Advertising, Sales and Two Regression Lines
A retailer has ten months of records on monthly advertising spend (x, ₹ lakh) and sales (y, ₹ lakh). The summary totals are: n = 10, Σx = 50, Σy = 400, Σx² = 280, Σy² = 16,600, Σxy = 2,120. Advertising ranged from ₹2 lakh to ₹9 lakh, and sales from ₹28 lakh to ₹56 lakh. The marketing head wants three things before the annual budget is fixed: a measure of how strongly the two variables move together, an equation to forecast sales from a proposed advertising level, and a view on whether doubling the budget to ₹12 lakh would reliably deliver about ₹68 lakh of sales. The finance director is more interested in working backwards: if the company needs ₹60 lakh of sales, how much advertising should be budgeted? The two directors have asked for one consistent statistical basis rather than two ad hoc calculations, and the retailer has no plans to change pricing or product mix.
1. Compute the correlation coefficient and the regression equation of sales on advertising, and interpret both.
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Means: x̄ = 50/10 = 5; ȳ = 400/10 = 40.
Sxx = Σx² − n x̄² = 280 − 10(25) = 30 Syy = 16,600 − 10(1,600) = 600 Sxy = 2,120 − 10(5)(40) = 120
r = Sxy ÷ √(Sxx × Syy) = 120 ÷ √(30 × 600) = 120 ÷ 134.164 = 0.894.
Regression of y on x: b = Sxy ÷ Sxx = 120 ÷ 30 = 4 a = ȳ − b x̄ = 40 − 4(5) = 20 y = 20 + 4x
Interpretation: r = 0.894 indicates a strong positive linear association. r² = 0.80, so about 80% of the variation in monthly sales is explained by the linear relationship with advertising, leaving 20% to other factors. The slope of 4 means that, within the observed range, each additional ₹1 lakh of advertising is associated with ₹4 lakh higher sales. This is an association, not proof of a ₹4 return on every advertising rupee, because both variables may respond to seasonality or promotion timing.
2. Forecast sales at ₹8 lakh of advertising. How should the directors treat the proposal to spend ₹12 lakh for about ₹68 lakh of sales?
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At x = 8: y = 20 + 4(8) = ₹52 lakh. This is an interpolation, since ₹8 lakh lies inside the observed ₹2–9 lakh range, and the prediction is reasonably supported.
At x = 12: the equation gives y = 20 + 48 = ₹68 lakh. The arithmetic is correct but the use is not. Three cautions apply:
- Extrapolation: ₹12 lakh is 33% above the highest observed spend. The fitted straight line cannot show diminishing returns, saturation or competitor response, all of which typically appear at higher spend levels.
- Fit is not perfect: r² = 0.80, so 20% of sales variation is unexplained and the forecast has a non-trivial standard error, which widens further outside the data range.
- Reverse causality and confounding: advertising budgets may be set in months when demand is already expected to be high.
A ₹12 lakh commitment should not be signed off on this equation. A staged test, for example ₹10 lakh for two months with a control region, would generate the data needed before scaling. The ₹68 lakh figure should be presented as a mechanical extrapolation, not a forecast.
3. The finance director needs ₹60 lakh of sales. Compute the advertising budget using the appropriate regression line and explain why inverting the sales-on-advertising equation gives a different, less defensible answer.
Read the worked answer
To predict x from y, use the regression of x on y, not a rearranged y-on-x line.
b_xy = Sxy ÷ Syy = 120 ÷ 600 = 0.2 x = x̄ + b_xy (y − ȳ) = 5 + 0.2(60 − 40) = 5 + 4 = ₹9.0 lakh.
The sales-on-advertising line would give x = (60 − 20) ÷ 4 = ₹10.0 lakh. The two answers differ by ₹1 lakh because the two regression lines are different unless r = ±1; the y-on-x line minimises vertical deviations and is not algebraically invertible for prediction in the other direction. Each line minimises errors in its own direction.
The defensible budget is ₹9.0 lakh, but it still needs a caveat: ₹60 lakh is above the observed maximum of ₹56 lakh, and ₹9 lakh sits at the top edge of the advertising range. A point estimate without a prediction interval overstates precision. The directors should treat ₹9 lakh as a planning starting point and monitor actual sales before committing the full amount.