Reading 3.1.2 Q

$\DeclareMathOperator{\Var}{Var}$ $\DeclareMathOperator{\Cov}{Cov}$ $\DeclareMathOperator{\Span}{Span}$

This section is dense. You want to have seen it before you hear it. Answer the following reading questions.

• What is the population regression line?
• What is the difference between the population regression line and the least sqaures line?
• What does $\hat{\mu}$ mean?
• What is the standard error $SE(\hat{\mu})$?
• Can you prove the formula given for the variance of $\hat{\mu}$? Try it.
• What disclaimer appears footnote 2? What do you make of it?
• You do not have to try to prove the formulas in (3.8) - we will do that as part of our homework. If you were studying this book on your own, you would attempt to prove each formula as you read it.
• The text says that the assumption that errors are “uncorrelated and have common variance $\sigma^2$” is “clearly not true in Figure 3.1”. What do the authors see in Figure 3.1 that makes this so “clear”? Explain.
• What characteristic of the independent variable observations ($x_i$) makes $SE(\hat{\beta}_1)$ small?
• What characteristic of the independent variable observations makes the standard error of $\hat{\beta}_0$ the same as the standard error of the estimate of the population mean ($\hat{\mu}$)?
• What is the definition of a confidence interval? (The assumption that you mean 95% confidence is standard in this stats usage.)
• What is the definition of a $t$-statistic for an estimate?
• What is a $p$-value? How does it relate to a large value for $|t|$?

More to learn, not required

• TODO: Make our own simulated data and analyze different runs. (Create our own graphs like Figure 3.3.)
• TODO: Demonstrate least squares is unbiased estimator.
• TODO: Discuss the footnote 3 about the approximations for the confidence interval in class. Also to mention: $\alpha = 1-\text{confidence}$.