Построение регрессионной модели системы двух случайных величин

 

Номера, которые необходимо выбрать, для выполнения лабораторной работы №4 определяются с помощью двух последних цифр шифра студента +15 значений. Например, если две последние цифры шифра студента 22, то выбираем с 22 по 36 строку.

Значения Х выбираются всеми студентами со 2-го столбца (вес железобетонных перекрытий), значения У выбираем по первой букве фамилии.

 

 

Номер п/п Вес железобетонных перекрытий кг/м кв   Длина плиты перекрытия, м А-Л Время установки плиты перекрытия, час. М-Т Стоимость строительно-монтажных работ, млн р. У-Я
516,148 30,31 4,1 33,2
512,475 30,4 4,185 32,8
443,568 31,1 4,098 26,6
510,058 30,35 4,19 32,6
488,541 30,7 4,156 30,7
541,694 30,1 4,225 35,5
449,666 31,1 4,108 27,2
472,208 30,9 3,95 29,2
553,791 29,85 4,2 36,5
507,401 30,45 4,18 32,4
480,709 30,79 4,145
404,602 31,7 4,05 23,1
468,393 31,1 4,13 28,9
487,242 30,65 4,154 30,6
400,322 31,8 4,04 22,7
462,801 30,95 4,122 28,4
429,344 31,35 4,274 25,3
503,57 30,46 4,175
578,028 29,56 4,274 38,7
475,214 3,97 29,5
611,563 29,2 4,32 41,7
478,877 30,75 4,143 29,8
514,042 30,33 4,189
585,644 29,47 4,285 39,4
524,349 31,5 4,2 33,9
500,753 30,5 4,17 31,8
532,163 30,11 4,21 34,6
529,632 30,2 4,3 34,4
404,673 31,65 4,05 23,1
405,141 31,64 4,05 23,2
479,527 30,75 4,146 29,9
473,668 30,9 4,137 29,3
468,759 30,87 4,13 28,9
439,92 31,22 4,09 26,3
542,804 30,2 4,22 35,6
507,974 30,4 4,181 32,4
493,446 30,65 4,162 31,1
520,758 30,3 4,2 33,6
555,651 30,5 4,244 36,7
354,368 32,25 3,987 18,6
487,813 30,8 4,156 30,6
559,046 4,25
519,484 30,27 4,2 33,5
421,672 31,6 4,07 24,7
479,017 30,75 4,143 29,8
558,401 29,8 4,248
490,911 30,61 4,156 30,9
467,798 30,95 4,128 28,8
534,327 4,214 34,8
486,433 30,66 4,15 30,5
513,547 30,34 4,19 32,9
522,285 30,23 4,2 33,7
533,934 29,5 4,22 34,8
493,953 30,6 4,162 31,2
436,678 31,7 4,09
455,322 31,05 4,01 27,7
513,041 30,34 4,19 32,9
506,337 30,42 4,18 32,3
571,162 29,65 4,265 38,1
473,051 30,82 4,136 29,3
502,705 30,47 4,174 31,9
522,44 31,2 4,2 33,7
553,71 29,86 4,24 36,5
30,1 4,238 36,3
523,344 30,22 4,2 33,8
491,266 30,6 4,16 30,9
532,988 30,1 4,2 34,7
545,561 29,95 4,231 35,8
519,683 30,26 4,19 33,5
592,408 30,1 4,294
447,626 31,2 4,1
484,998 30,68 4,15 30,3
603,083 29,7 4,31
530,224 30,14 4,21 34,4
518,926 30,27 4,19 33,4
471,808 30,84 4,137 29,2
538,127 30,04 4,22 35,1
504,49 30,45 4,18 32,1
498,215 30,52 4,168 31,5
457,645 31,01 4,214 27,9
471,706 4,137 29,2
604,033 29,5 4,314 41,1
435,876 31,27 4,09 25,9
547,073 30,1 4,233 35,9
488,658 30,64 4,156 30,7
483,232 30,7 4,15 30,2
525,513 30,2 4,21
437,864 31,25 4,09 26,1
523,714 30,2 4,2 33,8
615,479 29,1 4,325 42,1
388,428 31,8 4,09 21,7
421,66 31,44 4,07 24,6
517,403 30,29 4,19 33,3
575,794 30,1 4,271 38,5
485,776 30,67 4,156 30,4
497,492 30,53 4,167 31,5
446,804 31,14 4,1 26,9
475,051 30,8 4,138 29,5
528,184 30,16 4,208 34,2
545,557 30,9 4,234 35,8

