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Question
The following data relate to advertisement expenditure (in lakh of rupees) and their corresponding sales (in crores of rupees)
| Advertisement expenditure | 40 | 50 | 38 | 60 | 65 | 50 | 35 |
| Sales | 38 | 60 | 55 | 70 | 60 | 48 | 30 |
Estimate the sales corresponding to advertising expenditure of ₹ 30 lakh.
Solution
| X | Y | X2 | Y2 | XY |
| 40 | 38 | 1600 | 1444 | 1520 |
| 50 | 60 | 2500 | 3600 | 3000 |
| 38 | 55 | 1444 | 3025 | 2090 |
| 60 | 70 | 3600 | 4900 | 4200 |
| 65 | 60 | 4225 | 3600 | 3900 |
| 50 | 48 | 2500 | 2304 | 2400 |
| 35 | 32 | 1225 | 900 | 1050 |
| 338 | 361 | 17094 | 19773 | 18160 |
N = 7, ΣX = 338, ΣY = 361, ΣX2 = 17094, ΣY2 = 19773, ΣXY = 18160.
`bar"X" = (sum"X")/"N" = 338/7` = 48.29
`bar"Y" = (sum"Y")/"N" = 361/7` = 51.57
byx = `("N"sum"XY" - (sum"X")(sum"Y"))/("N"sum"X"^2 - (sum"X")^2)`
= `(7(18160) - (388)(361))/(7(17094) - (338)^2)`
= `(127120 - 122018)/(119658 - 114244)`
= `5102/5414`
= 0.942
Regression equation of Y on X
`"Y" - bar"Y" = "b"_"yx"("X" - bar"X")`
Y – 51.57 = 0.942(X – 48.29)
Y – 51.57 = 0.942X – 0.942 × 48.29
Y – 51.57 = 0.942X – 45.48918
Y = 0.942X + 51.57 – 48.29
Y = 0.942X + 6.081
To find the sales, when the advertising is X = ₹ 30 lakh in the above equation we get,
Y = 0.942(30) + 6.081
= 28.26 + 6.081
= 34.341
= ₹ 34.34 crores
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