Q.5 (A) (i) Compute the appropriate regression equation for the following data : X is the independent variable and Y is the dependent variable.
| X | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Y | 5 | 7 | 9 | 11 | 13 |
Best answer
Answer: \(Y=2X+3\)
Since Y is dependent, the appropriate equation is the regression of Y on X. \(n=5,\ \bar{x}=3,\ \bar{y}=9\).
| x | y | \(x-\bar{x}\) | \(y-\bar{y}\) | \((x-\bar{x})^2\) | \((x-\bar{x})(y-\bar{y})\) |
|---|---|---|---|---|---|
| 1 | 5 | -2 | -4 | 4 | 8 |
| 2 | 7 | -1 | -2 | 1 | 2 |
| 3 | 9 | 0 | 0 | 0 | 0 |
| 4 | 11 | 1 | 2 | 1 | 2 |
| 5 | 13 | 2 | 4 | 4 | 8 |
| \(\Sigma x=15\) | \(\Sigma y=45\) | 0 | 0 | 10 | 20 |
\(b_{YX}=\frac{\Sigma(x-\bar{x})(y-\bar{y})}{\Sigma(x-\bar{x})^2}=\frac{20}{10}=2\)
\(Y-\bar{y}=b_{YX}(X-\bar{x})\Rightarrow Y-9=2(X-3)\Rightarrow Y=2X+3\).
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