3.5 Assessing Accuracy of Coefficient Estimates

\[Y = \beta_{0} + \beta_{1}X + \epsilon\]

• RSE = residual standard error
• Estimate of \(\sigma\)

\[\mathrm{RSE} = \sqrt{\frac{\mathrm{RSS}}{n - 2}}\] \[\mathrm{SE}(\hat\beta_0)^2 = \sigma^2 \left[\frac{1}{n} + \frac{\bar{x}^2}{\sum_{i=1}^n (x_i - \bar{x})^2}\right],\ \ \mathrm{SE}(\hat\beta_1)^2 = \frac{\sigma^2}{\sum_{i=1}^n (x_i - \bar{x})^2}\]

• 95% confidence interval: a range of values such that with 95% probability, the range will contain the true unknown value of the parameter
  • If we take repeated samples and construct the confidence interval for each sample, 95% of the intervals will contain the true unknown value of the parameter

\[\hat\beta_1 \pm 2\ \cdot \ \mathrm{SE}(\hat\beta_1)\] \[\hat\beta_0 \pm 2\ \cdot \ \mathrm{SE}(\hat\beta_0)\] Learning Objectives:

• Estimate the standard error of regression coefficients. ✔️