 

 


ПРИЛОЖЕНИЕ А (справочное) Критические точки распределения Стьюдента

Степени свободы n Уровень значимости a (односторонняя критическая область)
0.2 0.1 0.05 0.025 0.01 0.005 0.001 0.0005
1.376 3.078 6.314 12.706 31.821 63.656 318.29 636.58
1.061 1.886 2.920 4.303 6.965 9.925 22.328 31.600
0.978 1.638 2.353 3.182 4.541 5.841 10.214 12.924
0.941 1.533 2.132 2.776 3.747 4.604 7.173 8.610
0.920 1.476 2.015 2.571 3.365 4.032 5.894 6.869
0.906 1.440 1.943 2.447 3.143 3.707 5.208 5.959
0.896 1.415 1.895 2.365 2.998 3.499 4.785 5.408
0.889 1.397 1.860 2.306 2.896 3.355 4.501 5.041
0.883 1.383 1.833 2.262 2.821 3.250 4.297 4.781
0.879 1.372 1.812 2.228 2.764 3.169 4.144 4.587
0.876 1.363 1.796 2.201 2.718 3.106 4.025 4.437
0.873 1.356 1.782 2.179 2.681 3.055 3.930 4.318
0.870 1.350 1.771 2.160 2.650 3.012 3.852 4.221
0.868 1.345 1.761 2.145 2.624 2.977 3.787 4.140
0.866 1.341 1.753 2.131 2.602 2.947 3.733 4.073
0.865 1.337 1.746 2.120 2.583 2.921 3.686 4.015
0.863 1.333 1.740 2.110 2.567 2.898 3.646 3.965
0.862 1.330 1.734 2.101 2.552 2.878 3.610 3.922
0.861 1.328 1.729 2.093 2.539 2.861 3.579 3.883
0.860 1.325 1.725 2.086 2.528 2.845 3.552 3.850
0.859 1.323 1.721 2.080 2.518 2.831 3.527 3.819
0.858 1.321 1.717 2.074 2.508 2.819 3.505 3.792
0.858 1.319 1.714 2.069 2.500 2.807 3.485 3.768
0.857 1.318 1.711 2.064 2.492 2.797 3.467 3.745
0.856 1.316 1.708 2.060 2.485 2.787 3.450 3.725
0.856 1.315 1.706 2.056 2.479 2.779 3.435 3.707
0.855 1.314 1.703 2.052 2.473 2.771 3.421 3.689
0.855 1.313 1.701 2.048 2.467 2.763 3.408 3.674
0.854 1.311 1.699 2.045 2.462 2.756 3.396 3.660
0.854 1.310 1.697 2.042 2.457 2.750 3.385 3.646
0.851 1.303 1.684 2.021 2.423 2.704 3.307 3.551
0.849 1.299 1.676 2.009 2.403 2.678 3.261 3.496
0.848 1.296 1.671 2.000 2.390 2.660 3.232 3.460
0.847 1.294 1.667 1.994 2.381 2.648 3.211 3.435
0.846 1.292 1.664 1.990 2.374 2.639 3.195 3.416
0.846 1.291 1.662 1.987 2.368 2.632 3.183 3.402
0.845 1.290 1.660 1.984 2.364 2.626 3.174 3.390
0.844 1.287 1.655 1.976 2.351 2.609 3.145 3.357
0.843 1.286 1.653 1.972 2.345 2.601 3.131 3.340
0.842 1.283 1.648 1.965 2.334 2.586 3.107 3.310
¥ 0.842 1.282 1.645 1.960 2.326 2.576 3.090 3.290
Степени свободы 0.4 0.2 0.1 0.05 0.02 0.01 0.002 0.001
n Уровень значимости a (двусторонняя критическая область)

