Question

From the data given below:

Marks in Economics:	25	28	35	32	31	36	29	38	34	32
Marks in Statistics:	43	46	49	41	36	32	31	30	33	39

Find

1. The two regression equations,
2. The coefficient of correlation between marks in Economics and Statistics,
3. The mostly likely marks in Statistics when the marks in Economics is 30.

Answer 1

Marks in Economics (X)	Marks in Statistics (Y)	x = `"X" - bar"X"`	y = `"Y" - bar"Y"`	x2	y2	xy
25	43	− 7	5	49	25	− 35
28	46	− 4	8	16	64	− 32
35	49	3	11	9	121	33
32	41	0	3	0	9	0
31	36	− 1	− 2	1	4	2
36	32	4	− 6	16	36	− 24
29	31	− 3	− 7	9	49	21
38	30	6	− 8	36	64	− 48
34	33	2	− 5	4	25	− 10
32	39	0	1	0	1	0
320	380	0	0	140	398	− 93

N = 10, ∑X = 320, ∑Y = 280, ∑x² = 140, ∑y² = 398, ∑xy = − 93, `bar"X" = 320/100` = 32, `bar"Y" = 380/100` = 38

(a) Regression equation of X on Y.

b_xy = `"r"(sigma_"x")/(sigma_"y") = (sum"xy")/(sum"y"^2) = (-93)/398` = − 0.234

`"X" - bar"X" = "b"_"xy"("Y" - bar"Y")`

X − 32 = − 0.234(Y − 38)

X = − 0.234Y + 8.892 + 32

X = − 0.234Y + 40.892

Regression equation of Y on X.

`"Y" - bar"Y" = "b"_"xy"("X" - bar"X")`

b_yx = `"r"(sigma_"x")/(sigma_"y") = (sum"xy")/(sum"y"^2) = (-93)/140` = − 0.664

Y − 38 = − 0.664(X − 32)

Y = − 0.664X + 21.248 + 38

Y = − 0.664X + 59.248

(b) Coefficient of correlation (r) = `±sqrt("b"_"xy" xx "b"_"yx")`

= `sqrt((-0.234)(-0.664))`

= − 0.394

(c) When X = 30, Y = ?

Y = − 0.664(30) + 59.248

= − 19.92 + 59.248

= 39.328