The paper contains an analysis of the possibilities of mathematical statistics as a section of mathematics for solving the applied ecological and economic problem: utilization of small electrochemical power source (batteries). The statistical samples of the investigated phenomena are analyzed. Trend lines have been constructed, resulting in projected needs of the population of Ukraine and the world in the volume of battery disposal. A statistical test of the null hypothesis that the content of nickel in the batteries has a Poisson distribution is carried out. The implementation of this project is proposed at State Enterprise "Argentum" the only Ukrainian battery recycling company located in Lviv.
Significant general problem of deterioration of life in large cities, including pollution by automobile exhaust gases from air, toxic waste from the territories of landfills, underground water of cities and villages of Ukraine, emission of harmful substances from factories and factories, is the accumulation of such dangerous small batteries, accumulators to mobile phones or laptops, and also energy-saving bulbs, to which we are all accustomed and do not even notice the dangers. The problem of economic evaluation of the volume of utilization of small batteries by means of mathematical statistics is devoted to this article.
The main tasks of mathematical statistics are statistical verification of hypotheses, estimation of the distribution of statistical probabilities and its parameters, studying statistical dependence, determining the basic numerical characteristics of random samples. The term "statistics" means practical activities in collecting, processing, analyzing quantitative information that characterizes a certain aspect of public life (trade, production, population, education, etc.) [14]. In general, mathematical statistics is a section of mathematics in which the probabilistic laws of mass phenomena are studied on the basis of experimental data, statistical verification of hypotheses is carried out, and the distribution of statistical probabilities is evaluated, that is, the systematization of data for scientific and practical conclusions is carried out [12].
Mathematical statistics solves two categories of problems [13]: 1) statistical estimation (point, interval) of parameters of the general population; 2) checking the truthfulness of statistical hypotheses about the meaning of the parameters of the general population or the law of distribution of the feature of the general population on the basis of processing the results of the sample.
The range of statistical methods is extremely wide. The most important area of application of statistics is the economy. As an example of an actual task, it may be a prediction of subsequent economic development based on economic data obtained in the presence of random perturbations.
Forecasting is based on keeping the general tendency of the development of phenomena in time, therefore in practice the forecasting process is reduced to the choice based on the data of past periods of analytical dependencies of the investigated parameter on the factors influencing and extrapolation of these dependencies on the future. The forecast of the indicator is obtained by substituting the necessary value of the factor into the obtained regression equation. Thus, the predictive value is a point estimate of the average value of the indicator for these levels of factors [7].
By means of mathematical statistics, the following results were established. Among the 750 pcs. sold for 1 month of batteries with the corresponding content of nickel in the groups of the following quantities of ni were sold during the month (Table 1). i! where λ = M (X) is an unknown parameter.
To test, we push out and check the hypothesis about the law of distribution of the number of sold batteries with nickel content for a month if the significance level is α = 0.01. Let the random variable X be the number of batteries sold with the corresponding nickel content. We construct a frequency range (Fig. 1). From the form of the frequency range and the contents of the random variable X, we assume that X is distributed according to the Poisson law.
Consequently, we propose a null hypothesis H0: (1):
i ( = ) = e , = 0, 1, 2, . . ., Nickel content 4 5
The point estimate for the Poisson distribution parameter λ is a selective mean x B . In this case: λ* = x B = (424 • 0 + 233 • 1 + 68 • 2 + 20 • 3 + 1 • 4 + 1 • 5) / 750= 438 / 750=0.584 = 0.6.
To calculate the theoretical probability pі by Poisson's formula (2), take into account that λ = 0.6: (2) (3) (4) (5) (6) . pi 0.6i
e 0.6 , і = 0, 1, 2, 3.
i!
Since the last two variants of the variable X in the statistical distribution of the sample have frequencies less than 5, and the sum of these frequencies is also less than 5, they are combined with the variant X = 3. We obtain the value p0 - p2 by the formulas (3) - (5):
0.60 1 . p0 P( X 0) . p1 P( X 1) . p2 P( X 2)
0! 0.61
1! 0.62 e0.6
0,549 , e0.6 e0.6 0.6 e0.6
0,329 , e0.6 0.36
0,099 , 3 (ni ni ')2 emp 2
ni ' 2! 2e0.6
Note also that the last probability p3 will be determined as an addition to one: P3 = P (X> 3) = 1 0.549 - 0.329 - 0.099 ≈ 0.023.
To calculate Kemp = χ2emp we use the formula (6):
i0 , where n0 = 424, n1 = 233, n2 = 68, n3 = 22, and the theoretical frequencies n'i are determined from the equality n'i = npi: The calculated calculations allowed to determine that χ 2емп = 2,97.
Based on the sample data, we estimated the parameter λ and S = 1 , and m = 4 (after combining the last three variants of the sign). Therefore, the number of degrees of freedom k = 4 - 1 - 1 = 2. From the table of critical points of distribution χ2 for a = 0.01 i k = 2 we find the critical value of the criterion χ 2kp = 9.2 .
Since χ2емп = 2,97< χ2kp = 9.2, then the hypothesis H0 is formulated that the number of sold batteries with nickel content within a month has a Poisson distribution at the significance level of α = 0.01 , is accepted because it does not contradict the statistical data.
Let's consider the problem of the formation of SEPSs used in Ukraine. Batteries in Ukraine are not manufactured, therefore, they are exported from abroad. To do this, the collection of data on the import of small chemical sources of electricity across the border of Ukraine was carried out. According to the Customs Service of Ukraine in the last years (2008-2020), 304,94 million units were imported into Ukraine in 2020 (Table 2). Since there are no statistical observations in Ukraine regarding the volumes of battery sales, we will assume that all the quantities of imported batteries have been sold and therefore used.