 

 

ПРИЛОЖЕНИЕ Б (справочное) Критические точки распределения Фишера

(число степеней свободы бóльшей дисперсии – n1. меньшей – n2)

Уровень значимости a = 0.05
n2 n1
¥
161.45 199.50 215.71 224.58 230.16 233.99 238.88 243.90 246.47 249.05 251.77 254.31
18.513 19.000 19.164 19.247 19.296 19.329 19.371 19.412 19.433 19.454 19.476 19.496
10.128 9.552 9.277 9.117 9.013 8.941 8.845 8.745 8.692 8.638 8.581 8.526
7.709 6.944 6.591 6.388 6.256 6.163 6.041 5.912 5.844 5.774 5.699 5.628
6.608 5.786 5.409 5.192 5.050 4.950 4.818 4.678 4.604 4.527 4.444 4.365
5.987 5.143 4.757 4.534 4.387 4.284 4.147 4.000 3.922 3.841 3.754 3.669
5.591 4.737 4.347 4.120 3.972 3.866 3.726 3.575 3.494 3.410 3.319 3.230
5.318 4.459 4.066 3.838 3.688 3.581 3.438 3.284 3.202 3.115 3.020 2.928
5.117 4.256 3.863 3.633 3.482 3.374 3.230 3.073 2.989 2.900 2.803 2.707
4.965 4.103 3.708 3.478 3.326 3.217 3.072 2.913 2.828 2.737 2.637 2.538
4.844 3.982 3.587 3.357 3.204 3.095 2.948 2.788 2.701 2.609 2.507 2.404
4.747 3.885 3.490 3.259 3.106 2.996 2.849 2.687 2.599 2.505 2.401 2.296
4.667 3.806 3.411 3.179 3.025 2.915 2.767 2.604 2.515 2.420 2.314 2.206
4.600 3.739 3.344 3.112 2.958 2.848 2.699 2.534 2.445 2.349 2.241 2.131
4.543 3.682 3.287 3.056 2.901 2.790 2.641 2.475 2.385 2.288 2.178 2.066
4.494 3.634 3.239 3.007 2.852 2.741 2.591 2.425 2.333 2.235 2.124 2.010
4.451 3.592 3.197 2.965 2.810 2.699 2.548 2.381 2.289 2.190 2.077 1.960
4.414 3.555 3.160 2.928 2.773 2.661 2.510 2.342 2.250 2.150 2.035 1.917
4.381 3.522 3.127 2.895 2.740 2.628 2.477 2.308 2.215 2.114 1.999 1.878
4.351 3.493 3.098 2.866 2.711 2.599 2.447 2.278 2.184 2.082 1.966 1.843
4.325 3.467 3.072 2.840 2.685 2.573 2.420 2.250 2.156 2.054 1.936 1.812
4.301 3.443 3.049 2.817 2.661 2.549 2.397 2.226 2.131 2.028 1.909 1.783
4.279 3.422 3.028 2.796 2.640 2.528 2.375 2.204 2.109 2.005 1.885 1.757
4.260 3.403 3.009 2.776 2.621 2.508 2.355 2.183 2.088 1.984 1.863 1.733
4.242 3.385 2.991 2.759 2.603 2.490 2.337 2.165 2.069 1.964 1.842 1.711
4.225 3.369 2.975 2.743 2.587 2.474 2.321 2.148 2.052 1.946 1.823 1.691
4.210 3.354 2.960 2.728 2.572 2.459 2.305 2.132 2.036 1.930 1.806 1.672
4.196 3.340 2.947 2.714 2.558 2.445 2.291 2.118 2.021 1.915 1.790 1.654
4.183 3.328 2.934 2.701 2.545 2.432 2.278 2.104 2.007 1.901 1.775 1.638
4.171 3.316 2.922 2.690 2.534 2.421 2.266 2.092 1.995 1.887 1.761 1.622
4.085 3.232 2.839 2.606 2.449 2.336 2.180 2.003 1.904 1.793 1.660 1.509
4.034 3.183 2.790 2.557 2.400 2.286 2.130 1.952 1.850 1.737 1.599 1.438
4.001 3.150 2.758 2.525 2.368 2.254 2.097 1.917 1.815 1.700 1.559 1.389
3.978 3.128 2.736 2.503 2.346 2.231 2.074 1.893 1.790 1.674 1.530 1.353
3.960 3.111 2.719 2.486 2.329 2.214 2.056 1.875 1.772 1.654 1.508 1.325
3.947 3.098 2.706 2.473 2.316 2.201 2.043 1.861 1.757 1.639 1.491 1.302
3.936 3.087 2.696 2.463 2.305 2.191 2.032 1.850 1.746 1.627 1.477 1.283
3.904 3.056 2.665 2.432 2.274 2.160 2.001 1.817 1.711 1.590 1.436 1.223
3.888 3.041 2.650 2.417 2.259 2.144 1.985 1.801 1.694 1.572 1.415 1.189
3.860 3.014 2.623 2.390 2.232 2.117 1.957 1.772 1.664 1.539 1.376 1.113
¥ 3.841 2.996 2.605 2.372 2.214 2.099 1.938 1.752 1.644 1.517 1.350 1.000

