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:

  1. Calculate the means of X and Y.
  2. Calculate the standard deviations of X and Y.
  3. Calculate the correlation coefficient between X and Y.
  4. Calculate the regression coefficient of Y on X (byx).
  5. 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
Wrote answer · 3/14/2025
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