Реферат: Банковская система 10

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   3. Рассчитайте характеристики интервального ряда распределения: среднюю арифметическую, среднее квадратическое отклонение, коэффициент вариации.

   4. Вычислите среднюю арифметическую по исходным данным, сравните ее с аналогичным показателем, рассчитанным в п.3 для интервального ряда распределения. Объясните причину их расхождения.

   Сделайте выводы по результатам выполнения задания.
1. Построим статистический ряд распределения банков по работающим активам. Проранжируем исходные данные по предложенному признаку:
|№ банка п/п       |Работающие активы     |Прибыль             |№ банка п/п       |Работающие активы       |Прибыль           |

|31                |4493                  |165                 |20                |14288                   |93                |

|16                |4780                  |134                 |10                |14389                   |224               |

|3                 |5728                  |68                  |12                |14389                   |348               |

|19                |6585                  |105                 |21                |14691                   |329               |

|17                |7024                  |121                 |13                |15076                   |64                |

|34                |7442                  |204                 |11                |15676                   |203               |

|9                 |7560                  |215                 |36                |15683                   |189               |

|33                |9350                  |77                  |5                 |17213                   |146               |

|7                 |9387                  |111                 |30                |17961                   |166               |

|6                 |9711                  |70                  |32                |18785                   |112               |

|35                |10038                 |78                  |22                |19243                   |269               |

|4                 |10085                 |213                 |28                |20968                   |282               |

|15                |10146                 |153                 |27                |21337                   |237               |

|25                |10225                 |62                  |24                |21902                   |139               |

|18                |10668                 |94                  |14                |24089                   |240               |

|8                 |11908                 |153                 |26                |24888                   |441               |

|23                |11971                 |451                 |1                 |25946                   |110               |

|29                |14124                 |91                  |2                 |26110                   |538               |

|Итого:                                                                           |503859                  |6695              |
Образуем 5 групп с равными интервалами длиной [pic] млн. руб. по изучаемому признаку:
|Группы        |Группы банков по работающим активам, млн.руб.     |№ банка         |Работающие активы, млн.руб.            |

|              |                                                  |31              |4493                                   |

|              |                                                  |16              |4780                                   |

|              |                                                  |3               |5728                                   |

|I             |4493-8816,4                                       |19              |6585                                   |

|              |                                                  |17              |7024                                   |

|              |                                                  |34              |7442                                   |

|              |                                                  |9               |7560                                   |

|              |Итого:                                            |7               |43612                                  |

|              |                                                  |33              |9350                                   |

|              |                                                  |7               |9387                                   |

|              |                                                  |6               |9711                                   |

|              |                                                  |35              |10038                                  |

|II            |8816,4-13139,8                                    |4               |10085                                  |

|              |                                                  |15              |10146                                  |

|              |                                                  |25              |10225                                  |

|              |                                                  |18              |10668                                  |

|              |                                                  |8               |11908                                  |

|              |                                                  |23              |11971                                  |

|              |Итого:                                            |10              |103489                                 |

|              |                                                  |29              |14124                                  |

|              |                                                  |20              |14288                                  |

|              |                                                  |10              |14389                                  |

|              |                                                  |12              |14389                                  |

|III           |13139,8-17463,2                                   |21              |14691                                  |

|              |                                                  |13              |15076                                  |

|              |                                                  |11              |15676                                  |

|              |                                                  |36              |15683                                  |

|              |                                                  |5               |17213                                  |

|              |Итого:                                            |9               |135529                                 |

|              |                                                  |30              |17961                                  |

|              |                                                  |32              |18785                                  |

|IV            |17463,2-21786,6                                   |22              |19243                                  |

|              |                                                  |28              |20968                                  |

|              |                                                  |27              |21337                                  |

|              |Итого:                                            |5               |98294                                  |

|              |                                                  |24              |21902                                  |

|              |                                                  |14              |24089                                  |

|V             |21786,6-26110                                     |26              |24888                                  |

|              |                                                  |1               |25946                                  |

|              |                                                  |2               |26110                                  |

|              |Итого:                                            |5               |122935                                 |
В результате группировки получили следующий ряд распределения:
|Группы         |Группы банков по                                              |Число банков                          |

|               |работающим активам, млн.руб.                                  |                                      |

|I              |4493-8816,4                                                   |7                                     |

|II             |8816,4-13139,8                                                |10                                    |

|III            |13139,8-17463,2                                               |9                                     |

|IV             |17463,2-21786,6                                               |5                                     |

|V              |21786,6-26110                                                 |5                                     |
2. Построим кумуляту и найдем графически значение медианы:
|Группы         |Группы банков по                                              |Число банков                          |Накопленные частоты                   |