 

Продолжение приложения Б

Уровень значимости a = 0.01
n2 n1
¥
4052.2 4999.3 5403.5 5624.3 5764.0 5859.0 5981.0 6106.7 6170.0 6234.3 6302.3 6365.6
98.502 99.000 99.164 99.251 99.302 99.331 99.375 99.419 99.437 99.455 99.477 99.499
34.116 30.816 29.457 28.710 28.237 27.911 27.489 27.052 26.826 26.597 26.354 26.125
21.198 18.000 16.694 15.977 15.522 15.207 14.799 14.374 14.154 13.929 13.690 13.463
16.258 13.274 12.060 11.392 10.967 10.672 10.289 9.888 9.680 9.466 9.238 9.020
13.745 10.925 9.780 9.148 8.746 8.466 8.102 7.718 7.519 7.313 7.091 6.880
12.246 9.547 8.451 7.847 7.460 7.191 6.840 6.469 6.275 6.074 5.858 5.650
11.259 8.649 7.591 7.006 6.632 6.371 6.029 5.667 5.477 5.279 5.065 4.859
10.562 8.022 6.992 6.422 6.057 5.802 5.467 5.111 4.924 4.729 4.517 4.311
10.044 7.559 6.552 5.994 5.636 5.386 5.057 4.706 4.520 4.327 4.115 3.909
9.646 7.206 6.217 5.668 5.316 5.069 4.744 4.397 4.213 4.021 3.810 3.602
9.330 6.927 5.953 5.412 5.064 4.821 4.499 4.155 3.972 3.780 3.569 3.361
9.074 6.701 5.739 5.205 4.862 4.620 4.302 3.960 3.778 3.587 3.375 3.165
8.862 6.515 5.564 5.035 4.695 4.456 4.140 3.800 3.619 3.427 3.215 3.004
8.683 6.359 5.417 4.893 4.556 4.318 4.004 3.666 3.485 3.294 3.081 2.868
8.531 6.226 5.292 4.773 4.437 4.202 3.890 3.553 3.372 3.181 2.967 2.753
8.400 6.112 5.185 4.669 4.336 4.101 3.791 3.455 3.275 3.083 2.869 2.653
8.285 6.013 5.092 4.579 4.248 4.015 3.705 3.371 3.190 2.999 2.784 2.566
8.185 5.926 5.010 4.500 4.171 3.939 3.631 3.297 3.116 2.925 2.709 2.489
8.096 5.849 4.938 4.431 4.103 3.871 3.564 3.231 3.051 2.859 2.643 2.421
7.945 5.719 4.817 4.313 3.988 3.758 3.453 3.121 2.941 2.749 2.531 2.305
7.823 5.614 4.718 4.218 3.895 3.667 3.363 3.032 2.852 2.659 2.440 2.211
7.636 5.453 4.568 4.074 3.754 3.528 3.226 2.896 2.716 2.522 2.300 2.064
7.562 5.390 4.510 4.018 3.699 3.473 3.173 2.843 2.663 2.469 2.245 2.006
7.314 5.178 4.313 3.828 3.514 3.291 2.993 2.665 2.484 2.288 2.058 1.805
7.171 5.057 4.199 3.720 3.408 3.186 2.890 2.563 2.382 2.183 1.949 1.683
7.077 4.977 4.126 3.649 3.339 3.119 2.823 2.496 2.315 2.115 1.877 1.601
7.011 4.922 4.074 3.600 3.291 3.071 2.777 2.450 2.268 2.067 1.826 1.540
6.963 4.881 4.036 3.563 3.255 3.036 2.742 2.415 2.233 2.032 1.788 1.494
6.925 4.849 4.007 3.535 3.228 3.009 2.715 2.389 2.206 2.004 1.759 1.457
6.895 4.824 3.984 3.513 3.206 2.988 2.694 2.368 2.185 1.983 1.735 1.427
6.807 4.749 3.915 3.447 3.142 2.924 2.632 2.305 2.122 1.918 1.665 1.331
6.763 4.713 3.881 3.414 3.110 2.893 2.601 2.275 2.091 1.886 1.629 1.279
6.686 4.648 3.821 3.357 3.054 2.838 2.547 2.220 2.036 1.829 1.566 1.164
¥ 6.635 4.605 3.782 3.319 3.017 2.802 2.511 2.185 2.000 1.791 1.523 1.000

