Section 2.1: Inferences Concerning \(\beta_{1}\)

1. State the mean and variance of \(b_{1}\) under the SLR model.
2. Explain why the variance of \(b_{1}\) depends on: the sample size, the noise variance, and the spacing of the predictor values.
3. State the distribution of \(b_{1}\) under the SLRGN model.
4. State the distribution of \(b_{1}\) after studentization under the SLRGN model.
5. Construct a confidence interval for \(\beta_{1}\) under the SLRGN model.
6. Perform a hypothesis test for \(\beta_{1}\) under the SLRGN model.

Section 2.2: Inferences Concerning \(\beta_{0}\)

1. State the mean and variance of \(b_{0}\) under the SLR model.
2. Explain why the variance of \(b_{0}\) depends on: the sample size, the noise variance, the spacing of the predictor values, and the sample mean of the predictors.
3. State the distribution of \(b_{0}\) under the SLRGN model.
4. State the distribution of \(b_{0}\) after studentization under the SLRGN model.
5. Construct a confidence interval for \(\beta_{0}\) under the SLRGN model.
6. Perform a hypothesis test for \(\beta_{0}\) under the SLRGN model.

R

1. Compute the estimates of the standard errors for \(b_{0}\) and \(b_{1}\) directly from the sample statistics of a data frame.
2. Extract the estimates of the standard errors \(b_{0}\) and \(b_{1}\) using summary and an output from lm.
3. Construct confidence intervals for \(\beta_{0}\) and \(\beta_{1}\) under the SLRGN model directly from the sample statistics of a data frame.
4. Construct confidence intervas for \(\beta_{0}\) and \(\beta_{1}\) under the SLRGN model using confint and an output from lm.
5. Perform hypothesis tests for \(\beta_{0}\) and \(\beta_{1}\) under the SLRGN model directly from the sample statistics of a data frame.