Maharashtra State BoardHSC Commerce 12th Board Exam
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Obtain the two regression equations and estimate the test score when the productivity index is 75. - Mathematics and Statistics

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Sum

The following table gives the aptitude test scores and productivity indices of 10 workers selected at random.

Aptitude score (X) 60 62 65 70 72 48 53 73 65 82
Productivity Index (Y) 68 60 62 80 85 40 52 62 60 81

Obtain the two regression equations and estimate the test score when the productivity index is 75.

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Solution

Here, X = Aptitude score, Y = Productivity index

X = xi Y =yi `"x"_"i" - bar"x"` `bar"y"_"i" - bar"y"` `("x"_"i" - bar"x")^2` `("y"_"i" - bar"y")^2` `("x"_"i" - bar"x")("y"_"i" - bar"y")`
60 68 -5 3 25 9 -15
62 60 -3 -5 9 25 15
65 62 0 -3 0 9 0
70 80 5 15 25 225 75
72 85 7 20 49 400 140
48 40 -17 -25 289 625 425
53 52 -12 -13 144 169 156
73 62 8 -3 64 9 -24
65 60 0 -5 0 25 0
82 81 17 16 289 256 272
650 650 - - 894 1752 1044

From the table, we have

n = 10, ∑ xi = 650,  ∑ yi = 650

∴ `bar"x" = (sum "x"_"i")/"n" = 650/10 = 65`

`bar"y" = (sum "y"_"i")/"n" = 650/10 = 65`

Since the mean of X and Y are whole numbers, we will use the formula

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

From the table, we have

`sum ("x"_"i" - bar"x")("y"_"i" - bar"y") = 1044, sum ("x"_"i" - bar"x")^2 = 894, sum ("y"_"i" - bar"y") = 1752`

`"b"_"XY" = (sum ("x"_"i" - bar"x")("y"_"i" - bar"y"))/(sum("x"_"i" - bar"x")^2) = 1044/1752 = 0.59`

Now, `"a"' = bar"x" - "b"_"XY" bar"y"`

= 65 - 0.59 × 65 = 65 - 38.35  = 26.65

∴ The regression equation of Aptitude score (X) on productivity index (Y) is

X = a' + bXY Y

∴ X = 26.65 + 0.59 Y

For Y = 75,

X = 26.65 + 0.59 × 75 = 26.65 + 44.25 = 70.9

∴ The test score is 70.9 when productivity index is 75

Concept: Types of Linear Regression
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