Maharashtra State BoardHSC Commerce 12th Board Exam
Advertisement Remove all ads

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

Advertisement Remove all ads
Advertisement Remove all ads
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 productivity index of a worker whose test score is 95.

Advertisement Remove all ads

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"_"YX" = (sum ("x"_"i" - bar"x")("y"_"i" - bar"y"))/(sum("x"_"i" - bar"x")^2) = 1044/894 = 1.16`

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

= 65 - 1.16 × 65 = 65 - 75.4 = - 10.4

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

Y = a + bYX X

∴ Y = - 10.4 + 1.16 X

For X = 95,

Y = - 10.4 + 1.16(95) = - 10.4 + 110.2 = 99.8

∴ The productivity index of worker with a test score of 95 is 99.8.

Concept: Types of Linear Regression
  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×