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
  Is there an error in this question or solution?
Chapter 3: Linear Regression - Exercise 3.1 [Page 41]

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