Sixteen batches of the plastic were made, and from each batch one test item was molded. Each test item was randomly assigned to one of the four predetermined time levels, and the hardness was measured after the assigned elapsed time. The results are shown below; X is the elapsed time in hours, and Y is hardness in Brinell units. Assume the following regression model YiZQ-i'Xi-l-Ei, 1:1,...110 where oz and ,6 are unknown constants and 6,; 3251 N (0, a2), a > 0 holds true. i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 X z- 16 16 16 16 24 24 24 24 32 32 32 32 40 40 40 K; 199 205 196 200 218 220 215 223 237 234 235 230 250 248 253 where 221:1 xi = 448, 221:1211' = 3609, 2:133? = 13824, 221:1 y? = 819499, and 2;; say.- = 103656. a. Find the least square estimates of the unknown parameters a and 6. Provide a 95% condence interval for 6. Interpret your interval estimate. (4 pts) b. Test whether or not there is a linear association between the hardness of the plastic and the elapsed time at the 5 percent signicance level. State the null and alternative hypotheses, decision rule, P-value of the test, and conclusion. (4 pts) c. Obtain the Type 11 error of your test in part (b) if H A : ,3 = 2.5. Assume Var(b) = 0.008. (4 pts) d. Obtain a 95 percent condence interval for the mean hardness of molded items with an elapsed time of 30 hours. Interpret your condence interval. (4 pts) e. Obtain a 95 percent prediction interval for the mean hardness of 10 newly molded test items, each with an elapsed time of 30 hours. (4 pts) 16 40 246