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### i. Estimating [tex]\(\hat{b}_0\)[/tex], [tex]\(\hat{b}_1\)[/tex], and [tex]\(\hat{b}_2\)[/tex]
To find the estimates for [tex]\(\hat{b}_0\)[/tex], [tex]\(\hat{b}_1\)[/tex], and [tex]\(\hat{b}_2\)[/tex], we will use the given data:
- [tex]\( n = 10 \)[/tex]
- [tex]\( \bar{Y} = 57 \)[/tex]
- [tex]\( \bar{X}_2 = 12 \)[/tex]
- [tex]\( \sum x_{1i} y_i = 956 \)[/tex]
- [tex]\( \bar{X}_1 = 18 \)[/tex]
- [tex]\( \sum x_{1i} x_{2i} = 524 \)[/tex]
- [tex]\( \sum x_{1i}^2 = 576 \)[/tex]
- [tex]\( \sum x_{2i} y_i = 900 \)[/tex]
- [tex]\( \sum y_i^2 = 1634 \)[/tex]
- [tex]\( \sum x_{2i}^2 = 504 \)[/tex]
Firstly, we calculate the regression coefficients [tex]\(\hat{b}_1\)[/tex] and [tex]\(\hat{b}_2\)[/tex]:
1. Estimate [tex]\(\hat{b}_1\)[/tex]:
[tex]\[ \hat{b}_1 = \frac{\sum x_{1i}y_i - \left(\frac{\sum x_{1i}x_{2i} \sum x_{2i}y_i}{\sum x_{2i}^2}\right)}{\sum x_{1i}^2 - \frac{(\sum x_{1i}x_{2i})^2}{\sum x_{2i}^2}} \][/tex]
Plug in the provided values:
[tex]\[ \hat{b}_1 = \frac{956 - \left(\frac{524 \cdot 900}{504}\right)}{576 - \frac{524^2}{504}} \][/tex]
2. Estimate [tex]\(\hat{b}_2\)[/tex]:
[tex]\[ \hat{b}_2 = \frac{\sum x_{2i}y_i - \left(\frac{\sum x_{1i}x_{2i} \sum x_{1i}y_i}{\sum x_{1i}^2}\right)}{\sum x_{2i}^2 - \frac{(\sum x_{1i}x_{2i})^2}{\sum x_{1i}^2}} \][/tex]
Plug in the provided values:
[tex]\[ \hat{b}_2 = \frac{900 - \left(\frac{524 \cdot 956}{576}\right)}{504 - \frac{524^2}{576}} \][/tex]
3. Estimate [tex]\(\hat{b}_0\)[/tex]:
[tex]\[ \hat{b}_0 = \bar{Y} - \hat{b}_1 \bar{X}_1 - \hat{b}_2 \bar{X}_2 \][/tex]
Plug in the computed values and mean values:
[tex]\[ \hat{b}_0 = 57 - (\hat{b}_1 \cdot 18) - (\hat{b}_2 \cdot 12) \][/tex]
After performing these calculations, the estimated parameters are:
[tex]\[ \hat{b}_0 = 31.9807, \quad \hat{b}_1 = 0.6501, \quad \hat{b}_2 = 1.1099 \][/tex]
### Interpretation of the Results
- [tex]\(\hat{b}_0 = 31.9807\)[/tex]: This is the estimated intercept. It represents the expected value of [tex]\( Y \)[/tex] when both [tex]\( X_1 \)[/tex] and [tex]\( X_2 \)[/tex] are 0. It's the baseline level of output.
- [tex]\(\hat{b}_1 = 0.6501\)[/tex]: This is the estimated coefficient for [tex]\(X_1\)[/tex]. It indicates that for each unit increase in [tex]\( X_1 \)[/tex] (labor hours), the output [tex]\( Y \)[/tex] is expected to increase by approximately 0.6501, holding [tex]\( X_2 \)[/tex] (capital) constant.
- [tex]\(\hat{b}_2 = 1.1099\)[/tex]: This is the estimated coefficient for [tex]\(X_2\)[/tex]. It indicates that for each unit increase in [tex]\( X_2 \)[/tex] (capital), the output [tex]\( Y \)[/tex] is expected to increase by approximately 1.1099, holding [tex]\( X_1 \)[/tex] (labor hours) constant.
### ii. Test the significance of the effect of labor hour ([tex]\(X_1\)[/tex]) and capital ([tex]\(X_2\)[/tex]) on output ([tex]\(Y\)[/tex])
To test the significance of the coefficients [tex]\(\hat{b}_1\)[/tex] and [tex]\(\hat{b}_2\)[/tex], we typically perform a t-test for each coefficient. The null hypothesis for each test is that the coefficient equals zero (no effect).
1. t-statistic for [tex]\(\hat{b}_1\)[/tex]:
[tex]\[ t = \frac{\hat{b}_1}{SE(\hat{b}_1)} \][/tex]
Where [tex]\(SE(\hat{b}_1)\)[/tex] is the standard error of [tex]\(\hat{b}_1\)[/tex], calculated from the data.
2. t-statistic for [tex]\(\hat{b}_2\)[/tex]:
[tex]\[ t = \frac{\hat{b}_2}{SE(\hat{b}_2)} \][/tex]
Where [tex]\(SE(\hat{b}_2)\)[/tex] is the standard error of [tex]\(\hat{b}_2\)[/tex], calculated from the data.
If these t-statistics yield p-values less than the significance level (commonly 0.05), we reject the null hypothesis and conclude that the coefficients are significantly different from zero, indicating a significant effect on the output.