 

Multiple Regression - A.Col_2

Dependent variable: A.Col_2

Independent variables:

A.Col_3

 

    Standard T  
Parameter Estimate Error Statistic P-Value
CONSTANT 3439,34 147,937 23,2486 0,0000
A.Col_3 -96,2044 4,85659 -19,809 0,0000

 

Analysis of Variance

Source Sum of Squares Df Mean Square F-Ratio P-Value
Model 32211,1 32211,1 392,40 0,0000
Residual 1067,14 82,0879    
Total (Corr.) 33278,2      

 

R-squared = 96,7933 percent

R-squared (adjusted for d.f.) = 96,5466 percent

Standard Error of Est. = 9,06024

Mean absolute error = 6,45345

Durbin-Watson statistic = 1,98052 (P=0,5328)

Lag 1 residual autocorrelation = 0,00286229

 

The StatAdvisor

The output shows the results of fitting a multiple linear regression model to describe the relationship between A.Col_2 and 1 independent variables. The equation of the fitted model is

 

A.Col_2 = 3439,34 - 96,2044*A.Col_3

 

Since the P-value in the ANOVA table is less than 0,05, there is a statistically significant relationship between the variables at the 95,0% confidence level.

 

The R-Squared statistic indicates that the model as fitted explains 96,7933% of the variability in A.Col_2. The adjusted R-squared statistic, which is more suitable for comparing models with different numbers of independent variables, is 96,5466%. The standard error of the estimate shows the standard deviation of the residuals to be 9,06024. This value can be used to construct prediction limits for new observations by selecting the Reports option from the text menu. The mean absolute error (MAE) of 6,45345 is the average value of the residuals. The Durbin-Watson (DW) statistic tests the residuals to determine if there is any significant correlation based on the order in which they occur in your data file. Since the P-value is greater than 0,05, there is no indication of serial autocorrelation in the residuals at the 95,0% confidence level.

 

In determining whether the model can be simplified, notice that the highest P-value on the independent variables is 0,0000, belonging to A.Col_3. Since the P-value is less than 0,05, that term is statistically significant at the 95,0% confidence level. Consequently, you probably don't want to remove any variables from the model.