There are two questions. Please answer all questions and show your steps clearly. Correct your numerical answers to 4 decimal places. You may use SAS and/ or R to do the calculations. Please include the program codes in your answer script. Combine all materials in a single pdf le with naming: [AMSZ320 Assignment 1] - StudentID.pdf (e.g. [AMSZ320 Assignment 1] - s123456.pdf) and submit it via Moodle. Question 1. Consider the following hypothetical data of rainfall and crop yield Rainfall 1.593 2.192 1.952 2.093 1.963 1.349 1.857 1.211 1.546 2.618 (a) [5 marks] Obtain a scattered plot for the variables rainfall (xaxis) and crop yield (yaxis). Consider the simple linear regression model (Yield) = [80 + [61 (Rainfall) + Error. (0.1) Here, we assume that the error terms are independent Normal random variables N (0, a2) . (b) [5 marks] Obtain the least square estimate of ([30, [81). (c) [5 marks] Obtain 95% condence intervals for ,8.) and ,81. (d) [5 marks] Give an estimate of 02. (e) [5 marks] Construct the corresponding ANOVA table. Consider another regression model (Yield) 2 a0 + aRainfali) + a2(Rainfall)2 + Error. (0.2) Here, we assume that the error terms are independent Normal random variables N (0, 72) . (f) [5 marks] Obtain the least square estimate of (0:0, (11,052). (Yield) = a0 + almar'nfau) + a2(Raz'nfall)2 + Error. (0.2) Here, we assume that the error terms are independent Normal random variables N (0, 72) . (f ) [5 marks] Obtain the least square estimate of (a0, a1, (12). (g) [5 marks] Obtain 95% condence interval for cm, a1 , and 052. (h) [5 marks] Give an estimate of 72. (i) [5 marks] Construct the corresponding ANOVA table. (j) [5 marks] Which model (0.1) or (0.2) do you prefer? Give reasons. (k) [5 marks] Perform a formal Ftest comparing the models (0.1) and (0.2). Write down your null and alternative hypothesis clearly. Question 2. We are interested in the relationship between the response Y and the predictor X. You are given the following table: n ?=1 Xi 221:1 XE 21:1 XE 21:1 X? We are interested in predicting the value of Y when X = 30. (a) [20 marks] Suppose that the true relationship between Y and X is given by Y = 16+3X+Error. Here, the error terms are independent N (0, 02) random variables. Pretend that you do not know the above relationship. You do the prediction by regressing Y against X . Denote the predicted value by 1),, and the true value by yp . Find the expectation E (yp ap) and the variance Ver(yp Q?) . 2 You may leave the notation a in your answer. (b) [20 marks] The true relationship between Y and X assumed in part (a) can be rewritten as Y: 16+3X+0-X2+Error. Now, you do the prediction by regressing Y against both X and X 2 . Denote the predicted value by 9,, and the true value by yp. Find the expectation E(yp 13p) and the variance Var(yp Q?) . 2 You may leave the notation a in your answer. (c) [5 marks] Compare your answers in (a) and (b). If you are very condent that the relationship between Y and X is a straight line, which method, (a) or (b), should you use