# Obtain the two regression equations and estimate the test score when the productivity index is 75. - Mathematics and Statistics

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.

#### 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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Chapter 3: Linear Regression - Exercise 3.1 [Page 41]

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