|               |работающим активам, млн.руб.                                  |                                      |                                      |

|I              |4493-8816,4                                                   |7                                     |7                                     |

|II             |8816,4-13139,8                                                |10                                    |17                                    |

|III            |13139,8-17463,2                                               |9                                     |26                                    |

|IV             |17463,2-21786,6                                               |5                                     |31                                    |

|V              |21786,6-26110                                                 |5                                     |36                                    |
[pic]
Таким образом, половина банков имеет работающие активы более 13610 млн. руб., а другая половина менее 13610 млн. руб..
Построим гистограмму и найдем графически значение моды:
 [pic]
Таким образом, наибольшая часть банков имеет 12050 млн. руб. работающих активов.
3. Рассчитаем характеристики ряда.
|Группы        |Группы банков по                       |Число банков, [pic] |[pic]         |[pic]           |[pic]                       |

|              |работающим активам, млн.руб.           |                    |              |                |                            |

|I             |4493-8816,4                            |7                   |6654,7        |46582,9         |400705195,8175              |

|II            |8816,4-13139,8                         |10                  |10978,1       |109781          |105141305,0250              |

|III           |13139,8-17463,2                        |9                   |15301,5       |137713,5        |10514130,5025               |

|IV            |17463,2-21786,6                        |5                   |19624,9       |98124,5         |146029590,3125              |

|V             |21786,6-26110                          |5                   |23948,3       |119741,5        |473135872,6125              |

|              |Итого:                                 |36                  |              |511943,4        |1135526094,2700             |
Среднее арифметическое: [pic] млн. руб., т.е. в среднем работающие активы на один банк составляют 14220,65 млн. руб.

Среднее квадратическое отклонение: [pic] млн. руб.

Коэффициент вариации: [pic], т.е. совокупность можно считать неоднородной по работающим активам, т.к.39,5%>33%.

4. Вычислим среднюю арифметическую по исходным данным: [pic] млн. руб.

Средняя арифметическая интервального ряда распределения больше средней арифметической по исходным данным на 224,567 млн. руб. Причина их расхождения в том, что средняя арифметическая интервального ряда распределения рассчитана по формуле средней арифметической взвешенной, которая показывает нам среднюю из вариантов, которые повторяются различное число раз, т.е. имеют различный вес. В качестве весов выступает численность банков в каждой из пяти групп совокупности.
Задание 2.
По исходным данным:

   1. Установите наличие и характер связи между работающими активами и прибылью, образовав 5 групп с равными интервалами по обоим признакам методами:

         • Аналитической группировки.

         • Корреляционной таблицы.

         • Оценить тесноту связи, рассчитав коэффициент детерминации и эмпирическое корреляционное отношение.
1. По исходным данным установим наличие и характер связи между работающими активами и прибылью:

а) составим аналитическую группировку:
|Группы        |Группы банков по работающим активам, млн.руб.     |№ банка         |Работающие активы, млн.руб.        |Прибыль,             |

|              |                                                  |                |                                   |млн. руб.            |

|              |                                                  |31              |4493                               |165                  |

|              |                                                  |16              |4780                               |134                  |

|              |                                                  |3               |5728                               |68                   |

|I             |4493-8816,4                                       |19              |6585                               |105                  |

|              |                                                  |17              |7024                               |121                  |

|              |                                                  |34              |7442                               |204                  |

|              |                                                  |9               |7560                               |215                  |

|              |Итого:                                            |7               |43612                              |1012                 |

|              |                                                  |33              |9350                               |77                   |

|              |                                                  |7               |9387                               |111                  |

|              |                                                  |6               |9711                               |70                   |

|              |                                                  |35              |10038                              |78                   |

|II            |8816,4-13139,8                                    |4               |10085                              |213                  |

|              |                                                  |15              |10146                              |153                  |

|              |                                                  |25              |10225                              |62                   |

|              |                                                  |18              |10668                              |94                   |

|              |                                                  |8               |11908                              |153                  |

|              |                                                  |23              |11971                              |451                  |

|              |Итого:                                            |10              |103489                                 продолжение
--PAGE_BREAK--|1462                 |

|              |                                                  |29              |14124                              |91                   |

|              |                                                  |20              |14288                              |93                   |

|              |                                                  |10              |14389                              |224                  |

|              |                                                  |12              |14389                              |348                  |

|III           |13139,8-17463,2                                   |21              |14691                              |329                  |

