Exercise №2. Estimation of numerical characteristics
Statistic parameters (ex.1) varies in some range with an standard deviation(“ a standard mistake ”):
1) For arithmetic mean at absence correlation inside lines (autocorrelation)
(1)
At presence correlation inside lines (autocorrelation) and small autocorrelation coefficient application of dependence (23) is supposed
(2)
2) For coefficient of variation Cv at absence autocorrelation inside lines
(3)
At presence autocorrelation inside lines in case Cs=2Cv
(4)
3) For factor of asymmetry at Cs=2Cv and absence autocorrelation inside lines (formula of Reznikovsy)
(5)
4) For correlation coefficient (formula of Romanovsky)
(6)
At n> 25 use of dependence (7) is supposed
(7)
For autocorrelation coefficient the formula changes a little
(8).
It is given: the data of observation (tab. 1; ex. №1); a significance level – 5 %; normal distrbution.
The task:
1) Define a significance level and a confidence interval;
2) To calculate error of estimation of for arithmetic mean, Cv, Cs, r(autocorrelation coefficient.);
3) To define borders of a confidential interval for arithmetic mean and r (autocorrelation coefficient).
Table 1. Mid-annual water runoff, m3/s
| Год | Р1 | Р2 | Р3 | Р4 | Р5 | Р6 | Р7 | Р8 | Р9 | Р10 | Р11 | Р12 | Р13 | Р14 | Р15 | Р16 |
| -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 5.3 | -1 | -1 | -1 | -1 | -1 | -1 | ||
| -1 | -1 | -1 | -1 | -1 | -1 | 7.2 | -1 | -1 | -1 | -1 | -1 | -1 | ||||
| -1 | -1 | -1 | -1 | -1 | -1 | 13.0 | -1 | -1 | -1 | -1 | -1 | -1 | ||||
| -1 | -1 | -1 | -1 | -1 | -1 | 9.4 | -1 | -1 | -1 | -1 | -1 | -1 | ||||
| -1 | -1 | -1 | -1 | -1 | -1 | 7.6 | -1 | -1 | -1 | -1 | -1 | -1 | ||||
| -1 | -1 | -1 | -1 | -1 | -1 | 15.3 | -1 | -1 | -1 | -1 | -1 | -1 | ||||
| -1 | -1 | -1 | -1 | -1 | -1 | 10.0 | -1 | -1 | -1 | -1 | -1 | -1 | ||||
| -1 | -1 | -1 | -1 | -1 | -1 | 10.6 | -1 | -1 | -1 | -1 | -1 | -1 | ||||
| -1 | -1 | -1 | -1 | -1 | -1 | 7.9 | -1 | -1 | -1 | -1 | -1 | -1 | ||||
| -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | |||||
| -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | |||||
| -1 | -1 | -1 | -1 | -1 | -1 | 14.3 | -1 | -1 | -1 | -1 | -1 | |||||
| -1 | -1 | -1 | -1 | -1 | -1 | 14.7 | -1 | -1 | -1 | -1 | -1 | |||||
| -1 | -1 | -1 | -1 | -1 | -1 | 10.8 | -1 | -1 | -1 | -1 | -1 | |||||
| -1 | -1 | -1 | -1 | -1 | -1 | 8.3 | -1 | -1 | -1 | -1 | -1 | |||||
