The data set contained in the file "cutaneous.txt" are part of an assay on cutaneous melanoma, a type of malignant cancer, for the evaluation of postoperative treatment performance with a high dose of interferon alpha-2b as a drug to prevent recurrence (Ibrahim et al., 2001). Patients were allowed to enter the study from 1991 to 1995 and were followed up until 1998. The data set contains information on 417 patients and present their survival times, representing the time in years (t) until the patients death or the censoring time. The other covariate information available for patients is the nodule category x, taking the values 1, 2, 3 or 4, which categorizes patient based on the severity of the disease. (i) The clinician assumes that the standard deviation of the lifetime of patients is 1.5 years and believes that the true mean lifetime of patients is at most 4.5 years. (a) State the null and the alternative hypotheses [2 marks] (b) Carry out a suitable test of hypothesis to see if the clinicians conjecture about the mean lifetime is correct. Use the p-value method and the critical value method and use = 10%. Draw appropriate conclusion in each case [15 marks] 1 (c) Using your alternative hypothesis in (a), suggest a set of 20 admissible alternative values of the mean lifetime and draw the power curve under these alternatives. Interpret the power curve [8 marks] (ii) Split the overall data set into four individual data sets stratified by the nodule category (x). So, the first data set should contain the lifetimes corresponding to x = 1 and so on the fourth data set should contain the lifetimes corresponding to x = 4. Assuming the standard deviation of lifetimes for all these four data sets to be equal to 1.5 years, test at 10% level of significance the following (in each case, specify the null and the alternative hypotheses and draw appropriate conclusion) (a) The average lifetime of patients corresponding to nodule category 1 is exactly equal to 3.5 years [5 marks] (b) The average lifetime of patients corresponding to nodule category 2 is exactly equal to 3.25 years [5 marks] (c) The average lifetime of patients corresponding to nodule category 3 is exactly equal to 3 years [5 marks] (d) The average lifetime of patients corresponding to nodule category 4 is exactly equal to 2.5 years [5 marks] (e) From the test results of (a) - (d) above what can you conclude about the lifetime of patients belonging to different nodule categories

Patient t x 1 0.72279 3 2 6.29979 1 3 6.62286 3 4 3.47159 4 5 7.01164 1 6 6.64750 1 7 6.97604 2 8 0.67077 4 9 6.88022 1 10 6.60917 1 11 1.06776 4 12 6.32991 2 13 6.32170 1 14 1.50308 3 15 0.77207 3 16 1.78234 4 17 6.65845 2 18 3.53730 1 19 5.74949 2 20 3.81383 1 21 5.97673 2 22 1.68925 1 23 5.73854 3 24 5.87543 1 25 1.57426 1 26 5.56879 3 27 1.57153 3 28 2.43669 1 29 5.78782 3 30 1.59890 2 31 0.60780 2 32 5.67556 1 33 0.24093 4 34 5.54689 3 35 6.04517 4 36 1.90007 2 37 2.13279 4 38 0.35592 2 39 2.43395 4 40 1.45380 3 41 5.24846 2 42 1.53046 4 43 5.32786 3 44 0.93361 4 45 2.22313 3 46 4.94456 1 47 5.07598 4 48 5.10609 1 49 5.05133 4 50 5.73032 2 51 1.24572 4 52 1.41273 3 53 0.94182 2 54 5.46475 2 55 1.62902 3 56 5.41547 2 57 5.63997 2 58 3.14853 1 59 5.63450 3 60 1.69199 4 61 5.13347 3 62 3.24162 3 63 5.24298 2 64 5.31143 4 65 2.27515 1 66 5.33060 1 67 2.56810 2 68 5.04312 3 69 4.95003 1 70 4.98015 3 71 1.54415 3 72 4.72005 4 73 5.15811 1 74 5.06776 1 75 1.71389 3 76 5.15264 3 77 1.92197 4 78 4.68720 3 79 4.95825 1 80 4.14784 2 81 1.44559 2 82 4.73922 2 83 5.10883 1 84 4.36961 2 85 3.33470 4 86 3.30459 4 87 3.61123 2 88 0.58590 3 89 1.48665 4 90 4.63792 1 91 4.32033 2 92 4.46270 3 93 4.90349 1 94 2.19576 1 95 1.18549 4 96 4.42710 2 97 1.53593 2 98 4.37782 2 99 4.33676 1 100 4.50103 3 101 4.46543 1 102 2.79808 1 103 4.71458 3 104 3.03080 3 105 3.33744 2 106 2.77892 1 107 4.13963 1 108 4.48734 1 109 0.76660 1 110 4.19439 2 111 4.57769 1 112 1.53320 2 113 1.48118 4 114 4.49555 3 115 4.46270 2 116 4.49008 1 117 4.07392 3 118 4.07666 2 119 3.39493 3 120 3.96167 1 121 3.93977 3 122 4.18617 1 123 2.00958 4 124 