Question

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.

Answer 1

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, ∑ x_i = 650,  ∑ y_i = 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' + b_XY 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