To estimate the regression model (Y_i = \alpha + \beta X_i + u_i), where (u_i) is a stochastic error term, you need to find the values of (\alpha) and (\beta) that minimize the sum of squared residuals.
The model can be estimated using the method of least squares.
The formulas for (\beta) and (\alpha) are given by:
[
\beta = \frac{\sum_{i=1}^{n} (X_i – \bar{X})(Y_i – \bar{Y})}{\sum_{i=1}^{n} (X_i – \bar{X})^2}
]
[
\alpha = \bar{Y} – \beta \bar{X}
]
where (\bar{X}) and (\bar{Y}) are the sample means of X and Y, respectively.
Let’s calculate (\alpha) and (\beta) using the provided data:
[ \bar{X} = \frac{\sum_{i=1}^{n} X_i}{n} ]
[ \bar{Y} = \frac{\sum_{i=1}^{n} Y_i}{n} ]
[ \beta = \frac{\sum_{i=1}^{n} (X_i – \bar{X})(Y_i – \bar{Y})}{\sum_{i=1}^{n} (X_i – \bar{X})^2} ]
[ \alpha = \bar{Y} – \beta \bar{X} ]
Let’s calculate these values:
[ \bar{X} = \frac{5+6+7+8+9+11+12+13+14+15}{10} = \frac{110}{10} = 11 ]
[ \bar{Y} = \frac{14+16+21+23+28+24+29+28+32+35}{10} = \frac{250}{10} = 25 ]
[ \beta = \frac{\sum_{i=1}^{10} (X_i – 11)(Y_i – 25)}{\sum_{i=1}^{10} (X_i – 11)^2} ]
[ \beta = \frac{(5-11)(14-25) + (6-11)(16-25) + \ldots + (15-11)(35-25)}{(5-11)^2 + (6-11)^2 + \ldots + (15-11)^2} ]
After calculating (\beta), you can substitute it into the formula for (\alpha):
[ \alpha = \bar{Y} – \beta \bar{X} ]
Now, once you have the estimated values of (\alpha) and (\beta), you can find the predicted values (\hat{Y_i}) and the residuals (u_i) for each observation.
For part (b), the percentage variation in Y explained by X is given by the coefficient of determination ((R^2)). It is calculated as:
[ R^2 = \frac{\text{SSR}}{\text{SST}} ]
where SSR is the sum of squared residuals and SST is the total sum of squares.
[ R^2 = 1 – \frac{\text{SSR}}{\text{SST}} ]
You can then express (R^2) as a percentage to find the percentage of variation in Y explained by X.