Section 23 Simple Linear Regression: Estimates & SE
23.2 \(\beta_1\) Estimates & SE
\[ \large \hat \beta_1 = \frac{\sum\limits_{i=1}^{n}(x_i-\bar x)(y_i-\bar y)}{\sum\limits_{i=1}^{n}(x_i-\bar x)^2} = \frac{S_{xy}}{S_{xx}} \]
\[ \large Var(\hat \beta_1) = \frac{\sigma^2}{\sum\limits_{i=1}^{n}(x_i-\bar x)^2} = \frac{\sigma^2}{S_{xx}} \]
\[ \large SE(\hat \beta_1) = \sqrt {Var(\hat \beta_1)} \]
95% Confidence Interval:
\[ \large CI_{0.95}(\hat \beta_1) = \left[ \hat\beta_1 \pm t_{0.025, df_{residual}} * SE(\hat\beta_1) \right]\]
23.3 \(\beta_0\) Estimates & SE
\[ \large \hat \beta_0 = \bar y - \hat \beta_1 \bar x \]
\[ \large Var(\hat \beta_0) = \sigma^2 \left[ \frac{1}{n} + \frac{\bar x^2}{\sum\limits_{i=1}^{n}(x_i-\bar x)^2} \right] = \sigma^2 \left[ \frac{1}{n} + \frac{\bar x^2}{S_{xx}} \right] \]
\[ \large SE(\hat \beta_0) = \sqrt {Var(\hat \beta_0)} \]
95% Confidence Interval:
\[ \large CI_{0.95}(\hat \beta_0) = \left[ \hat\beta_0 \pm t_{0.025, df_{residual}} * SE(\hat\beta_0) \right]\]
Here,
\[ \large \bar{x} = \frac{1}{n}\sum\limits_{i=1}^{n} x_{i} \]
\[ \large \bar{y} = \frac{1}{n}\sum\limits_{i=1}^{n} y_{i} \]
\[ \large \sigma^2 = Var(\epsilon) \]