Statistics
Regression
Calculate the regression coefficient and obtain the lines of regression of Y on X for the following data: X Y 1 9 2 8 3 10 4 12 5 11 6 13 7 14
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Calculate the regression coefficient and obtain the lines of regression of Y on X for the following data: X Y 1 9 2 8 3 10 4 12 5 11 6 13 7 14
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To calculate the regression coefficient and lines of regression, we'll perform the following steps:
- Calculate the means of X and Y.
- Calculate the standard deviations of X and Y.
- Calculate the correlation coefficient between X and Y.
- Calculate the regression coefficient of Y on X (byx).
- Determine the line of regression of Y on X.
1. Calculate the means of X and Y:
- Mean of X (X̄) = (1 + 2 + 3 + 4 + 5 + 6 + 7) / 7 = 28 / 7 = 4
- Mean of Y (Ȳ) = (9 + 8 + 10 + 12 + 11 + 13 + 14) / 7 = 77 / 7 = 11
2. Calculate the standard deviations of X and Y:
- First, calculate the deviations from the mean for X and Y.
X | Y | x = X - X̄ | y = Y - Ȳ | x2 | y2 | xy |
---|---|---|---|---|---|---|
1 | 9 | -3 | -2 | 9 | 4 | 6 |
2 | 8 | -2 | -3 | 4 | 9 | 6 |
3 | 10 | -1 | -1 | 1 | 1 | 1 |
4 | 12 | 0 | 1 | 0 | 1 | 0 |
5 | 11 | 1 | 0 | 1 | 0 | 0 |
6 | 13 | 2 | 2 | 4 | 4 | 4 |
7 | 14 | 3 | 3 | 9 | 9 | 9 |
Totals | 28 | 28 | 26 |
- Standard deviation of X (σx) = √[Σ(x2) / n] = √(28 / 7) = √4 = 2
- Standard deviation of Y (σy) = √[Σ(y2) / n] = √(28 / 7) = √4 = 2
3. Calculate the correlation coefficient (r):
- r = Σ(xy) / [n * σx * σy] = 26 / (7 * 2 * 2) = 26 / 28 ≈ 0.9286
4. Calculate the regression coefficient of Y on X (byx):
- byx = r * (σy / σx) = 0.9286 * (2 / 2) = 0.9286
5. Determine the line of regression of Y on X:
- The regression equation is: Y = a + byx * X
- Where 'a' can be found using the formula: a = Ȳ - byx * X̄
- a = 11 - 0.9286 * 4 = 11 - 3.7144 = 7.2856
- So, the regression line of Y on X is: Y = 7.2856 + 0.9286X
Results:
- Regression Coefficient of Y on X (byx): 0.9286
- Line of Regression of Y on X: Y = 7.2856 + 0.9286X