Advertising and sales: fitting a line, reading r², and the trap of inverting a regression
A wholesale distributor of school stationery has eight months of records on monthly advertising spend (X, in ₹ thousands) and monthly sales (Y, in ₹ thousands). The clerk who kept the data has left, but the summary totals survive: n = 8, ΣX = 80, ΣY = 400, ΣX² = 900, ΣY² = 21,800, ΣXY = 4,400. Monthly advertising over the period ranged between ₹4,000 and ₹16,000. The sales head wants three things before the new session's budget is fixed: a line he can quote, a single number that says how much of the movement in sales advertising actually explains, and an estimate of the advertising needed to reach monthly sales of ₹70,000. The accountant, working from the same totals, has already written down X = (Y − 10)/4 by inverting the sales equation, and intends to use it for the ₹70,000 estimate.
1. Fit the regression of sales on advertising and interpret both coefficients.
Read the worked answer
Mean X = 80/8 = 10; mean Y = 400/8 = 50.
b_yx = [nΣXY − ΣX·ΣY]/[nΣX² − (ΣX)²] = [8(4,400) − 80(400)]/[8(900) − 80²] = [35,200 − 32,000]/[7,200 − 6,400] = 3,200/800 = 4.
a = Ȳ − b·X̄ = 50 − 4(10) = 10.
Regression equation: Y = 10 + 4X, i.e. sales (₹'000) = 10 + 4 × advertising (₹'000).
Interpretation of b = 4: within the observed range of advertising (₹4,000–₹16,000), an extra ₹1,000 of advertising is associated with about ₹4,000 more monthly sales on average. The word 'associated' matters: this is not proof that advertising caused the increase.
Interpretation of a = 10 is the weaker claim. It says the model predicts ₹10,000 of sales at zero advertising, but zero is far outside the observed range and is an extrapolation; the intercept is best read as the anchor that positions the line and should not be quoted as a business fact.
2. Compute the coefficient of determination and explain what it does and does not say about the relationship.
Read the worked answer
r = [nΣXY − ΣX·ΣY] / √{[nΣX² − (ΣX)²][nΣY² − (ΣY)²]}
Numerator = 3,200. Denominators: 8(900) − 80² = 800; 8(21,800) − 400² = 174,400 − 160,000 = 14,400.
r = 3,200 / √(800 × 14,400) = 3,200 / √(1,15,20,000) = 3,200 / 3,394.11 ≈ 0.943
r² ≈ 0.889, so about 88.9% of the variation in monthly sales is accounted for by the linear relationship with advertising; roughly 11.1% is left to price, season, school-term timing, competitor activity and plain noise.
What this does say: the linear fit is strong and positive, and quoting a single slope is defensible. What it does not say: (a) it does not establish causation - advertising budgets may follow expected sales rather than drive them; (b) r is symmetric in X and Y but the regression of Y on X is not, so r² says nothing about how well the reverse line fits; (c) 88.9% of a small eight-observation sample can be fragile, and the relationship is only validated over ₹4,000–₹16,000 of spend - predicting sales at ₹25,000 of advertising would be extrapolation beyond the data that produced r.
3. Evaluate the accountant's inverted equation X = (Y − 10)/4 for estimating the advertising needed for ₹70,000 of sales, and give the correct estimate.
Read the worked answer
The accountant has solved Y = 10 + 4X for X, which assumes the line of Y on X is the same as the line of X on Y. It is not. Two distinct least-squares lines exist, and they coincide only when |r| = 1; here r ≈ 0.943.
Correct X-on-Y regression:
b_xy = [nΣXY − ΣX·ΣY]/[nΣY² − (ΣY)²] = 3,200/14,400 = 0.2222
a = X̄ − b_xy·Ȳ = 10 − 0.2222(50) = 10 − 11.111 = −1.111
X = −1.111 + 0.2222Y. At Y = 70: X = −1.111 + 15.556 = 14.44, i.e. about ₹14,440 of monthly advertising.
The accountant's inversion gives (70 − 10)/4 = 15, i.e. ₹15,000 - about ₹560 (3.7%) too high. The error arises because the reverse line minimises deviations measured along the other axis and therefore rotates towards the mean of Y.
Both figures lie inside the observed spend range of ₹4,000–₹16,000, so no extrapolation warning is needed here - but the ₹70,000 sales target itself must be checked against observed sales before it is quoted, and any single-point forecast should be presented as a central estimate, not a promise.