4.21903 2 125 4.02190 2 126 1.96851 3 127 2.08350 4 128 3.50719 1 129 3.76728 1 130 1.15264 3 131 0.81314 2 132 4.06845 2 133 2.98973 2 134 1.34155 2 135 3.84942 2 136 2.22040 1 137 3.74538 2 138 3.60301 4 139 0.68446 4 140 1.06229 2 141 3.55921 1 142 3.38672 2 143 3.46612 1 144 3.53730 2 145 3.68241 2 146 3.57563 4 147 0.96099 2 148 1.42368 2 149 2.15469 1 150 3.61670 2 151 3.23614 1 152 0.92539 1 153 0.95551 4 154 3.21150 2 155 3.20329 2 156 3.44422 2 157 2.64203 3 158 1.94114 3 159 0.98015 2 160 0.95551 4 161 3.10198 2 162 3.20055 1 163 2.96509 2 164 3.12389 2 165 2.37919 2 166 2.92676 4 167 3.06639 3 168 3.04723 3 169 2.58453 1 170 2.05886 4 171 0.79124 1 172 2.89665 1 173 3.12663 3 174 0.34771 4 175 5.99863 4 176 5.96851 4 177 6.29432 1 178 6.58179 3 179 6.67488 2 180 0.61328 2 181 0.87611 2 182 5.84531 2 183 6.10541 3 184 1.70294 3 185 6.01506 1 186 1.66461 2 187 2.11088 4 188 1.59617 2 189 0.77207 3 190 1.40726 2 191 1.68104 2 192 1.60438 4 193 1.81793 3 194 3.41136 2 195 5.19370 4 196 5.95483 4 197 5.71116 2 198 3.39220 1 199 1.88638 1 200 3.96167 1 201 1.00753 4 202 2.08624 1 203 1.62902 2 204 6.03970 3 205 1.20465 3 206 3.30732 1 207 5.50308 4 208 1.73580 1 209 3.72348 1 210 2.07529 3 211 0.82136 3 212 5.80698 1 213 2.87201 2 214 1.57974 2 215 4.98289 2 216 5.77139 4 217 0.96372 1 218 1.32512 2 219 5.22656 1 220 1.51403 1 221 0.67625 4 222 1.01574 3 223 5.04312 4 224 5.22382 2 225 4.88433 2 226 0.45996 3 227 5.01300 1 228 5.09240 1 229 5.40452 2 230 2.86379 2 231 4.98289 2 232 5.08966 4 233 0.80219 4 234 3.65777 2 235 4.80493 2 236 2.40383 1 237 4.17248 2 238 1.73854 2 239 0.60233 3 240 5.16359 2 241 1.29500 2 242 4.80767 4 243 5.14990 4 244 4.52293 1 245 4.75565 2 246 4.27105 1 247 0.52567 3 248 5.01574 1 249 2.51608 1 250 4.80219 1 251 0.85969 2 252 4.73648 2 253 4.92266 3 254 4.48460 3 255 0.45996 2 256 4.85147 1 257 3.30732 1 258 4.49281 1 259 0.47912 3 260 3.88227 1 261 4.68172 2 262 4.59959 3 263 3.50445 4 264 4.34497 3 265 1.67009 3 266 2.40110 4 267 4.34497 3 268 4.61328 2 269 3.44148 1 270 3.73717 2 271 3.22793 4 272 4.12868 4 273 0.54757 4 274 1.21287 2 275 4.14784 3 276 1.60712 4 277 4.54209 4 278 2.02053 1 279 0.16975 4 280 4.26557 2 281 4.20808 1 282 1.33881 4 283 3.51814 3 284 0.59685 4 285 3.29637 1 286 4.12047 1 287 1.76318 1 288 3.03901 3 289 2.25873 3 290 0.55578 2 291 1.24298 2 292 2.81725 1 293 4.05749 2 294 3.74538 1 295 2.86653 3 296 3.61396 1 297 4.03285 2 298 2.80082 3 299 1.47296 2 300 0.74196 4 301 4.06297 2 302 3.98905 2 303 3.57016 4 304 3.58385 1 305 3.59754 4 306 1.85900 2 307 0.85969 4 308 1.63723 1 309 1.69747 1 310 1.50856 3 311 3.86858 3 312 3.65777 1 313 1.99589 3 314 0.33128 3 315 3.63860 3 316 3.60301 2 317 2.54620 1 318 2.17385 1 319 3.52088 2 320 2.82272 2 321 2.63655 2 322 2.62560 1 323 2.80356 2 324 3.54552 2 325 0.99384 4 326 2.71869 1 327 0.91992 4 328 1.85079 2 329 0.91170 4 330 5.12526 3 331 5.79877 2 332 3.20055 2 333 4.43258 2 334 1.15811 1 335 0.26557 3 336 5.49760 4 337 1.24298 4 338 5.50308 2 339 5.65092 2 340 3.49350 2 341 0.14784 3 342 5.93018 2 343 0.78303 2 344 1.92471 2 345 3.11020 2 346 4.93361 3 347 1.70294 4 348 1.73580 4 349 1.43737 4 350 2.24778 2 351 5.40999 4 352 4.26283 1 353 4.73922 3 354 4.47639 1 355 0.53662 3 356 0.97741 1 357 5.07598 4 358 5.00205 2 359 3.05818 2 360 4.41342 1 361 1.91923 3 362 0.88980 4 363 1.31691 4 364 4.07392 2 365 2.68036 1 366 3.89049 2 367 4.04654 3 368 4.49281 3 369 1.54415 3 370 2.09172 4 371 3.80287 2 372 3.15674 2 373 1.13895 3 374 2.30801 2 375 4.64887 2 376 0.28474 2 377 1.26762 4 378 3.70431 1 379 3.25804 3 380 4.13689 3 381 0.42437 4 382 2.93771 2 383 1.83710 2 384 4.00274 3 385 3.08556 1 386 3.65229 3 387 3.78919 1 388 1.51129 2 389 3.37577 1 390 2.99521 3 391 1.31964 1 392 3.41136 2 393 0.29295 2 394 2.93224 1 395 2.89117 2 396 3.01985 1 397 3.17317 2 398 3.25257 1 399 2.73785 2 400 0.88433 4 401 3.36208 1 402 0.68994 4 403 2.98700 1 404 3.26899 4 405 3.22245 4 406 1.78234 2 407 1.80698 2 408 2.88296 3 409 2.71595 2 410 3.19233 4 411 2.94319 1 412 0.73374 4 413 2.77070 1 414 1.36893 1 415 2.67762 2 416 2.77618 2 417 2.97878 1