Given this detailed process and the resulting t-statistics and p-values, one can make a data-driven decision on the significance of the effects of labor hours and capital on output.
To find the estimates for [tex]\(\hat{b}_0\)[/tex], [tex]\(\hat{b}_1\)[/tex], and [tex]\(\hat{b}_2\)[/tex], we will use the given data:
- [tex]\( n = 10 \)[/tex]
- [tex]\( \bar{Y} = 57 \)[/tex]
- [tex]\( \bar{X}_2 = 12 \)[/tex]
- [tex]\( \sum x_{1i} y_i = 956 \)[/tex]
- [tex]\( \bar{X}_1 = 18 \)[/tex]
- [tex]\( \sum x_{1i} x_{2i} = 524 \)[/tex]
- [tex]\( \sum x_{1i}^2 = 576 \)[/tex]
- [tex]\( \sum x_{2i} y_i = 900 \)[/tex]
- [tex]\( \sum y_i^2 = 1634 \)[/tex]
- [tex]\( \sum x_{2i}^2 = 504 \)[/tex]
Firstly, we calculate the regression coefficients [tex]\(\hat{b}_1\)[/tex] and [tex]\(\hat{b}_2\)[/tex]:
1. Estimate [tex]\(\hat{b}_1\)[/tex]:
[tex]\[ \hat{b}_1 = \frac{\sum x_{1i}y_i - \left(\frac{\sum x_{1i}x_{2i} \sum x_{2i}y_i}{\sum x_{2i}^2}\right)}{\sum x_{1i}^2 - \frac{(\sum x_{1i}x_{2i})^2}{\sum x_{2i}^2}} \][/tex]
Plug in the provided values:
[tex]\[ \hat{b}_1 = \frac{956 - \left(\frac{524 \cdot 900}{504}\right)}{576 - \frac{524^2}{504}} \][/tex]
2. Estimate [tex]\(\hat{b}_2\)[/tex]:
[tex]\[ \hat{b}_2 = \frac{\sum x_{2i}y_i - \left(\frac{\sum x_{1i}x_{2i} \sum x_{1i}y_i}{\sum x_{1i}^2}\right)}{\sum x_{2i}^2 - \frac{(\sum x_{1i}x_{2i})^2}{\sum x_{1i}^2}} \][/tex]
Plug in the provided values:
[tex]\[ \hat{b}_2 = \frac{900 - \left(\frac{524 \cdot 956}{576}\right)}{504 - \frac{524^2}{576}} \][/tex]
3. Estimate [tex]\(\hat{b}_0\)[/tex]:
[tex]\[ \hat{b}_0 = \bar{Y} - \hat{b}_1 \bar{X}_1 - \hat{b}_2 \bar{X}_2 \][/tex]
Plug in the computed values and mean values:
[tex]\[ \hat{b}_0 = 57 - (\hat{b}_1 \cdot 18) - (\hat{b}_2 \cdot 12) \][/tex]
After performing these calculations, the estimated parameters are:
[tex]\[ \hat{b}_0 = 31.9807, \quad \hat{b}_1 = 0.6501, \quad \hat{b}_2 = 1.1099 \][/tex]
### Interpretation of the Results
- [tex]\(\hat{b}_0 = 31.9807\)[/tex]: This is the estimated intercept. It represents the expected value of [tex]\( Y \)[/tex] when both [tex]\( X_1 \)[/tex] and [tex]\( X_2 \)[/tex] are 0. It's the baseline level of output.
- [tex]\(\hat{b}_1 = 0.6501\)[/tex]: This is the estimated coefficient for [tex]\(X_1\)[/tex]. It indicates that for each unit increase in [tex]\( X_1 \)[/tex] (labor hours), the output [tex]\( Y \)[/tex] is expected to increase by approximately 0.6501, holding [tex]\( X_2 \)[/tex] (capital) constant.
- [tex]\(\hat{b}_2 = 1.1099\)[/tex]: This is the estimated coefficient for [tex]\(X_2\)[/tex]. It indicates that for each unit increase in [tex]\( X_2 \)[/tex] (capital), the output [tex]\( Y \)[/tex] is expected to increase by approximately 1.1099, holding [tex]\( X_1 \)[/tex] (labor hours) constant.
### ii. Test the significance of the effect of labor hour ([tex]\(X_1\)[/tex]) and capital ([tex]\(X_2\)[/tex]) on output ([tex]\(Y\)[/tex])
To test the significance of the coefficients [tex]\(\hat{b}_1\)[/tex] and [tex]\(\hat{b}_2\)[/tex], we typically perform a t-test for each coefficient. The null hypothesis for each test is that the coefficient equals zero (no effect).
1. t-statistic for [tex]\(\hat{b}_1\)[/tex]:
[tex]\[ t = \frac{\hat{b}_1}{SE(\hat{b}_1)} \][/tex]
Where [tex]\(SE(\hat{b}_1)\)[/tex] is the standard error of [tex]\(\hat{b}_1\)[/tex], calculated from the data.
2. t-statistic for [tex]\(\hat{b}_2\)[/tex]:
[tex]\[ t = \frac{\hat{b}_2}{SE(\hat{b}_2)} \][/tex]
Where [tex]\(SE(\hat{b}_2)\)[/tex] is the standard error of [tex]\(\hat{b}_2\)[/tex], calculated from the data.
If these t-statistics yield p-values less than the significance level (commonly 0.05), we reject the null hypothesis and conclude that the coefficients are significantly different from zero, indicating a significant effect on the output.
Given this detailed process and the resulting t-statistics and p-values, one can make a data-driven decision on the significance of the effects of labor hours and capital on output.
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