5.6: End-of-Chapter Materials
- Page ID
- 57729
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Here are the expected materials to supplement the chapter.
R Functions
We used no new R functions in this chapter, so there are none to cover here.
Exercises
- Prove that the covariance between our \(b_1\) estimator and \(\bar{y}\) is 0.
- Prove that the OLS estimators \(b_0\) and MSE are independent.
- Prove that the OLS estimators \(b_1\) and MSE are independent.
- Prove that the ratio \(T = \frac{\displaystyle b_0 - \beta_0}{\displaystyle \sqrt{ MSE \left( \frac{1}{n}+\frac{\bar{x}^2}{S_{xx}}\right) } }\) follows a Student's t distribution with \(n-p\) degrees of freedom.
- Prove that the ratio \(T = \frac{\displaystyle \hat{y} - \hat{y}_0}{\displaystyle \sqrt{ MSE \left( \frac{1}{n}+\frac{(x-\bar{x})^2}{S_{xx}}\right) } }\) follows a Student's t distribution with \(n-p\) degrees of freedom.
- Prove that the endpoints of a central confidence interval for \(\beta_0\) are defined by \(b_0 \pm t_{\alpha/2,n-p}\sqrt{MSE\left( \frac{1}{n} + \frac{\bar{x}^2}{S_{xx}}\right)}\).
- Prove that the endpoints of a confidence interval for \(\hat{y}\) are defined by \(b_0 + b_1 x \pm t_{\alpha/2,n-p}\sqrt{MSE\left( \frac{1}{n} + \frac{(x-\bar{x})^2}{S_{xx}}\right)}\).
- Prove that the endpoints of a prediction interval for \(y_{new}\) are defined by \(b_0 + b_1 x \pm t_{\alpha/2,n-p}\sqrt{MSE\left( 1 + \frac{1}{n} + \frac{(x-\bar{x})^2}{S_{xx}}\right)}\).
Theory Readings
- R. B. Bapat (2000). Linear Algebra and Linear Models (Second ed.). New York: Springer.
- William G. Cochrane (1934). "The distribution of quadratic forms in a normal system, with applications to the analysis of covariance." Mathematical Proceedings of the Cambridge Philosophical Society. 30(2): 178–191.
- Franklin A. Graybill and David C. Bowden (1968). "Linear Segment Confidence Bands for Simple Linear Models." Journal of the American Statistical Association, 62(318): 403–408.
- Henry Scheffé (1959). The Analysis of Variance. New York: Wiley.
- Holbrook Working and Harold Hotelling (1929). "Applications of the Theory of Error to the Interpretation of Trends." Journal of the American Statistical Association, 24(165A): 73–85.