|              |                                                  |13              |15076                              |64                   |

|              |                                                  |11              |15676                              |203                  |

|              |                                                  |36              |15683                              |189                  |

|              |                                                  |5               |17213                              |146                  |

|              |Итого:                                            |9               |135529                             |1687                 |

|              |                                                  |30              |17961                              |166                  |

|              |                                                  |32              |18785                              |112                  |

|IV            |17463,2-21786,6                                   |22              |19243                              |269                  |

|              |                                                  |28              |20968                              |282                  |

|              |                                                  |27              |21337                              |237                  |

|              |Итого:                                            |5               |98294                              |1066                 |

|              |                                                  |24              |21902                              |139                  |

|              |                                                  |14              |24089                              |240                  |

|V             |21786,6-26110                                     |26              |24888                              |441                  |

|              |                                                  |1               |25946                              |110                  |

|              |                                                  |2               |26110                              |538                  |

|              |Итого:                                            |5               |122935                             |1468                 |
Таким образом, получаем аналитическую группировку:
|№ группы      |Группы банков по работающим  |Число банков,              |Работающие активы, млн.руб.              |Прибыль, млн. руб.                 |

|              |активам, млн.руб.            |ед., [pic]                 |                                         |                                   |

|              |                             |                           |всего по группе     |на 1 банк           |всего по группе   |на 1 банк        |

|I             |4493-8816,4                  |7                          |43612               |6230,29             |1012              |144,57           |

|II            |8816,4-13139,8               |10                         |103489              |10348,90            |1462              |146,20           |

|III           |13139,8-17463,2              |9                          |135529              |15058,78            |1687              |187,44           |

|IV            |17463,2-21786,6              |5                          |98294               |19658,80            |1066              |213,20           |

|V             |21786,6-26110                |5                          |122935              |24587,00            |1468              |293,60           |

|Итого:                                     |36                         |                    |13996,08            |                  |185,97           |
Из таблицы видно, что между рассматриваемыми величинами существует прямая зависимость, т.е. чем больше работающие активы, тем больше прибыль и наоборот.
б) корреляционная таблица:

[pic] млн. руб.
|Группы банков                         |Группы банков                                                                                       |

|по кредитам, млн.руб.                 |по прибыли, млн.руб.                                                                                |

|                                      |62-157,2         |157,2-252,4         |252,4-347,6          |347,6-442,8          |442,8-538        |

|4493-8816,4                           |16               |31                  |-                    |-                    |-                |

|                                      |3                |34                  |                     |                     |                 |

|                                      |19               |10                  |                     |                     |                 |

|                                      |17               |                    |                     |                     |                 |

|8816,4-13139,8                        |33               |4                   |-                    |-                    |23               |

|                                      |7                |                    |                     |                     |                 |

|                                      |6                |                    |                     |                     |                 |

|                                      |35               |                    |                     |                     |                 |

|                                      |15               |                    |                     |                     |                 |

|                                      |25               |                    |                     |                     |                 |

|                                      |18               |                    |                     |                     |                 |

|                                      |8                |                    |                     |                     |                 |

|13139,8-17463,2                       |29               |10                  |21                   |12                   |-                |

|                                      |20               |11                  |                     |                     |                 |

|                                      |13               |36                  |                     |                     |                 |

|                                      |5                |                    |                     |                     |                 |

|17463,2-21786,6                       |32               |30                  |22                   |-                    |-                |

|                                      |                 |27                  |28                   |                     |                 |

|21786,6-26110                         |                 |                    |                     |                     |                 |
Получили:
|Группы банков                      |Группы банков                                                                                           |Итого           |

|по кредитам, млн.руб.              |по прибыли, млн.руб.                                                                                    |                |

|                                   |62-157,2        |157,2-252,4            |252,4-347,6          |347,6-442,8          |442,8-538          |                |

|4493-8816,4                        |4               |3                      |-                    |-                    |-                  |5               |

|8816,4-13139,8                     |8               |1                      |-                    |-                    |1                  |10              |

|13139,8-17463,2                    |4               |3                      |1                    |1                    |-                  |9               |

|17463,2-21786,6                    |1               |2                      |2                    |-                    |-                  |5               |

|21786,6-26110                      |2               |1                      |-                    |1                    |1                  |5               |

