Question 4 (b)
Coefficient of determination and standard error of estimate from explained and unexplained variation
4 marks
- Time to write
- About 8 minutes
- Topic
- Regression
Question
Question 4 (b) · 4 marksIn a regression equation of Y on X based on 11 observations, the explained and unexplained variations were found to be 18 and 27 respectively. Find out the coefficient of determination and the standard error of estimate of Y on X.
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
- Find outFormula 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 Total variation and the coefficient of determination 2 marks 3 min
- 2 Standard error of estimate of Y on X 2 marks 3 min
- Check: arithmetic, units (₹, %), final answer underlined 1 min
Try it first. About 8 minutes for 4 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.
Working table (variations of Y about its mean, n = 11 observations)
| Part of the variation in Y | Value |
|---|---|
| Explained by the regression of Y on X | 18 |
| Unexplained (residual), Σ(Y − Yc)² | 27 |
| Total variation | 45 |
The coefficient of determination is the ratio of the explained variation to the total variation:
r² = Explained / Total = 18 / (18 + 27) = 18 / 45 = 0.4.
So 40% of the variation in Y is accounted for by its linear relationship with X, and the remaining 60% (the coefficient of non-determination, 1 − r² = 27 / 45 = 0.6) is due to other factors. The correlation coefficient is r = √0.4 = 0.6325 in size; its sign is that of the regression coefficient, which the question does not give.
Standard error of estimate of Y on X
The standard error of estimate measures the scatter of the observed Y values about the regression line, so it is built from the unexplained variation. With a sample of n observations and a fitted line that uses up two constants (a and b), the residual variation is divided by n − 2:
Syx = √[Σ(Y − Yc)² / (n − 2)] = √[27 / (11 − 2)] = √(27 / 9) = √3 = 1.7321 (in the unit of Y).
The syllabus names the standard error of estimate within regression analysis. The question gives the number of observations together with the two variations, which points to this sample form.
The other reading. If the residual variation is divided by n instead (the descriptive form Syx = σy √(1 − r²)), the figure is Syx = √(27 / 11) = 1.5667; the same figure follows from σy² = 45 / 11 and √(1 − r²) = √0.6. The coefficient of determination, 0.4, is the same under both readings.
Where the 4 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Total variation and the coefficient of determination | 2 |
| 2 | Standard error of estimate of Y on X | 2 |
| Total | 4 |
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