Question 3 (a)
Regression of sales on advertising: equations, r², an estimate and the standard error
10 marks
- Time to write
- About 20 minutes
- Topic
- Regression
Question
Question 3 (a) · 10 marksA retail company wishes to analyze how advertising expenditure affects its monthly sales revenue. The marketing team collected data for the past ten months on the advertising budget (in thousand rupees) and the matching sales revenue (in thousand rupees), as shown below:
| Advertising Expenditure ('000 ₹) | 8 | 11 | 14 | 24 | 16 | 18 | 19 | 26 | 19 | 20 |
|---|---|---|---|---|---|---|---|---|---|---|
| Sales ('000 ₹) | 62 | 85 | 96 | 70 | 80 | 90 | 80 | 81 | 75 | 84 |
Calculate the following:
Two regression equations,
The coefficient of determination between advertising expenditure and sales,
How will sales change if the advertising expenditure is Rs. 35,000?
Find out the standard error of estimate for advertising expenditure on sales.
Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.
Before you write
Read off the question and its marks
Read from the question
- Calculate…How…?Find…3 questions in all: answer every one.
- CalculateFormula first, then the working, then the answer with its unit (₹, %, units).
What the paper leaves open
As printed, the table gives advertising expenditure and sales both in thousand rupees, and item (iii) asks about an amount stated in rupees.
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Working table, means and regression coefficients 3 marks 5 min
- 2 The two regression equations 2 marks 4 min
- 3 Coefficient of determination with interpretation 2 marks 3 min
- 4 Estimate of sales 1½ marks 3 min
- 5 Standard error of estimate 1½ marks 3 min
- Check: arithmetic, units (₹, %), final answer underlined 1 min
Try it first. About 20 minutes for 10 marks. Write it out, then open the solution.
Solution
Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.
Method. Both regression lines are fitted by least squares and pass through the means (x̄, ȳ). Working from deviations about the actual means, byx = Σdxdy / Σdx² and bxy = Σdxdy / Σdy², and r is the geometric mean of the two regression coefficients, with their common sign. X is advertising expenditure and Y is sales, both in ₹'000.
Working table (x̄ = 175 / 10 = 17.5; ȳ = 803 / 10 = 80.3; dx = X − 17.5, dy = Y − 80.3)
| X | Y | dx | dy | dx² | dy² | dx·dy |
|---|---|---|---|---|---|---|
| 8 | 62 | −9.5 | −18.3 | 90.25 | 334.89 | 173.85 |
| 11 | 85 | −6.5 | 4.7 | 42.25 | 22.09 | −30.55 |
| 14 | 96 | −3.5 | 15.7 | 12.25 | 246.49 | −54.95 |
| 24 | 70 | 6.5 | −10.3 | 42.25 | 106.09 | −66.95 |
| 16 | 80 | −1.5 | −0.3 | 2.25 | 0.09 | 0.45 |
| 18 | 90 | 0.5 | 9.7 | 0.25 | 94.09 | 4.85 |
| 19 | 80 | 1.5 | −0.3 | 2.25 | 0.09 | −0.45 |
| 26 | 81 | 8.5 | 0.7 | 72.25 | 0.49 | 5.95 |
| 19 | 75 | 1.5 | −5.3 | 2.25 | 28.09 | −7.95 |
| 20 | 84 | 2.5 | 3.7 | 6.25 | 13.69 | 9.25 |
| 175 | 803 | 0 | 0 | 272.5 | 846.1 | 33.5 |
(i) The two regression equations
- byx = 33.5 / 272.5 = 0.1229; bxy = 33.5 / 846.1 = 0.0396.
- Y on X (sales on advertising): Y − 80.3 = 0.1229 (X − 17.5), that is Y = 78.15 + 0.1229X.
- X on Y (advertising on sales): X − 17.5 = 0.0396 (Y − 80.3), that is X = 14.32 + 0.0396Y.
(ii) Coefficient of determination
r = Σdxdy / √(Σdx² × Σdy²) = 33.5 / √(272.5 × 846.1) = 0.0698 (check: √(0.1229 × 0.0396) = 0.0698).
r² = (0.0698)² = 0.0049. The coefficient of determination is the share of the total variance that the relationship explains: advertising explains only about 0.49% of the month-to-month variation in sales; 99.51% is due to other factors.
(iii) Sales when advertising is ₹35,000
The data are in ₹'000 (see the paper note), so X = 35 is used in the line of Y on X: Y = 78.15 + 0.1229 × 35 = 82.45, that is estimated sales of about ₹82,450, only ₹2,150 above average sales of ₹80,300. With r² this small, the estimate adds almost nothing to simply using the mean, and X = 35 also lies outside the observed range of 8 to 26, so it should not be relied on.
(iv) Standard error of estimate
σx = √(272.5 / 10) = 5.22; σy = √(846.1 / 10) = 9.20.
- Advertising estimated from sales (X on Y), as the item is worded: Sxy = σx √(1 − r²) = 5.22 × √(1 − 0.0049) = 5.21 (₹'000).
- Sales estimated from advertising (Y on X): Syx = σy √(1 − r²) = 9.20 × √(1 − 0.0049) = 9.18 (₹'000).
Each standard error is almost as large as the SD of the variable itself, confirming that the regression line improves the estimate very little.
Overall comment. On these figures advertising and sales are practically uncorrelated; the company should not expect higher advertising alone to raise sales, and should look at other factors (price, season, competition).
Where the 10 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Working table, means and regression coefficients | 3 |
| 2 | The two regression equations | 2 |
| 3 | Coefficient of determination with interpretation | 2 |
| 4 | Estimate of sales at ₹35,000 | 1.5 |
| 5 | Standard error of estimate | 1.5 |
| Total | 10 |
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