Question 3 OR (c)
Correlation and regression of sales on incentives: coefficients, equations, estimate, standard errors
14 marks
- Time to write
- About 28 minutes
- Topic
- Regression
Question
Question 3 OR (c) · 14 marksPQR Ltd. wants to study how its sales performance relates to employee incentives. The firm's HR collects data for the last 9 months of the total sales (in thousands of rupees) achieved by sales representatives and of the total employee incentives (in thousands of rupees), as shown below:
| Incentive ('000 ₹) | 14 | 18 | 27 | 23 | 29 | 24 | 16 | 21 | 18 |
|---|---|---|---|---|---|---|---|---|---|
| Sales ('00000 ₹) | 34 | 38 | 52 | 41 | 55 | 47 | 36 | 44 | 43 |
You are to:
Compute the correlation coefficient and the regression coefficients.
Determine the two regression equations.
What will the value of Sales be if the incentive is ₹32,000?
Determine the standard errors of estimate of incentive on sales and of sales on incentive.
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
- Compute…Determine…What…?4 questions in all: answer every one.
- ComputeFormula first, then the working, then the answer with its unit (₹, %, units).
What the paper leaves open
As printed, the text gives sales in thousands of rupees while the table heading gives Sales in '00000 rupees; the heading is kept as printed.
Write it in this order
- Read the data; note exactly what is asked 2 min
- 1 Working table and totals 2 marks 4 min
- 2 Correlation coefficient and regression coefficients 4 marks 7 min
- 3 The two regression equations 3 marks 5 min
- 4 Estimate of sales 2 marks 3 min
- 5 Standard errors of estimate 3 marks 5 min
- Check: arithmetic, units (₹, %), final answer underlined 2 min
Try it first. About 28 minutes for 14 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. X is the incentive (₹'000) and Y the sales, in the unit of the table heading (see the paper note). Karl Pearson's coefficient is worked from the totals of the data, r = [nΣXY − ΣXΣY] / √{[nΣX² − (ΣX)²][nΣY² − (ΣY)²]}; the regression coefficients use the same totals, and r² = byx × bxy.
| Month | X | Y | X² | Y² | XY |
|---|---|---|---|---|---|
| 1 | 14 | 34 | 196 | 1,156 | 476 |
| 2 | 18 | 38 | 324 | 1,444 | 684 |
| 3 | 27 | 52 | 729 | 2,704 | 1,404 |
| 4 | 23 | 41 | 529 | 1,681 | 943 |
| 5 | 29 | 55 | 841 | 3,025 | 1,595 |
| 6 | 24 | 47 | 576 | 2,209 | 1,128 |
| 7 | 16 | 36 | 256 | 1,296 | 576 |
| 8 | 21 | 44 | 441 | 1,936 | 924 |
| 9 | 18 | 43 | 324 | 1,849 | 774 |
| Total | 190 | 390 | 4,216 | 17,300 | 8,504 |
n = 9; X̄ = 190 / 9 = 21.11; Ȳ = 390 / 9 = 43.33.
- nΣX² − (ΣX)² = 9 × 4,216 − 190² = 37,944 − 36,100 = 1,844.
- nΣY² − (ΣY)² = 9 × 17,300 − 390² = 1,55,700 − 1,52,100 = 3,600.
- nΣXY − ΣXΣY = 9 × 8,504 − 190 × 390 = 76,536 − 74,100 = 2,436.
(i) Correlation coefficient and regression coefficients
- r = 2,436 / √(1,844 × 3,600) = 2,436 / 2,576.48 = 0.9455.
- byx (sales on incentive) = 2,436 / 1,844 = 1.3210.
- bxy (incentive on sales) = 2,436 / 3,600 = 0.6767.
Check: √(1.3210 × 0.6767) = 0.9455. Incentives and sales are very highly, positively correlated.
(ii) The two regression equations
- Y on X: Y − 43.33 = 1.3210 (X − 21.11), so Y = 15.44 + 1.3210X.
- X on Y: X − 21.11 = 0.6767 (Y − 43.33), so X = −8.21 + 0.6767Y.
(iii) Sales when the incentive is ₹32,000
X = 32 in the line of Y on X: Y = 15.44 + 1.3210 × 32 = 57.72, in the unit of the sales column. Read with the text (sales in ₹'000) this is about ₹57,720; read with the table heading (₹'00000) it is ₹57.72 lakh. X = 32 lies a little beyond the observed incentives (14 to 29), so the estimate assumes the straight-line relation continues.
(iv) Standard errors of estimate
σx = √1,844 / 9 = 4.7713 and σy = √3,600 / 9 = 6.6667; √(1 − r²) = √(1 − 0.8940) = 0.3256. The standard error is the unexplained part of each variable's spread, (1 − r²) being the share of variance the regression leaves unexplained.
- Incentive on sales (X estimated from Y): Sxy = σx √(1 − r²) = 4.7713 × 0.3256 = 1.55 (₹'000).
- Sales on incentive (Y estimated from X): Syx = σy √(1 − r²) = 6.6667 × 0.3256 = 2.17 (sales units).
Both errors are small beside the spread of each variable, so estimates from either line are fairly reliable.
Where the 14 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Working table and totals | 2 |
| 2 | Correlation coefficient and regression coefficients | 4 |
| 3 | The two regression equations | 3 |
| 4 | Estimate of sales at ₹32,000 | 2 |
| 5 | Standard errors of estimate | 3 |
| Total | 14 |
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