| -1 | -1 | 119.71 | -1 | -1 | -1 | -1 | 5.1 | -1 | -1 | -1 | -1 | 77.7 | -1 | |||
| -1 | -1 | 88.36 | -1 | -1 | -1 | -1 | 2.4 | -1 | -1 | -1 | -1 | 85.4 | -1 | |||
| -1 | -1 | -1 | -1 | 516.0 | 55.42 | 3.6 | -1 | -1 | -1 | 6.11 | -1 | |||||
| -1 | -1 | -1 | -1 | 408.0 | 43.72 | 2.4 | -1 | -1 | -1 | 5.61 | 14.8 | |||||
| 339.0 | -1 | -1 | 25.6 | 447.0 | 48.07 | 2.6 | -1 | -1 | -1 | 7.87 | 14.4 | |||||
| 400.0 | -1 | -1 | -1 | 456.0 | 49.66 | 3.1 | -1 | -1 | 16.2 | 5.89 | 13.5 | |||||
| 497.0 | -1 | 343.14 | -1 | 417.0 | 72.78 | 5.5 | -1 | -1 | 33.7 | 14.66 | 29.8 | |||||
| 447.0 | -1 | 280.27 | 48.7 | 520.0 | 60.98 | 3.7 | -1 | 58.9 | 23.6 | 11.60 | 18.4 | |||||
| 564.0 | -1 | 476.52 | 60.7 | 687.0 | 109.46 | -1 | -1 | 44.6 | 20.36 | 38.4 | ||||||
| 676.0 | 501.43 | 58.3 | 593.0 | 129.82 | 11.8 | 33.6 | 39.9 | 26.22 | ||||||||
| 543.0 | 90.9 | 320.89 | 43.5 | 513.0 | 86.74 | -1 | 22.2 | 82.2 | 31.8 | 17.12 | 30.8 | |||||
| 551.0 | 81.7 | 365.19 | 50.1 | 482.0 | 60.51 | 3.6 | 13.2 | 23.9 | 10.31 | 15.4 | ||||||
| 503.0 | 54.3 | 269.93 | 42.2 | 453.0 | 61.60 | 3.7 | 15.5 | 69.1 | 24.6 | 8.37 | 21.3 | |||||
| 423.0 | 72.7 | 284.93 | 37.2 | 412.0 | 67.93 | 7.2 | 19.5 | 69.2 | 24.6 | 23.98 | 87.4 | 23.5 | ||||
| 416.0 | 59.1 | 276.48 | 49.2 | 601.0 | 55.77 | 3.6 | 14.6 | 55.1 | 20.5 | 8.56 | 20.1 | |||||
| 520.0 | 258.38 | 474.0 | 77.15 | 4.6 | 18.1 | 73.5 | 28.5 | 13.53 | 27.5 | |||||||
| 441.0 | 27.7 | 155.97 | 37.7 | 411.0 | 27.13 | 1.0 | 5.07 | 9.72 | 3.83 | 82.5 | 8.35 | |||||
| 343.0 | 39.8 | 208.02 | 38.8 | 455.0 | 28.22 | 0.7 | 5.44 | 26.9 | 9.76 | 3.24 | 8.91 | |||||
| 526.0 | 318.68 | 45.2 | 470.0 | 70.28 | 4.5 | 17.6 | 70.4 | 25.6 | 14.48 | 21.9 | ||||||
| 468.0 | 507.57 | 57.5 | 531.0 | 80.60 | 7.8 | 20.6 | 76.6 | 33.2 | 10.94 | 29.9 | ||||||
| 595.0 | 347.23 | 71.7 | 614.0 | 117.23 | 9.0 | 28.4 | 17.72 | 37.7 | ||||||||
| 610.0 | 88.7 | -1 | 618.0 | 112.48 | 11.6 | 26.3 | 96.7 | 38.2 | 23.29 | 39.4 | ||||||
| 633.0 | 93.4 | -1 | 66.9 | 629.0 | 119.78 | 12.9 | 29.6 | 40.4 | 34.43 | 40.6 | ||||||
| 490.0 | 77.5 | -1 | 68.6 | 704.0 | 69.41 | 3.0 | 12.3 | 74.8 | 10.39 | 27.7 | ||||||
| 563.0 | 99.5 | -1 | 56.7 | 675.0 | 103.16 | 5.8 | 20.5 | 33.8 | 21.58 | 32.4 | ||||||
| 347.0 | 83.8 | -1 | 42.7 | 443.0 | 48.09 | 2.5 | 9.38 | 37.2 | 14.9 | 7.49 | 15.3 | |||||
| 392.0 | 66.7 | -1 | 44.6 | 518.0 | 53.90 | 4.2 | 10.7 | 35.9 | 14.4 | 12.40 | ||||||
