Following data are given for marks in subject A and B in a certain examination : (10)
SUBJECT A SUBJECT B
MEAN MARKS 36 85
STANDARD DEVIATION 11 8
Coefficient of correlation between A and B = ±0.66
i) Determine the two equations of regression
ii) Calculate the expected marks in A corresponding to 75 marks obtained in B.
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Theory:
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Covariance Cov(X, Y) = σ_X * σ_Y * Corr(X, Y) = E [(X - X_bar) (Y - Y_bar) ]
Slope of the Linear regression line: beta β = Covariance (X, Y) / variance(X)
β = σ_X * σ_Y * Corr(X, Y) / σ_X² = Corr(X, Y) * σ_Y / σ_X
α = alpha = Y_bar - β * X_bar = Y intercept of the line.
equation of linear regression: Y = β * X + α
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The given problem:
Here X is the variable denoting the marks in subject A and Y is the variable denoting marks in subject B.
Given data: X_bar = 36 , Y_bar = 85, σ_X = 11 , σ_Y = 8
and Corr(X, Y) = +0.66 or -0.66
So β = 0.66 * 8 / 11 = 0.48
α = alpha = 85 - 0.48 * 36 = 67.72
=> Equation: Y = 0.48 X + 67.72 , OR, B = 0.48 A + 67.72 ---- (1)
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I am not sure of the following. I am taking X as variable for the marks in subject B and Y as the variable for the marks in subject A. But correlation coefficient remains the same as: Corr(X,Y) = Corr(Y, X).
β = 0.66 * 11 / 8 = 0.9075
α = 36 - 0.9075 * 85 = - 41.1375
=> equation is: Y = 0.9075 X - 41.1375.
writing in terms of A and B, A = 0.9075 B - 41.1375. --- (2)
============================
marks obtained in subject B = 75.
As per (1), 75 = 0.48 A + 67.72
A = 7.28 /0.48 = 15.17 marks
as per (2) , A = 0.9075 * 75 - 41.1375 = 26.925 marks
I am not really too sure. Please verify.
=========
Covariance Cov(X, Y) = σ_X * σ_Y * Corr(X, Y) = E [(X - X_bar) (Y - Y_bar) ]
Slope of the Linear regression line: beta β = Covariance (X, Y) / variance(X)
β = σ_X * σ_Y * Corr(X, Y) / σ_X² = Corr(X, Y) * σ_Y / σ_X
α = alpha = Y_bar - β * X_bar = Y intercept of the line.
equation of linear regression: Y = β * X + α
================
The given problem:
Here X is the variable denoting the marks in subject A and Y is the variable denoting marks in subject B.
Given data: X_bar = 36 , Y_bar = 85, σ_X = 11 , σ_Y = 8
and Corr(X, Y) = +0.66 or -0.66
So β = 0.66 * 8 / 11 = 0.48
α = alpha = 85 - 0.48 * 36 = 67.72
=> Equation: Y = 0.48 X + 67.72 , OR, B = 0.48 A + 67.72 ---- (1)
========================
I am not sure of the following. I am taking X as variable for the marks in subject B and Y as the variable for the marks in subject A. But correlation coefficient remains the same as: Corr(X,Y) = Corr(Y, X).
β = 0.66 * 11 / 8 = 0.9075
α = 36 - 0.9075 * 85 = - 41.1375
=> equation is: Y = 0.9075 X - 41.1375.
writing in terms of A and B, A = 0.9075 B - 41.1375. --- (2)
============================
marks obtained in subject B = 75.
As per (1), 75 = 0.48 A + 67.72
A = 7.28 /0.48 = 15.17 marks
as per (2) , A = 0.9075 * 75 - 41.1375 = 26.925 marks
I am not really too sure. Please verify.
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