Question 1 (20 marks)Simon Dlamini is enrolled for a course in quantitative methods at his home university. Simon lives on the Atlantic seaboard and loves surfing. Every swell is a good reason not to study – and as a result he is not performing too well in his course.
The course is assessed via three assessments: a take home assignment (20%), a mid-term test (30%) and a final test (50%). He has already received his home assignment and mid-term test results and knows what he should get for his final test in order to pass the course. Simon managed to get his hand on the results of a previous class, and based on these results he would like to estimate what his final test mark could be. The results of the previous class are shown in the table below.
Max = 20 |
Max = 30 |
Max = 50 |
Max = 20 |
Max = 30 |
Max = 50 |
Student no |
Assignment |
Midterm |
Final test |
Student no |
Assignment |
Midterm |
Final test |
1 |
12 |
11 |
25 |
18 |
14 |
20 |
44 |
2 |
15 |
28 |
45 |
19 |
11 |
20 |
43 |
Question 1 (20 marks) Simon Dlamini is enrolled for a course in quantitative methods at his home university. Simon lives on the Atlantic seaboard and loves surfing. Every swell is a good reason not to study – and as a result he is not performing too well in his course. The course is assessed via three assessments: a take home assignment (20%), a mid-term test (30%) and a final test (50%). He has already received his home assignment and mid-term test results and knows what he should get for his final test in order to pass the course. Simon managed to get his hand on the results of a previous class, and based on these results he would like to estimate what his final test mark could be. The results of the previous class are shown in the table below. Max = 20 Max = 30 Max = 50 Max = 20 Max = 30 Max = 50 Student no Assignment Midterm Final test Student no Assignment Midterm Final test 1 12 11 25 18 14 20 44 2 15 28 45 19 11 20 43 3 13 19 37 20 13 15 32 4 19 26 38 21 15 10 26 5 20 22 40 22 19 30 48 6 11 13 22 23 17 16 38 7 15 10 32 24 13 24 29 8 10 11 30 25 18 20 34 9 13 16 27 26 10 16 27 10 14 15 34 27 12 10 25 11 19 16 40 28 11 18 30 12 17 16 28 29 12 13 25 13 10 12 24 30 14 15 20 14 12 22 26 31 15 22 40 15 10 13 29 32 12 12 24 16 16 23 45 33 18 21 33 17 13 28 46 34 14 23 37 Use the data in the sheet named “Simon” and perform the analysis and interpretation asked below. a. Decide which linear regression model (with the assignment grade (A) or Midterm (M) or both as independent variable) is the better model to describe the final test score and motivate your selection. The remaining answers must be based on the model that you have selected in a. b. What is the total variation in final test grade that is explained by the independent variable(s) that you have selected? c. Is the overall regression model statistically significant? Test at the 5% level of significance using the model that you have selected in a. For this test formulate the appropriate null and alternative hypothesis, determine the region of acceptance, use an appropriate test statistic and draw the statistical and management conclusions. d. Write down the linear regression equation in algebraic format using A for the assignment grade, M for the midterm grade and F for the final test grade. e. Compute Simon’s expected final test grade (out of 50) if he scored 12/20 for his assignment and 13/30 for his midterm test. f. Compute the 95% confidence interval of the mean test grade for Simon (with his 12/20 assignment grade and 13/30 midterm test grade). Comment on his likelihood of passing the course. Based on: Keller, G 2009. Managerial Statistics, 8th ed. Mason, OH: South Western Cengage Learning, p 688. Question 2 (20 marks) The Super Corporation has three branch plants with excess production capacity. Fortunately, the corporation has a new product ready to begin production, and all three plants have the capability to produce this product, so some of the excess capacity can be used in this way. This product can be made in three sizes – large, medium and small – that yield a net unit profit of R850, R700 and R625, respectively. Plants 1, 2 and 3 respectively have the capacity to produce 750, 900 and 450 units per day of this new product, regardless of the size or combination of sizes involved. The amount of available in-process storage space also imposes a limitation on the production of the new product. Plants 1, 2, and 3 have 1 400, 1 000, and 600 m², respectively, of in-process storage space available for a day’s production of this product. Each unit of the large, medium, and small sizes produced per day requires 2.0, 1.5 and 1.2 m², respectively. Sales forecasts indicate that if available, 900, 1 200 and 750 units of the large, medium and small sizes, respectively, could be sold per day, and management wishes to satisfy at least 60% of this daily forecast for each size. Management wishes to know how much of each of the sizes should be produced by each of the plants to maximise profit. Use the data in the sheet named “SuperCorp” and build an appropriate linear programming model and determine the optimal solution. Do not generate any reports. Write a brief report about your solution the management of Super Corporation. Question 3 (20 marks) Athlete’s Wearhouse is commemorating Wayde van Niekerk’s Olympic glory with a specially designed golf shirt for the current Christmas season. The shirts will be sold at R250 each, but at later dates in the season will be offered at a 25% discount, and towards the end of the summer season the discount will be increased to 60% in an attempt to clear all remaining stock. Demand at the full price of R250 is estimated to be normally distributed with a mean of 500 units and a standard deviation of 75 units. When the excess shirts are later offered at a discount of 25%, the additional demand is expected to be uniformly distributed with a minimum of 50 units and a maximum of 150 units. At the end of the season, when the remainder of the shirts are offered at a 60% discount, the shirts becomes such a bargain that the additional demand can be assumed to be unlimited. The cost of the shirts is dependent on the volume bought as depicted in the table below. Order quantity Unit cost 0 – 249 250 – 499 500 – 999 1000 and more R175.00 R166.00 R158.00 R150.00 Use the shell in the sheet named “Wayde” and build a simulation model to simulate the profit generated from sales of commemorative golf shirt in this Christmas season. Assume that stock that has not been sold at the initial price will be offered at successively deeper discounts. Determine the mean, standard deviation, minimum and maximum of the simulated profit as well as the probability of exceeding a profit of R50 000 Use the model to evaluate various order quantities and recommend an order size (to the closest 50 units) based on the analysis. Provide a motivation for your recommendation. Simon SimonMax = 20Max = 30Max = 50 Student noAssignmentMidtermFinal test QuestionAnswer1121125 a.2152845 b.3131937 c.: Hypothesis4192638 c.: Test statistic and region of acceptance5202240 c: Statistcal conclusion6111322 c: Managerial conclusion7151032 d.AM8101130 e.12139131627 f.10141534 11191640 12171628 13101224 14122226 15101329 16162345 17132846 18142044 19112043 20131532 21151026 22193048 23171638 24132429 25182034 26101627 27121025 28111830 29121325 30141520 31152240 32121224 33182133 34142337 SuperCorp Super Corporation0 LargeMediumSmallMinimumCapacityStorage Demand900120075060%Plant 17501400 Profit850700625Plant 29001000 Unit space2.01.51.2Plant 3450600 Wayde Athlete's Wearhouse0Data table Mean DemandStDev MeanStDevVolumeCostMax Selling priceR250.00Normal500750R175.00Min MinMax250R166.00P(>50000) Discount25%Uniform50150500R158.00Recommendation: Discount60%1000R150.00 Order ReplicationOrderUnit costRNNormal demandUnits soldRevenueUnits excessRNDemand @ discount 25%Units soldRevenueExcessRevenueTotal RevenueTotal costProfit 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500