Consider the following data set. X ## [1] -0. 6264538 0. 1836433 -0.8356286 1.5952808 0.3295078 -0.8204684 ## [7] 0. 4874291 0.7383247 0.5757814 -0.3053884 1.5117812 0.3898432 ## [13] -0. 6212406 -2. 2146999 1. 1249309 Y # # [1] -0. 9846143 0.2592747 -0.3096067 3.2141424 1.0881630 -0.3117252 ## [7] 1.5132799 1. 1820520 -1. 1256797 0. 1617432 2.2115430 0.4289693 ## [13] -2. 4026133 -3.8001999 2. 1053379 (a) Test the null hypothesis Ho : p =0 v.s. the alternative Ha : p # 0. (5 pts) (b) Fit a simple linear regression model for Y ~ X. Calculate Bo, B1. Also write down the fitted linear regression equation. (5 pts) (c) Fit the regression line by R, and use "residuals()" function to get all the residuals. Denote the i-th residual as ri = Yi - Yi, the difference between the i-th Y and the i-th Y. Calculate the sum of all r; and the sum of all r; * X;. What do you observe? (5 pts) (d) Calculate the estimate of residual standard deviation, s. (2 pts) (e) Test for the existence of the linear relationship, i.e. Ho : B1 =0 v.s. Ha : B1 # 0. Compare your result with the result in (a). (5 pts) (f) Construct a confidence interval for My and a prediction interval for y, given the new data r* = 0.5. Do they have the same center, which one is wider, and explain your findings. (5 pts)