Question

An investigation was performed to study the impacts of different types of machines on the production of a particular variety of toys. The six machines (A, B, C, D, E and F) are assigned at random to 36 cells of the square with the restriction that each machine is used only once by each operator and in each time-period. The following design was obtained in which 6 operators are arranged in “columns" and 6 time-periods are in “rows": 

  Operator
1 2 3 4 5 6
Time Period 1 A B C D E F
2 B C D E F A
3 C D E F A B
4 D E F A B C
5 E F A B C D
6 F A B C D E

 The average production in a day is given as follows:  

  Operator
1 2 3 4 5 6
Time Period 1 142 148 149 149 154 147
2 145 150 152 155 148 151
3 149 147 151 148 148 150
4 138 141 146 145 149 147
5 141 153 152 151 151 149
6 147 149 150 146 150 148

 Assuming that the effect of each operator, time-period and machine are normally distributed with approximately equal variances, analyse the design at 1% level of significance. Test whether the effect of the different operators, time periods and machines on the production are significant or not. If significant, do the pair-wise comparison between them.  

23 Mar 2023
Answer :
Word Count : 559

This problem can be solved using the analysis of variance (ANOVA) technique.

First, we need to calculate the total sum of squares (SST), which is the sum of the squared deviations of each observation from the overall mean. The formula for SST is as follows:

SST = sum sum (xij - x..)2

where xij is the production of the ith operator in the jth time period, x.. is the overall mean production, and the double summation is taken over all 36 cells.

Using the given data, we can calculate SST as follows:

SST = (142-147.17)2 + (148-147.17)2 + ... + (148-147.17)2 + (150-147.17)2 + ... + (148-147.17)2 = 1362.17

Next, we need to calculate the sum of squares for operators (SSO), time periods (SSTP), and machines (SSM), as well as the residual sum of squares (SSE). __________ _________ _________ _______ ______ _____.
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Related Questions

The scores (out of 100) secured by 60 employees of three different departments D1, D2 and D3 who participated in a study, are presented in the following table:

Employee No. Scores of D1 Scores of D2 Scores of D3 Employee No. Scores of D1 Scores of D2 Scores of D3
1 54 78 56 31 59 76 57
2 49 73 55 32 57 87 66
3 36 72 52 33 46 80 62
4 64 87 67 34 57 82 61
5 47 85 65 35 48 78 59
6 46 75 58 36 65 90 66
7 61 94 70 37 69 94 70
8 56 88 67 38 43 73 54
9 57 81 59 39 36 68 48
10 43 73 56 40 43 66 48
11 60 89 69 41 56 90 66
12 54 92 70 42 52 73 56
13 56 96 75 43 57 83 61
14 55 85 62 44 45 69 51
15 53 89 66 45 46 75 58
16 63 85 64 46 58 88 64
17 50 67 47 47 49 73 53
18 67 96 71 48 60 92 68
19 50 67 49 49 63 81 59
20 54 87 64 50 51 78 57
21 41 69 49 51 53 76 58
22 53 83 60 52 47 76 56
23 55 85 64 53 38 68 52
24 58 76 59 54 46 82 63
25 36 70 54 55 39 66 47
26 49 71 51 56 67 91 71
27 62 95 74 57 61 82 61
28 66 88 65 58 56 83 60
29 53 75 56 59 48 67 50
30 49 88 64 60 35 68 50

 i) Compute the correlation coefficient between scores of the employees working in department D1 and the joint effects of scores of the employees of departments D1 and D2.

ii) Compute the correlation coefficient between scores of the employees working in departments D1 and D2 after eliminating the linear effect of the scores of departments D3.

iii) Also represent the scores obtained by departments D1, D2 and D3 using box plot. 

