5.6: End-of-Chapter Materials
- Page ID
- 57729
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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.