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Run test : The non parametric test for randomness.

Critical Thinking: Was the draft lottery random?

In 1970, a lottery was used to determine who would be drafted into the U.S Army. The 366 dates in the year were placed individual capsules. First, the 31 January capsules were placed in a box; then the 29 February capsules were added and the two months were mixed. Then the 31 March capsules were added and the three were mixed. This process continued until all the months were included. The first capsule selected was September 14, so men born on that date were drafted first. The accompanying list shows the 366 dates in order of selection.

Analyzing the Results

a).Use the run test to test the sequence for randomness above and below the median of 183.5.

b).Use the Kruskal-Wallis test to test the claim that the 12 months had priority numbers drawn from the same population.

c).Calculate the 12 monthly means. Then plot those 12 means on a graph. (The horizontal scale lists the 12 months, and vertical scale ranges from 100 to 260.) Note any pattern suggesting that the original priority numbers were not randomly selected.

d).Based on the results from part (a), (b), and (c), decide whether this particular draft lottery was fair. Write a statement either supporting your position that it was a fair or explaining why you believe it was not fair. If you decide that the lottery was unfair, describe the process for selecting lottery numbers that would have been fair.

Jan: 305 159 251 215 101 224 306 199 194 325 329 221 318 238 017 121
235 140 058 280 186 337 118 059 052 092 355 077 349 164 211

Feb: 086 144 297 210 214 347 091 181 338 216 150 068 152 004 089 212
189 292 025 302 363 290 057 236 179 365 205 299 285

Mar:108 029 267 275 293 139 122 213 317 323 136 300 259 354 169 166
033 332 200 239 334 265 256 258 343 170 268 223 362 217 030

Apr: 032 271 083 081 269 253 147 312 219 218 014 346 124 231 273 148
260 090 336 345 062 316 252 002 351 340 074 262 191 208

May: 330 298 040 276 364 155 035 321 197 065 037 133 295 178 130 055
112 278 075 183 250 326 319 031 361 357 296 308 226 103 313

Jun: 249 228 301 020 028 110 085 366 335 206 134 272 069 356 180 274
073 341 104 360 060 247 109 358 137 022 064 222 353 209

Jul: 093 350 115 279 188 327 050 013 277 284 248 015 042 331 322 120
098 190 227 187 027 153 172 023 067 303 289 088 270 287 193

Aug: 111 045 261 145 054 114 168 048 106 021 324 142 307 198 102 044
154 141 311 344 291 339 116 036 286 245 352 167 061 333 011

Sep: 225 161 049 232 082 006 008 184 263 071 158 242 175 001 113 207
255 246 177 063 204 160 119 195 149 018 233 257 151 315

Oct: 359 125 244 202 024 087 234 283 342 220 237 072 138 294 171 254
288 005 241 192 243 117 201 196 176 007 264 094 229 038 079

Nov: 019 034 348 266 310 076 051 097 080 282 046 066 126 127 131 107
143 146 203 185 156 009 182 230 132 309 047 281 099 174

Dec: 129 328 157 165 056 010 012 105 043 041 039 314 163 026 320 096
304 128 240 135 070 053 162 095 084 173 078 123 016 003 100

Critical Thinking: Are the axial loads within statistical control?

Is the process of manufacturing cans proceeding as it should? Exercise 5 and 6 in Section 13-2 used process data from a New York company that manufactures 0.0109-inch thick aluminum cans for a major beverage supplier. Refer to data Set 20 in Appendix B and conduct an analysis of the process data for cans that are 0.0111 inches thick. The values in the data set are measured axial loads of cans, and the top lids are pressed into place with pressures that vary between 158 lb and 165 lb.

Analyzing the Results
Should you take any corrective action? Write a report summarizing your conclusions. Address not only the issue of statistical stability, but also the ability of cans to withstand the pressures applied when the top lids are pressed into place. Also compare the behavior of the 0.0109-inch cans and recommend which thickness should be used.

Data Set 20: Cans

CANS109 CANS111
270 287
273 216
258 260
204 291
254 210
228 272
282 260
278 294
201 253
264 292
265 280
223 262
274 295
230 230
250 283
275 255
281 295
271 271
263 268
277 225
275 246
278 297
260 302
262 282
273 310
274 305
286 306
236 262
290 222
286 276
278 270
283 280
262 288
277 296
295 281
274 300
272 290
265 284
275 304
263 291
251 277
289 317
242 292
284 215
241 287
276 280
200 311
278 283
283 293
269 285
282 276
267 301
282 285
272 277
277 270
261 275
257 290
278 288
295 287
270 282
268 275
286 279
262 300
272 293
268 290
283 313
256 299
206 300
277 265
252 285
265 294
263 262
281 297
268 272
280 284
289 291
283 306
263 263
273 304
209 288
259 256
287 290
269 284
277 307
234 273
282 283
276 250
272 244
257 231
267 266
204 504
270 284
285 227
273 269
269 282
284 292
276 286
286 281
273 296
289 287
263 285
270 281
279 298
206 289
270 283
270 247
268 279
218 276
251 288
252 284
284 301
278 309
277 284
208 284
271 286
208 303
280 308
269 288
270 303
294 306
292 285
289 289
290 292
215 295
284 283
283 315
279 290
275 247
223 268
220 283
281 305
268 279
272 287
268 285
279 298
217 279
259 274
291 205
291 302
281 296
230 282
276 300
225 284
282 281
276 279
289 255
288 210
268 279
242 286
283 293
277 285
285 288
293 289
248 281
278 297
285 314
292 295
282 257
287 298
277 211
266 275
268 247
273 279
270 303
256 286
297 287
280 287
256 275
262 243
268 274
262 299
293 291
290 281
274 303
292 269

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Solution Summary

The solution gives the details of run test for randomness. Null hypothesis, alternative hypothesis, test statistic, critical value, and p value are given with interpretations.

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