Question

The two regression lines were found to be 4X – 5Y + 33 = 0 and 20X – 9Y – 107 = 0. Find the mean values and coefficient of correlation between X and Y.

Answer 1

To get mean values we must solve the given lines.

4X – 5Y = – 33 ……(1)

20X – 9Y = 107 …….(2)

Equation (1) × 5

20X – 25Y = – 165

20X – 9Y = 107

Subtracting (1) and (2),

– 16Y = – 272

Y = `272/16` = 17

i.e., `bar"Y"` = 17

Using Y = 17 in (1) we get,

4X – 85 = – 33

4X = 85 – 33

4X = 52

X = 13

i.e., `bar"X"` = 13

Mean values are `bar"X"` = 13, `bar"Y"` = 17,

Let regression line of Y on X be

4X – 5Y + 33 = 0

5Y = 4X + 33

Y = `1/5` (4X + 33)

Y = `4/5"X" + 33/5`

Y = 0.8X + 6.6

∴ b_yx = 0.8

Let regression line of X on Y be

20X – 9Y – 107 = 0

20X = 9Y + 107

X = `1/20` (9Y + 107)

X = `9/20"Y" + 107/20`

X = 0.45Y + 5.35

∴ b_xy = 0.45

Coefficient of correlation between X and Y is

r = `± sqrt("b"_"yx" xx "b"_"xy")`

r = `± (0.8 xx 0.45)`

= ± 0.6

= 0.6

Both b_yx and b_xy is positive take positive sign.