Question 3 (b)
Means, correlation and standard error of estimate from two regression lines
9 marks
- Time to write
- About 18 minutes
- Topic
- Regression
Question
Question 3 (b) · 9 marksThe relationship between Output (Y) and Labour Hours (X) is shown by two regression lines:
2Y - X - 55 = 0 and 3Y - 2X - 40 = 0
Calculate the following:
The mean values of output and of labour hours.
The coefficient of correlation between labour hours and output.
The standard error of estimate of labour hours on output when the variance of output is 9.
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
- (i)–(iii)3 items, each marked on its own: give each its share.
- CalculateFormula first, then the working, then the answer with its unit (₹, %, units).
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Means from the two lines 3 marks 5 min
- 2 Identifying the lines and the correlation coefficient 3 marks 5 min
- 3 Σx and the standard error of estimate 3 marks 5 min
- Check: arithmetic, units (₹, %), final answer underlined 2 min
Try it first. About 18 minutes for 9 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.
(i) Means of output and labour hours
Both regression lines pass through the point of means (X̄, Ȳ), so solve 2Y − X = 55 … (1) and 3Y − 2X = 40 … (2).
From (1), X = 2Y − 55. Substituting in (2): 3Y − 2(2Y − 55) = 40, so −Y + 110 = 40 and Y = 70; then X = 140 − 55 = 85.
Mean output Ȳ = 70; mean labour hours X̄ = 85.
(ii) Coefficient of correlation
- Taking (1) as Y on X and (2) as X on Y: (1) gives Y = 27.5 + 0.5X, so byx = 0.5; (2) gives X = 1.5Y − 20, so bxy = 1.5. byx × bxy = 0.75 ≤ 1: admissible.
- The other way round: (2) as Y on X gives byx = 2/3 and (1) as X on Y gives bxy = 2, a product of 1.3333 > 1: impossible, since r² cannot exceed 1.
So r is the geometric mean of 0.5 and 1.5, with their common positive sign:
r = +√(0.5 × 1.5) = √0.75 = 0.8660. Output and labour hours are highly, positively correlated.
(iii) Standard error of estimate of labour hours on output
Variance of output = 9, so σy = 3. From bxy = r σx / σy: σx = bxy × σy / r = 1.5 × 3 / 0.8660 = 5.1962.
The standard error of estimate of labour hours (X) from output (Y): Sxy = σx √(1 − r²) = 5.1962 × √(1 − 0.75) = 5.1962 × 0.5 = 2.60 hours.
Labour hours estimated from the line of X on Y are expected to be within about 2.6 hours of the actual figure on average.
Where the 9 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Means from the two lines | 3 |
| 2 | Identifying the lines and the correlation coefficient | 3 |
| 3 | σx and the standard error of estimate | 3 |
| Total | 9 |
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