Question

Given the following information about the production and demand of a commodity.
Obtain the two regression lines:

	ADVERTISEMENT (x) (₹ in lakhs)	DEMAND (y) (₹ in lakhs)
Mean	10	90
Variance	9	144

Coefficient of correlation between x and y is 0.8.
What should be the advertising budget if the company wants to attain the sales target of ₹ 150 lakhs?

Answer 1

Given, `bar(x)` = 10, `bar(y)` = 90, `sigma_x^2` = 9, `sigma_y^2` = 144, r = 0.8

∴ `sigma_x` = 3, `sigma_y` = 12

b_yx = `"r" sigma_y/sigma_x = 0.8 xx 12/3` = 0.8 × 4 = 3.2

b_xy = `"r" sigma_x/sigma_y = 0.8 xx 3/12` = 0.8 × 0.25 = 0.2

The regression equation of Y on X is

`("Y" - bary) = "b"_(yx) ("X" - barx)`

∴ (Y – 90) = 3.2 (X – 10)

∴ Y – 90 = 3.2 X – 32

∴ Y = 3.2 X – 32 + 90

∴ Y = 3.2 X + 58    ......(i)

The regression equation of X on Y is

`("X" - barx) = "b"_(xy) ("Y" - bary)`

∴ (X – 10) = 0.2 (Y – 90)

∴ X – 10 = 0.2 Y – 18

∴ X = 0.2 Y – 18 + 10

∴ X = 0.2 Y – 8    ......(ii)

When the company wants to attain the sales target of ₹ 150 lakhs,

Put Y = 150 lakh in equation (ii)

∴ X = 0.2 × 150 – 8 = 30 – 8 = 22

∴ The advertising budget should be ₹ 22 lakhs if the company wants to attain the sales target of ₹ 150 lakhs.