The quality control manager does not have sufficient time to review all of these
ID: 3045355 • Letter: T
Question
The quality control manager does not have sufficient time to review all of these data. Rather, she would like to examine the proportions of defective items for a sample of these shipments.
a. Generate a random sample of size 25.
b. Using the sample generated in part a, calculate a 95% confidence interval for the mean proportion of defective items over all monthly shipments.
c. Interpret the 95% confidence interval constructed in part b.
d. Assume that the population consists of the proportion of defective items for each of the given 500 monthly shipments:
Does the 95% confidence interval contain the actual population mean in this case?If not, explain why not.
What proportion of many similarly constructed confidence intervals should include the true population mean?
Shipment Proportion 1 0.10 2 0.77 3 0.16 4 0.54 5 0.08 6 0.05 7 0.34 8 0.76 9 0.57 10 0.55 11 0.23 12 0.86 13 0.37 14 0.46 15 0.61 16 0.52 17 0.03 18 0.60 19 0.45 20 0.64 21 0.03 22 0.41 23 0.10 24 0.57 25 0.39 26 0.68 27 0.61 28 0.71 29 0.51 30 0.52 31 0.54 32 0.58 33 0.72 34 0.06 35 0.06 36 0.91 37 0.58 38 0.11 39 0.06 40 0.80 41 0.88 42 0.86 43 0.10 44 0.81 45 0.31 46 0.47 47 0.72 48 0.25 49 0.17 50 0.63 51 0.90 52 0.45 53 0.06 54 0.62 55 0.24 56 0.74 57 0.90 58 0.85 59 0.62 60 0.54 61 0.29 62 0.60 63 0.73 64 0.09 65 0.32 66 0.56 67 0.43 68 0.70 69 0.30 70 0.47 71 0.67 72 0.11 73 0.37 74 0.40 75 0.94 76 0.14 77 0.66 78 0.87 79 0.84 80 0.24 81 0.44 82 0.62 83 0.66 84 0.69 85 0.45 86 0.26 87 0.81 88 0.97 89 0.66 90 0.51 91 0.12 92 0.08 93 0.00 94 0.13 95 0.96 96 0.16 97 0.20 98 0.98 99 0.43 100 0.55 101 0.21 102 0.07 103 0.62 104 0.20 105 0.93 106 0.52 107 0.79 108 0.48 109 0.62 110 0.55 111 0.39 112 0.32 113 0.51 114 0.58 115 0.66 116 0.11 117 0.37 118 0.85 119 0.57 120 0.87 121 0.30 122 0.56 123 0.27 124 0.47 125 0.56 126 0.25 127 0.32 128 0.65 129 0.48 130 0.53 131 0.63 132 0.86 133 0.67 134 0.12 135 0.14 136 0.36 137 0.97 138 0.53 139 0.87 140 0.80 141 0.35 142 0.34 143 0.84 144 0.79 145 0.10 146 0.50 147 0.98 148 0.18 149 0.81 150 0.33 151 0.61 152 0.48 153 0.30 154 0.87 155 0.94 156 0.50 157 0.76 158 0.53 159 0.89 160 0.43 161 0.77 162 0.95 163 0.96 164 0.25 165 0.14 166 0.86 167 0.94 168 0.99 169 0.54 170 0.46 171 0.09 172 0.93 173 0.11 174 0.21 175 0.27 176 0.44 177 0.69 178 0.86 179 0.13 180 0.58 181 0.74 182 0.45 183 0.35 184 0.15 185 0.75 186 0.94 187 0.16 188 0.37 189 0.21 190 0.39 191 0.37 192 0.32 193 0.74 194 0.10 195 0.71 196 0.38 197 0.16 198 0.91 199 0.39 200 0.21 201 0.40 202 0.45 203 0.76 204 0.22 205 0.65 206 0.62 207 0.47 208 0.24 209 0.85 210 0.36 211 0.03 212 0.81 213 0.01 214 0.76 215 0.47 216 0.90 217 0.99 218 0.36 219 0.30 220 0.13 221 0.80 222 0.87 223 0.29 224 0.29 225 0.90 226 0.87 227 0.14 228 0.62 229 0.30 230 0.83 231 0.58 232 0.22 233 0.50 234 0.46 235 0.49 236 0.83 237 0.02 238 0.97 239 0.15 240 0.48 241 0.40 242 0.66 243 0.15 244 0.88 245 0.50 246 0.15 247 0.72 248 0.36 249 0.86 250 0.87 251 0.42 252 0.28 253 0.21 254 0.91 255 0.27 256 0.52 257 0.85 258 0.79 259 0.92 260 0.46 261 0.03 262 0.34 263 0.81 264 0.91 265 0.29 266 0.14 267 0.65 268 0.91 269 0.77 270 0.31 271 0.53 272 0.39 273 0.68 274 0.86 275 0.05 276 0.15 277 0.94 278 0.31 279 0.54 280 0.92 281 0.82 282 0.18 283 0.85 284 0.67 285 0.10 286 