Statistics Homework
Q1. Understanding the coefficient of determination R2. Say whether the following statements are true or false and explain why.
(a) A large R2 always means that the fitted linear regression line is a good fit of the data.
(b) A small R2 always means that the predictor and the response are not related.
(c) If all observations Yi fall on straight line and the line is not horizontal, then R2 = 1.
Q2. Data analysis, more on the height-weight dataset. Use the \weight full.txt" dataset analyzed in Homework. Assuming the Normal error model.
(a) You are going to meet a man whose height is 175cm. What is the 95% confidence interval for the expected weight of this man? And what is the 95% prediction interval for the weight of this man? Are they the same? Why?
(b) You are going to meet three men whose heights are 174cm, 175cm, and 179cm, respectively. What is the 95% simultaneous confidence intervals for the expected weights of these men? How about the 95% simultaneous prediction intervals for the heights of these men? (Use the Bonferroni procedure.)
(c) Comparing the Bonferroni procedure and the Working-Hotelling procedure. Under the scenario in part (b), construct the 95% simultaneous confidence intervals through the Working-Hotelling procedure. Compare to the corresponding intervals in part (b). Which ones are wider? How about if you are asked to construct the 95% simultaneous confidence intervals on the mean weights for 8 levels of height: 170cm, 172cm, 174cm, 175cm, 179cm, 182cm, 183cm, 186cm? Report and compare the 95% simultaneous confidence intervals based on the two procedures.
Q3. Rigorous proof. For the simple linear regression model: Yi = β0 + β1Xi + εi, i = 1, . . . , n, where εi are iid with mean 0 and variance σ2. Let β^0 and β^1 be the least squares estimators for β0 and β1, respectively, and Y^i = β^iXi be the fitted value.
(a) Show that i=1∑n(Yi - Y-)2 = i=1∑n(Y^i - Y-)2 + i=1∑n(Yi - Y^i)2.
(b) For any other linear unbiased estimator for β1, show that β^1 has the smallest variance, i.e., let β~1 = i=1∑nciYi, such that E(β~1) = β1, then Var(β~1) ≥ Var(β^1).
Attachment:- Weight Data.rar