| 494.0 | 73.5 | -1 | 56.4 | 631.0 | 66.25 | 5.8 | 13.1 | 48.7 | 10.72 | 19.7 | ||||||
| 606.0 | -1 | 61.2 | 663.0 | 105.88 | 11.0 | 27.8 | 88.3 | 31.8 | 30.22 | 32.8 | ||||||
| 368.00 | 68.4 | -1 | 54.9 | -1 | 63.12 | 3.5 | 10.5 | 62.8 | 26.5 | 8.82 | ||||||
| 327.83 | 60.1 | -1 | 40.3 | -1 | 41.77 | 1.8 | 7.87 | 41.4 | 19.6 | 8.11 | 14.3 | |||||
| 283.83 | 81.7 | -1 | 407.0 | 29.68 | 1.5 | 6.34 | 32.7 | 13.4 | 4.91 | 91.5 | 9.97 | |||||
| 427.50 | -1 | 54.9 | 552.0 | 44.13 | 3.5 | 10.6 | 47.8 | 22.9 | 9.07 | 15.1 | ||||||
| 510.17 | 96.8 | -1 | 63.9 | 86.83 | 7.75 | 19.4 | 81.6 | 35.9 | 20.60 | 28.2 | ||||||
| 478.58 | -1 | 50.6 | -1 | 178.97 | -1 | -1 | 156.18 | -1 | 44.25 | 61.9 | ||||||
| 433.33 | -1 | -1 | -1 | -1 | 123.88 | -1 | -1 | 106.48 | -1 | 40.00 | -1 | -1 | ||||
| 429.33 | -1 | -1 | -1 | -1 | 99.00 | -1 | -1 | 83.78 | -1 | 17.64 | -1 | -1 | ||||
| -1 | -1 | -1 | -1 | -1 | 104.00 | -1 | -1 | -1 | -1 | 27.61 | -1 | -1 | ||||
| 387.83 | -1 | 73.5 | -1 | -1 | -1 | 42.77 | -1 | -1 | 40.77 | -1 | 9.37 | -1 | -1 | |||
| 459.92 | -1 | -1 | -1 | -1 | 52.31 | -1 | -1 | 93.66 | -1 | 4.63 | -1 | -1 | ||||
| -1 | -1 | -1 | -1 | -1 | 32.79 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | ||||
| -1 | -1 | -1 | -1 | -1 | 50.49 | 3.61 | -1 | -1 | -1 | -1 | -1 | -1 | ||||
| -1 | -1 | -1 | 97.19 | 7.79 | -1 | -1 | -1 | -1 | -1 | -1 | ||||||
| -1 | -1 | 41.8 | -1 | 71.43 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | |||||
| -1 | -1 | -1 | 39.2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | ||||
| -1 | -1 | -1 | 48.8 | -1 | 100.00 | 15.2 | -1 | -1 | -1 | 35.8 | -1 | -1 | ||||
| -1 | -1 | -1 | -1 | -1 | -1 | 96.30 | 12.7 | 8.44 | 75.20 | -1 | 34.3 | -1 | -1 | |||
| -1 | -1 | -1 | -1 | -1 | -1 | 123.00 | -1 | -1 | 93.00 | -1 | -1 | -1 | ||||
| -1 | -1 | -1 | -1 | -1 | -1 | 103.00 | -1 | -1 | 111.00 | -1 | -1 | -1 | -1 | |||
| -1 | -1 | -1 | -1 | -1 | -1 | 34.30 | -1 | -1 | 54.70 | -1 | -1 | -1 | -1 | |||
| -1 | -1 | -1 | -1 | -1 | -1 | 88.30 | 5.7 | 5.17 | 119.00 | -1 | 16.8 | -1 | -1 | |||
| 361.00 | -1 | -1 | -1 | -1 | -1 | 139.00 | 20.1 | 13.9 | 163.00 | -1 | 45.9 | -1 | -1 | |||
| -1 | -1 | 72.8 | -1 | -1 | -1 | -1 | 75.8 | 6.88 | 5.2 | -1 | -1 | 29.8 | -1 | -1 | ||
| -1 | 54.7 | -1 | -1 | -1 | 49.4 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | ||||
| -1 | 77.8 | -1 | -1 | -1 | 67.20 | 6.4 | -1 | 85.70 | -1 | -1 | -1 | -1 |