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# Calculate the regression equations of X on Y and Y on X from the following data: - Mathematics and Statistics

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Sum

Calculate the regression equations of X on Y and Y on X from the following data:

 X 10 12 13 17 18 Y 5 6 7 9 13
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#### Solution

 X = xi Y = yi "x"_"i"^2 "y"_"i"^2 xi yi 10 5 100 25 50 12 6 144 36 72 13 7 169 49 91 17 9 289 81 153 18 13 324 169 234 70 40 1026 360 600

From the table, we have,

n = 5, ∑ xi = 70, ∑ yi = 40, ∑ xi yi = 600, sum"x"_"i"^2 = 1026, sum"y"_"i"^2 = 360

bar"x" = sum"x"_"i"/"n" = 70/5 = 14,

bar"y" = sum"y"_"i"/"n" = 40/5 = 8

Now, for regression equation of X on Y

"b"_"XY" = (sum"x"_"i" "y"_"i" - "n"  bar "x"  bar "y")/(sum "y"_"i"^2 - "n" bar"y"^2)

= (600 - 5 xx 14 xx 8)/(360 - 5(8)^2) = (600 - 560)/(360 - 320) = 40/40 = 1

Also, "a"' = bar"x" - "b"_"XY"  bar"y" = 14 - 1(8) = 14 - 8 = 6

∴ The regression equation of X on Y is

X = a' + bXYY

∴ X = 6 + Y

Now, for regression equation of Y on X

"b"_"YX" = (sum"x"_"i" "y"_"i" - "n" bar "x" bar "y")/(sum "x"_"i"^2 - "n"  bar"x"^2)

= (600 - 5(14)(8))/(1026 - 5(14)^2) = (600- 560)/(1026 - 980) = 40/46 = 0.87

Also, a = bar"y" - "b"_"YX"  bar"x"

= 8 - 0.87 xx 14 = 8 - 12.18 = - 4.18`

∴ The regression equation of Y on X is

Y = a + bYX X

∴ Y = - 4.18 + 0.87X

Concept: Types of Linear Regression
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