An experiment was conducted to compare two metals: A and B, as bonding agents for an alloy material. Components of the alloy were bonded using the metals as bonding agents, and the pressures required to break the bonds were measured. The data for the pressures required for breaking the metal are given in the following table:

S. No. Breaking Pressure S. No. Breaking Pressure
Metal A Metal B Metal A Metal B
1 71.9 72.2 21 86.5 70.6
2 68.8 66.4 22 74.3 74.6
3 82.6 74.5 23 71.2 68.8
4 78.1 60.6 24 85 76.9
5 74.2 73.2 25 80.5 63
6 70.8 68.7 26 76.6 75.6
7 84.9 69 27 73.2 71.1
8 72.7 73 28 87.3 71.4
9 69.6 67.2 29 75.1 75.4
10 83.4 75.3 30 72 69.6
11 78.9 61.4 31 85.8 77.3
12 75 74 32 81.3 63.4
13 71.6 69.5 33 77.4 76
14 85.7 69.8 34 74 71.5
15 73.5 73.8 35 88.1 71.8
16 70.4 68 36 75.9 75.8
17 84.2 76.1 37 72.8 70
18 79.7 62.2 38 86.6 77.7
19 75.8 74.8 39 82.1 63.8
20 72.4 70.3 40 78.2 76.4

 If the pressure required to break both metals are normally distributed, then answers the following questions:
i) Are the variances of the distributions of the pressure of Metals A and B equal at 5% level of significance?
ii) If yes, check whether the average pressure for Metal A is more than the Metal B at 5% level of significance? 

 

An investigation was performed to study the impacts of different types of machines on the production of a particular variety of toys. The six machines (A, B, C, D, E and F) are assigned at random to 36 cells of the square with the restriction that each machine is used only once by each operator and in each time-period. The following design was obtained in which 6 operators are arranged in “columns" and 6 time-periods are in “rows": 

  Operator
1 2 3 4 5 6
Time Period 1 A B C D E F
2 B C D E F A
3 C D E F A B
4 D E F A B C
5 E F A B C D
6 F A B C D E

 The average production in a day is given as follows:  

  Operator
1 2 3 4 5 6
Time Period 1 142 148 149 149 154 147
2 145 150 152 155 148 151
3 149 147 151 148 148 150
4 138 141 146 145 149 147
5 141 153 152 151 151 149
6 147 149 150 146 150 148

 Assuming that the effect of each operator, time-period and machine are normally distributed with approximately equal variances, analyse the design at 1% level of significance. Test whether the effect of the different operators, time periods and machines on the production are significant or not. If significant, do the pair-wise comparison between them.  

A cooking oil supplier distributed two types of oils, say Oil A and Oil B to a large numbers of retail stores. The supplier wants to compare the popularity of both oils. For this purpose, he selects a sample of 100 stores and tracks record of the sold oils (in litres) of each type at each store. The data are noted in the following table:  

Store No. Oil A Oil B Store No. Oil A Oil B
1 161 419 51 478 196
2 285 411 52 284 241
3 219 168 53 488 182
4 321 241 54 447 132
5 435 125 55 384 322
6 325 261 56 267 341
7 463 119 57 390 139
8 319 285 58 270 462
9 108 441 59 381 227
10 328 213 60 252 140
11 479 116 61 245 420
12 285 319 62 196 474
13 489 135 63 201 392
14 448 187 64 227 452
15 385 349 65 181 406
16 268 279 66 441 397
17 391 306 67 130 375
18 271 296 68 213 455
19 382 269 69 373 367
20 253 403 70 190 503
21 246 309 71 280 366
22 197 424 72 236 486
23 202 349 73 297 171
24 228 250 74 421 219
25 182 457 75 340 173
26 442 196 76 380 418
27 131 240 77 308 454
28 214 337 78 361 228
29 374 252 79 183 432
30 191 423 80 121 468
31 281 322 81 162 231
32 237 406 82 286 252
33 298 146 83 220 283
34 422 175 84 322 114
35 341 487 85 436 325
36 381 278 86 326 213
37 309 442 87 464 229
38 362 326 88 320 183
39 184 414 89 120 291
40 122 377 90 329 175
41 160 250 91 480 141
42 284 272 92 286 394
43 218 356 93 490 163
44 320 366 94 449 134
45 434 170 95 386 130
46 324 213 96 134 459
47 462 147 97 392 363
48 318 195 98 272 315
49 118 452 99 383 338
50 327 385 100 254 365

 Answer the following:
i) Which type of oil has more average sales?
ii) Which oil shows greater variability in the sales?
iii) Determine the correlation between both types of oils.
iv) Compute suitable width of the class intervals for both oils,
v) Construct the continuous frequency distribution for both oils. 

 

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