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Question
For 5 observations of pairs of (X, Y) of variables X and Y the following results are obtained. ∑X = 15, ∑Y = 25, ∑X2 = 55, ∑Y2 = 135, ∑XY = 83. Find the equation of the lines of regression and estimate the values of X and Y if Y = 8; X = 12.
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Solution
N = 5, ΣX = 15, ΣY = 25, ΣX2 = 55, ΣY2 = 135, ΣXY = 83, `bar"X" = 15/5` = 3, `bar"Y" = 25/5` = 5.
byx = `("N"sum"XY" - (sum"X")(sum"Y"))/("N"(sum"X"^2) - (sum"X")^2)`
= `(5(83) - (15)(25))/(5(55) - (15)^2)`
= `(415 - 375)/(275 - 225)`
= `40/50`
= 0.8
Regression line of Y on X:
`"Y" - bar"Y" = "b"_"yx"("X" - bar"X")`
Y – 5 = 0.8(X – 3)
Y = 0.8X – 2.4 + 5
Y = 0.8X + 2.6
When X = 12, Y = 0.8X + 2.6
Y = (0.8)12 + 2.6
= 9.6 + 2.6
= 12.2
bxy = `("N"sum"XY" - (sum"X")(sum"Y"))/("N"(sum"Y"^2) - (sum"Y")^2)`
= `(5(83) - (15)(25))/(5(135) - (25)^2)`
= `(415 - 375)/(675 - 625)`
= `40/50`
= 0.8
Regression line of X on Y:
`"X" - bar"X" = "b"_"xy"("Y" - bar"Y")`
X – 3 = 0.8(Y – 5)
X = 0.8Y – 4 + 3
X = 0.8Y – 1
When Y = 8, X = 0.8Y – 1
X = (0.8)8 – 1
= 6.4 – 1
= 5.4
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