Fig. 1 shows the distribution of volumes of sold batteries in Ukraine for 2008-2020, from which it is evident that there is a tendency for growth. A similar tendency to increase are the volumes for all countries of the world (Fig. 2).
MS Excel has a fairly large range of capabilities for predicting events based on existing data. One of the easiest ways of this prediction is to build a trend line. The choice can be made among the following types of functions of available data: linear, logarithmic, exponential, power or polynomial. By comparing R2 values for different lines, we can choose the type of graph that characterizes the data most accurately, that is, builds the most reliable forecast. Further, indicating the period for which the forecast is made, in our case it is 3 years, we received the results (Figures 4 - 5). We see that forecast values are for Ukraine (Fig. 4) 382 million pcs. and 487 million pcs. in 2019 and 2020, according to the polynomial trend, in which R2 is closest to the unit (R2 = 0.4102). This method of constructing a trend is rather approximate. . s 3c00 p n o i l 2il50 m , e 2i00 n a r k U y = -0,438x2 + 12,768x + 211,15
R² = 0,9501
Years
Exponential trYeenadrs
Recycling of storage batteries and batteries is carried out in order to reduce the amount of toxic substances in solid household waste. Used batteries include mercury, cadmium, lead, tin, nickel, zinc, magnesium and other chemical elements and compounds. On landfills, under the influence of atmospheric factors, the batteries rapidly collapse, the substances that are in their composition are evaporated and washed away. Due to water, air and soil, toxic metals fall into living organisms that cause damage to living organisms, impair reproductive capacity and cause genetic changes and cancer.
The share of batteries in household rubbish is 0.05% of the total weight of rubbish. The share of toxic substances from batteries in the same household rubbish is already 50%. In a year is formed, as scientists have investigated in the work [1]: 40 kg mercury, 160 kg of cadmium, 260 t of manganese compounds, 250 t of sodium chloride. Toxic substances from the landfill penetrate the soil, into water, into the air, and heavy metals from biological organisms are practically not derived and settle in bones and tissues, which leads to poisoning of the organism and genetic changes.
In the scientific literature [3] it is estimated that one spent battery can pollute about 20 m2 of land and 400 liters of surface water. The calculation of the total volume of possible pollution of the environment by the SEPS waste is given in Table 4.
Total amount of waste from batteries
2014 3223,68 2015 4900,0
2016 5163,28 2017 5540,0 109649,12 64473,68 98000,0 116465,86 110800,0
Total amount of metals contained in spent SEPS, kg 2013 1370,61 5482,46 877192,98 164473,68 109649,12 54824,56 274122,81 274122,81 54824,56 2014 805,92 3223,68 515789,47 96710,53 64473,68 32236,84 161184,21 161184,21 32236,84
2015 1225,00 4900,00 784000,00 147000,00 98000,00 49000,00 245000,00 245000,00 49000,00
2016 1290,82 5163,28 826124,80 154898,40 103265,60 51632,80 258164,00 258164,00 51632,80
2017 1385,00 5540,00 886400,00 166200,00 110800,00 55400,00 277000,00 258164,00 55400,00
2020 1524,70 6098,80 975808,00 182964,00 121976,00 60988,00 304940,00 304940,00 60988,00
On the other hand, for the industry, spent batteries are raw materials with a high concentration of valuable elements - non-ferrous metals and minerals. Therefore, it is more advisable to adjust the batteries for recycling than simply throwing them into common landfills. Ukraine has almost no reserves of non-ferrous metal ores, and those that open stocks of colored ores are low-concentrated and polymetallic. Since there are several other metals along with the base metal, such ores need to be enriched several times, and this affects the price of copper. SEPS contain dozens of times more nonferrous metals. The most used round batteries, depending on their size, contain about 16-20% zinc, 813% iron, 17-29% manganese, 23% nickel. Calculate the share in 1 kg of spent batteries of their components and the total volume of toxic substances that can enter the environment through the spent SEPS (Table 4). Consequently, spent SEPS are highly concentrated raw materials for the production of non-ferrous metals. The economic evaluation of the value of valuable substances (metals) that can be obtained during the processing (recycling) of spent SEPS is given in Table 5.
Consequently, spent batteries are a very valuable raw material for the production of non-ferrous metals and minerals for industrial purposes. Only in 2017 the Ukrainian industry could receive from the recycled batteries lithium, whose deposits are not in Ukraine, in the amount of 9 million UAH.
Total cost of substances (metals) contained in spent SEPS, mln. UAH 2013
The obtained results indicate that it is expedient to adjust the SEPS processing, since besides nonferrous metals, we will also receive an advantage in reducing the pollution of groundwater and the total areas of landfill. We recommend that the SEPS be processed in Ukraine at the State Enterprise "Argentum", which has appropriate equipment for this purpose.
The work described the composition of the SEPS, and the future prospects for solving this problem through the paths of alternative change. The results of solving the complex ecological and economic problem of utilization of small SEPS prove that it is expedient to adjust the processing of batteries, since in addition to non-ferrous metals; we also get an advantage in reducing the pollution of groundwater and the total areas of landfills. Also, by means of mathematical statistics, the statistical verification of the null hypothesis that the content of nickel in the batteries has a Poisson distribution is carried out. The results obtained confirm the correctness of the research.
The prospect of further development is the economic justification of the corresponding management decision and its adoption at the state level for the purpose of processing all the batteries in Ukraine at the State Enterprise "Argentum", which is located in the city of Lviv.
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