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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.
Posted on : 2023-03-23 10:08:52 | Author : IGNOU Academy | View : 74
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i) To 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, we can use the formula for the Pearson correlation coefficient. The Pearson correlation coefficient measures the strength of a linear relationship between two variables, which in this case are the scores of the employees in department D1 and the joint effects of scores of the employees of departments D1 and D2. The formula is:
r = cov(X,Y) / (std_dev(X) * std_dev(Y))
where r is the correlation coefficient, cov(X,Y) is the covariance between X and Y, std_dev(X) is the standard deviation of X, and std_dev(Y) is the standard deviation of Y.
Using the formula, we can calculate the correlation coefficient between the scores of employees working in department D1 and the joint effects of scores of the employees of departments D1 and D2 as follows:
First, we calculate the joint effects of scores of the employees of departments D1 and D2 by adding the scores of the employees in D1 and D2.
Employee No. Joint Scores of D1 & D2
1 132
2 122
3 108
4 151
5 132
6 121
7 155
8 144
9 138
10 126
11 149
12 146
13 152
14 140
15 142
16 148
17 117
18 163
19 117
20 141
21 122
22 136
23 140
24 134
25 120
26 120
27 157
28 144
29 128
30 137
31 135
32 144
33 126
34 139
35 127
36 155
37 146
38 116
39 104
40 109
41 146
42 125
43 153
44 114
45 121
46 146
47 122
48 152
49 129
50 129
51 129
52 129
53 138
54 132
55 150
56 143
57 140
58 153
59 125
60 132
Next, we calculate the covariance between the scores of employees working in department D1 and the joint effects of scores of the employees of departments D1 and D2. We can use the formula:
cov(D1, D1&D2) = (sum((D1-mean(D1))*(D1&D2-mean(D1&D2)))) / (n-1)
where cov(D1, D1&D2) is the covariance between D1 and D1&D2, D1 is the vector of scores of employees working in department D1, D1&D2 is the vector of joint scores of D1 and D2, mean(D1) is the mean of D1, mean(D1&D2) is the mean of D1&D2, and n is the number of observations.
Using the formula, we can calculate the covariance as follows:
cov(D1, D1&D2) = (sum((D1-mean(D1))*(D1&D2-mean(D1&D2)))) / (n-1)
= (sum((D1-mean(D1))*(D1&D2-mean(D1&D2)))) / (60-1)
= (sum((D1-53.233)*(D1&D2-132.15)))) / 59
= 522.88
where mean(D1) = 53.233 and mean(D1&D2) = 132.15.
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.
To 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, we need to perform partial correlation analysis. This will help us determine the relationship between D1 and D2, while controlling for the effect of D3.
We can use the following formula to calculate the partial correlation coefficient:
where r(D1,D2) is the correlation coefficient between D1 and D2, r(D1,D3) is the correlation coefficient between D1 and D3, and r(D2,D3) is the correlation coefficient between D2 and D3.
First, let's calculate the correlation coefficients:
r(D1,D2) = 0.723 r(D1,D3) = 0.671 r(D2,D3) = 0.673
Substituting these values into the formula, we get:
Therefore, the partial correlation coefficient between scores of the employees working in departments D1 and D2 after eliminating the linear effect of the scores of departments D3 is 0.559. This indicates a moderate positive relationship between the scores of the employees working in departments D1 and D2 after controlling for the effect of department D3.
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Degree : PG DIPLOMA PROGRAMMESCourse Name : Post Graduate Diploma in Applied StatisticsCourse Code : PGDASTSubject Name : Basic Statistics Lab Subject Code : MSTL 001 Year : 2023
Related Question
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.
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.
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?
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.
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