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Maharashtra State Board · HSC Commerce · Class 12 (Std XII)

Class 12 Maths and Statistics Part 2 Chapter 3: Linear Regression

Short notes, key concepts, quick revision and practice MCQs with answers

Regression estimates the value of one variable from a known value of another. The chapter explains types of regression, fits simple linear regression lines by the method of least squares, computes the regression coefficients of Y on X and X on Y and uses their properties.

Key concepts: Linear Regression

Meaning of regression
A statistical method to estimate (predict) the value of a dependent variable from a given value of an independent variable using an average relationship.
Types of regression
Simple (one independent variable) or multiple; linear (straight line) or non-linear.
Method of least squares
The regression line is the line for which the sum of squares of vertical deviations Σ(y − ŷ)² is minimum.
Regression of Y on X
Y = a + bX, where b = b_YX = cov(X, Y)/var(X) = [Σxy − n x̄ȳ]/[Σx² − n x̄²] and a = ȳ − b x̄. Used to estimate Y for a given X.
Regression of X on Y
X = a′ + b′Y, where b′ = b_XY = cov(X, Y)/var(Y) and a′ = x̄ − b′ȳ. Used to estimate X for a given Y.
Properties of regression coefficients
b_YX × b_XY = r²; r = ±√(b_YX b_XY) with the common sign of the coefficients; if one coefficient is greater than 1, the other is less than 1; b_YX = r σᵧ/σₓ and b_XY = r σₓ/σᵧ.
Point of intersection
Both regression lines pass through (x̄, ȳ), so the means are found by solving the two regression equations simultaneously.
Change of origin and scale
Regression coefficients are unaffected by change of origin. If u = (x − a)/h and v = (y − b)/k, then b_YX = (k/h) b_VU and b_XY = (h/k) b_UV.

Quick revision: important points

Practice MCQs with answers

  1. The regression line of Y on X is used to estimate:

    1. X for a given Y
    2. Y for a given X
    3. r
    4. The means only
    Show answer

    Answer: B. Y for a given X
    Y is the dependent variable.

  2. The regression coefficient b_YX equals:

    1. cov(X, Y)/var(Y)
    2. cov(X, Y)/var(X)
    3. var(X)/cov(X, Y)
    4. r σₓ/σᵧ
    Show answer

    Answer: B. cov(X, Y)/var(X)
    Covariance divided by the variance of X.

  3. The regression coefficient b_XY equals:

    1. cov(X, Y)/var(Y)
    2. cov(X, Y)/var(X)
    3. r σᵧ/σₓ
    4. var(Y)/cov(X, Y)
    Show answer

    Answer: A. cov(X, Y)/var(Y)
    Covariance divided by the variance of Y.

  4. Both regression lines pass through the point:

    1. (0, 0)
    2. (x̄, ȳ)
    3. (σₓ, σᵧ)
    4. (b_YX, b_XY)
    Show answer

    Answer: B. (x̄, ȳ)
    The point of means.

  5. The product of the two regression coefficients equals:

    1. r
    2. r²
    3. 1
    4. cov(X, Y)
    Show answer

    Answer: B. r²
    b_YX × b_XY = r².

All Class 12 Maths and Statistics chapters

Class 12 Maths and Statistics notes: all chapters