric ($r_{xy} = r_{yx}$) whereas regression lines are asymmetric.
Q4 (Assertion-Reason): Assertion (A): To estimate the value of $X$ corresponding to a known value of $Y$, we must use the regression line of $X$ on $Y$ rather than solving the regression line of $Y$ on $X$ for $X$. Reason (R): The regression line of $X$ on $Y$ minimizes the sum of squared horizontal deviations $\sum (x_i - \hat{x}_i)^2$, whereas the line of $Y$ on $X$ minimizes vertical deviations.
Solution: (A) Both Assertion and Reason are true and Reason provides the precise least-squares mathematical justification.
PART 2: Subjective Section (Q5 - Q10)
Q52 Marks
If the two regression lines are $8X - 10Y + 66 = 0$ and $40X - 18Y = 214$, find the mean of $X$ and the mean of $Y$.
State the condition under which the two regression lines coincide into a single line.
Show Step-by-Step Solution & Marking Scheme
Condition [2 Marks]:
The two regression lines coincide into a single straight line if and only if there is perfect linear correlation between the two variables ($r = +1$ or $r = -1$). In this case, all data points lie exactly on the line, leaving zero residual error in both directions.
Q73 Marks
Explain the Principle of Least Squares and illustrate the geometric difference between vertical residuals and horizontal residuals.
Show Step-by-Step Solution & Marking Scheme
• Principle of Least Squares [1.5 Marks]: The principle of least squares chooses the parameters of a curve such that the sum of the squares of the differences between observed values and estimated values is minimized: $\sum e_i^2 = \text{minimum}$.
• Geometric Difference [1.5 Marks]:
- Vertical Residuals: Distances measured parallel to the $Y$-axis ($y_i - \hat{y}_i$). Minimized when predicting $Y$ given $X$.
- Horizontal Residuals: Distances measured parallel to the $X$-axis ($x_i - \hat{x}_i$). Minimized when predicting $X$ given $Y$.
Q83 Marks
Show that if $r = 0$, the regression line of $Y$ on $X$ is parallel to the $X$-axis, and the regression line of $X$ on $Y$ is parallel to the $Y$-axis.
Show Step-by-Step Solution & Marking Scheme
• Regression equation of $Y$ on $X$ is $Y - \bar{Y} = r\frac{\sigma_y}{\sigma_x}(X - \bar{X})$ [1 Mark].
• When $r = 0$: $Y - \bar{Y} = 0 \implies \mathbf{Y = \bar{Y}}$. This is a horizontal line parallel to the $X$-axis at height $\bar{Y}$ [1 Mark].
• Regression equation of $X$ on $Y$ is $X - \bar{X} = r\frac{\sigma_x}{\sigma_y}(Y - \bar{Y})$.
• When $r = 0$: $X - \bar{X} = 0 \implies \mathbf{X = \bar{X}}$. This is a vertical line parallel to the $Y$-axis at position $\bar{X}$ [1 Mark].
• The two lines meet perpendicularly at $(\bar{X}, \bar{Y})$.
Q9 & Q10Case Study (4 Marks Total)
Case Context: Agricultural Irrigation & Crop Yield
An agronomist studies water irrigation volume ($X$ in kilolitres) and paddy yield ($Y$ in tonnes) across commercial farms. The regression lines are determined as:
• Paddy Yield on Irrigation: $Y = 0.25X + 1.5$
• Irrigation on Paddy Yield: $X = 1.44Y - 0.16$
Q9 (2 Marks): Find the mean irrigation volume $\bar{X}$ and the mean paddy yield $\bar{Y}$. Q10 (2 Marks): If an arid farm is allocated 12 kilolitres of water ($X = 12$), forecast its expected paddy yield.
Solution to Q10 [2 Marks]:
• To predict $Y$ from $X=12$, use line of $Y$ on $X$:
$$\hat{Y} = 0.25(12) + 1.5 = 3.0 + 1.5 = \mathbf{4.5 \text{ tonnes}}$$