0.55 287 0.53 288 0.88 289 0.90 290 0.27 291 0.24 292 0.95 293 0.49 294 0.53 295 0.70 296 0.39 297 0.02 298 0.61 299 0.81 300 0.50 301 0.95 302 0.53 303 0.80 304 0.60 305 0.72 306 0.31 307 0.35 308 0.55 309 0.81 310 0.40 311 0.31 312 0.33 313 0.70 314 0.23 315 0.04 316 0.22 317 0.14 318 0.41 319 0.17 320 0.88 321 0.17 322 0.74 323 0.74 324 0.92 325 0.04 326 0.89 327 0.80 328 0.98 329 0.25 330 0.52 331 0.53 332 0.68 333 0.38 334 0.82 335 0.12 336 0.96 337 0.06 338 0.16 339 0.09 340 0.61 341 0.08 342 0.92 343 0.98 344 0.00 345 0.09 346 0.69 347 0.04 348 0.40 349 0.86 350 0.37 351 0.05 352 0.24 353 0.71 354 0.17 355 0.03 356 0.49 357 0.97 358 0.81 359 0.87 360 0.68 361 0.69 362 0.65 363 0.82 364 0.44 365 0.64 366 0.52 367 0.96 368 0.65 369 0.84 370 0.23 371 0.67 372 0.46 373 0.61 374 0.24 375 0.68 376 0.31 377 0.05 378 0.88 379 0.80 380 0.58 381 0.26 382 0.64 383 0.45 384 0.69 385 0.01 386 0.23 387 0.87 388 0.52 389 0.34 390 0.77 391 0.82 392 0.54 393 0.49 394 0.39 395 0.33 396 0.75 397 0.63 398 0.36 399 0.18 400 0.43 401 0.06 402 0.82 403 0.00 404 0.09 405 0.75 406 0.71 407 0.03 408 0.64 409 0.73 410 0.75 411 0.47 412 0.13 413 0.97 414 0.35 415 0.55 416 0.07 417 0.55 418 0.21 419 0.88 420 0.65 421 0.77 422 0.38 423 0.09 424 0.68 425 0.00 426 0.56 427 0.28 428 0.44 429 0.62 430 0.61 431 0.11 432 0.03 433 0.65 434 0.77 435 0.30 436 0.68 437 0.80 438 0.08 439 0.91 440 0.13 441 0.60 442 0.19 443 0.39 444 0.69 445 0.21 446 0.71 447 0.49 448 0.40 449 0.62 450 0.54 451 0.79 452 0.40 453 0.19 454 0.66 455 0.54 456 0.11 457 0.05 458 0.71 459 0.67 460 0.52 461 0.35 462 0.38 463 0.43 464 0.18 465 0.56 466 0.14 467 0.91 468 0.34 469 0.24 470 0.96 471 0.79 472 0.79 473 0.77 474 0.41 475 0.87 476 0.00 477 0.20 478 0.39 479 0.06 480 0.59 481 0.84 482 0.91 483 0.13 484 0.23 485 0.07 486 0.74 487 0.89 488 0.95 489 0.47 490 0.21 491 0.23 492 0.33 493 0.15 494 0.85 495 0.94 496 0.04 497 0.62 498 0.18 499 0.39 500 0.85Explanation / Answer
#attaching the excel saved file
> data1=read.csv(file.choose(),header=T)
> attach(data1)
> names(data1)
[1] "Shipment" "Proportion"
a)
> samp=sample(1:500,25,replace=F)
> sampledata=data1[samp,]
> sampledata
Shipment Proportion
154 154 0.87
92 92 0.08
229 229 0.30
238 238 0.97
253 253 0.21
493 493 0.15
443 443 0.39
362 362 0.65
164 164 0.25
40 40 0.80
391 391 0.82
100 100 0.55
376 376 0.31
441 441 0.60
156 156 0.50
373 373 0.61
174 174 0.21
367 367 0.96
170 170 0.46
339 339 0.09
283 283 0.85
59 59 0.62
101 101 0.21
349 349 0.86
470 470 0.96
b)> sampprop=sampledata[,2]
> sampprop
[1] 0.87 0.08 0.30 0.97 0.21 0.15 0.39 0.65 0.25 0.80 0.82 0.55 0.31 0.60 0.50
[16] 0.61 0.21 0.96 0.46 0.09 0.85 0.62 0.21 0.86 0.96
> t.test(sampprop)
One Sample t-test
data: sampprop
t = 8.9066, df = 24, p-value = 4.485e-09
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
0.408106 0.654294
sample estimates:
mean of x
0.5312
c) The interval means that we are 95% confident that the actual population mean will fall in this interval.
d)> prop=data1[,2]
> t.test(prop)
One Sample t-test
data: prop
t = 39.664, df = 499, p-value < 2.2e-16
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
0.4776277 0.5274123
sample estimates:
mean of x
0.50252
Yes, the actual mean is contained in the confidence interval. 95% of confidence intervals should include the true population mean.
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