Question

You are given the following information about advertising expenditure and sales.

	Advertisement expenditure (₹ in lakh) (X)	Sales (₹ in lakh) (Y)
Arithmetic Mean	10	90
Standard Mean	3	12

Correlation coefficient between X and Y is 0.8

1. Obtain the two regression equations.
2. What is the likely sales when the advertising budget is ₹ 15 lakh?
3. What should be the advertising budget if the company wants to attain sales target of ₹ 120 lakh?

Answer 1

Given: `bar x = 10, bar y = 90, sigma_x = 3, sigma_y = 12`, r = 0.8

`"b"_"YX" = "r" sigma_y/sigma_x = 0.8 xx 12/3 = 0.8 xx 4` = 3.2

`"b"_"XY" = "r" sigma_x/sigma_y = 0.8 xx 3/12 = 0.8 xx 0.25` = 0.2

(i) The regression equation of Y on X is

`("Y" - bar y) = "b"_"YX" ("X" - bar x)`

∴ (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" - bar x) = "b"_"XY" ("Y" - bar y)`

∴ (X - 10) = 0.2 (Y - 90)

∴ X - 10 = 0.2 Y - 18

∴ X = 0.2 Y - 18 + 10

∴ X = 0.2 Y - 8         .....(ii)

(ii) For X = 15, from equation (i) we get

Y = 3.2 (15) + 58 = 48 + 58 = 106

∴ Likely sales is ₹ 106 lakh when advertising budget is ₹ 15 lakh.

(iii) For Y = 120, from equation (ii) we get

X = 0.2 (120) - 8 = 24 - 8 = 16

∴ To attain sales target of ₹ 120 lakh, advertising budget must be ₹ 16 lakh.