(See Steps) A linear model of the form y_i=β_1 x_i+β_0 for i=1,2,... , n observations (x_i, y_i), has the linear least squares solution for the
Question: A linear model of the form
\[\mathrm{y}_{\mathrm{i}}=\beta_{1} \mathrm{x}_{\mathrm{i}}+\beta_{0}\]for \(i=1,2,\ldots \), n observations \(\left(\mathrm{x}_{\mathrm{i}}, \mathrm{y}_{\mathrm{i}}\right)\), has the linear least squares solution for the parameters \(\beta_{1}\) and \(\beta_{0}\) of
\[\hat{\beta}_{1}=\frac{\sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)\left(x_{i}-\bar{x}\right)}{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}}\]and
\[\begin{aligned} & {{{\hat{\beta }}}_{0}}=\bar{y}-{{{\hat{\beta }}}_{1}}\overline{\text{x}} \\ & \overline{\text{x}}=\sum\limits_{\text{i}=1}^{\text{n}}{\frac{{{\text{x}}_{\text{i}}}}{\text{n}}}\text{ and }\overline{\text{y}}=\sum\limits_{\text{i}=1}^{\text{n}}{\frac{{{\text{y}}_{\text{i}}}}{\text{n}}} \\ \end{aligned}\]
The variance of the error,
\[\sigma^{2}=\operatorname{Var}\left[y_{i}-\left(\beta_{1} x_{i}+\beta_{0}\right)\right]\]may also be estimated by
\[\sigma^{2}=\frac{1}{n-2} \sum_{i=1}^{n}\left(y_{i}-\hat{\beta}_{0}-\hat{\beta}_{1} x_{i}\right)^{2}\]For the following two sets of data \(y_{i}\) and \(z_{i}\)
| \[{{X}_{i}}\] | \[{{Y}_{i}}\] | \[{{Z}_{i}}\] |
| 1 | 6.7 | 3.9 |
| 2 | 4.7 | 1.5 |
| 3 | 8.1 | -0.2 |
| 4 | 7.1 | 1 |
| 5 | 11.3 | 0.6 |
| 6 | 10.5 | -3.1 |
| 7 | 11.8 | -2.8 |
| 8 | 13.7 | -1.8 |
| 9 | 10.6 | -6 |
| 10 | 13.3 | -5 |
- Compute the regression coefficients \(\beta_{0}\) and \(\beta_{1}\) and \(\sigma^{2}\) for the two sets of data \(\left(\mathrm{x}_{\mathrm{i}}, \mathrm{y}_{\mathrm{i}}\right)\) and \(\left(\mathrm{x}_{\mathrm{i}}, \mathrm{z}_{\mathrm{i}}\right)\)
- Plot the raw data and the lines determined in a for both sets of data.
- Compute the correlation coefficient \(r_{x y}\) for the both sets of data.
- How do you interpret these correlation coefficients?
Deliverable: Word Document 
![[See Solution] For two joint normally distributed](/images/solutions/MC-solution-library-34128.jpg "[See Solution] For two joint normally distributed time series x_i")
[See Solution] For two joint normally distributed time series x_i
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