|Итого:                             |19              |10                     |3                    |2                    |2                  |36              |
Из таблицы видно (распределение банков не произошло по диагонали из левого верхнего угла в правый нижний угол), что между рассматриваемыми величинами существует слабая прямая зависимость.
2. Измерим тесноту корреляционной связи между названными признаками с использованием коэффициента детерминации и эмпирического корреляционного отношения.
Используем формулы: [pic], [pic], [pic] млн. руб.
|№ группы      |Группы банков               |Число банков,          |Прибыль от продаж на 1 банк, |[pic]             |[pic]                 |

|              |по кредитам, млн.руб.       |ед., [pic]             |млн. руб., [pic]             |                  |                      |

|I             |4493-8816,4                 |5                      |144,57                       |-41,40            |11998,18              |

|II            |8816,4-13139,8              |10                     |146,20                       |-39,77            |15818,2966            |

|III           |13139,8-17463,2             |9                      |187,44                       |1,47              |19,50694444           |

|IV            |17463,2-21786,6             |5                      |213,20                       |27,23             |3706,759414           |

|V             |21786,6-26110               |5                      |293,60                       |107,63            |57918,69275           |

|Итого:                                    |36                     |                             |                  |89461,43571           |
Тогда [pic].
[pic];   [pic]

|№                       |Прибыль                       |[pic]           |

|банка                   |млн. руб., [pic]              |                |

|1                       |110                           |12100           |

|2                       |538                           |289444          |

|3                       |68                            |4624            |

|4                       |213                           |45369           |

|5                       |146                           |21316           |

|6                       |70                            |4900            |

|7                       |111                           |12321           |

|8                       |153                           |23409           |

|9                       |215                           |46225           |

|10                      |224                           |50176           |

|11                      |203                           |41209           |

|12                      |348                           |121104          |

|13                      |64                            |4096            |

|14                      |240                           |57600           |

|15                      |153                           |23409           |

|16                      |134                           |17956           |

|17                      |121                           |14641           |

|18                      |94                            |8836            |

|19                      |105                           |11025           |

|20                      |93                            |8649            |

|21                      |329                           |108241          |

|22                      |269                           |72361           |

|23                      |451                           |203401          |

|24                      |139                           |19321           |

|25                      |62                            |3844            |

|26                      |441                           |194481          |

|27                      |237                           |56169           |

|28                      |282                           |79524           |

|29                      |91                            |8281            |

|30                      |166                           |27556           |

|31                      |165                           |27225           |

|32                      |112                           |12544           |

|33                      |77                            |5929            |

|34                      |204                           |41616           |

|35                      |78                            |6084            |

|36                      |189                           |35721           |

|Итого:                                                  |1720707         |
[pic] и эмпирический коэффициент детерминации [pic] или 18,8%.

Т.е. вариация прибыли на 18,8% обусловлена вариацией работающих активов.
Эмпирическое корреляционной отношение [pic], т.е. связь между рассматриваемыми признаками слабая.
Таким образом, исследования показали, что между рассматриваемыми величинами существует слабая зависимость.
Задание 3.
По результатам выполнения задания 1 с вероятностью 0,954 определите:

   1. Ошибку выборки средней величины работающих активов банка и границы, в которых будет находиться средняя величина работающих активов в генеральной совокупности.

   2. Ошибку выборки доли банков с величиной работающих активов 21902 млн. руб. и более и границы, в которых будет находиться генеральная доля.
Для выполнения этого задания все необходимые расчеты возьмем из задания 1 (выборочную среднюю, дисперсию), определим объем генеральной совокупности:

Среднее арифметическое: [pic] млн. руб., т.е. в среднем работающие активы на один банк составляют 14220,65 млн. руб.

Дисперсия: [pic] (млн. руб.)2.

Объем генеральной совокупности [pic] банков.

Среднюю ошибку выборочной средней рассчитаем по формуле для бесповторной выборки:

       [pic] млн. руб.

Для вероятности 0,954 находим из таблицы функции Лапласа, что t=2. Предельную ошибку выборки находим по формуле:

[pic].

Находим и границы, в которых будет находиться средняя величина работающих активов банка в генеральной совокупности:[pic]

14220,65-1843,7920 ≤ [pic] ≤ 14220,65+1843,7920  или   12376,8580≤ [pic] ≤ 16064,4420.
Таким образом, с вероятностью 0,954 можно утверждать, что границы, в которых будет находиться средняя величина работающих активов банка в генеральной совокупности от 12376,8580 до 16064,4420 млн. руб… Ошибка выборки равна 1843,7920 млн. руб..
Выборочная доля банков с величиной работающих активов 21902 млн. руб. и более равна:

[pic] или 14%.

Среднюю ошибку выборочной доли находим по формуле бесповторного отбора:
    продолжение
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