STATISTICAL METHODS FOR EVALUATING EXPERIMENTAL DATA ON THE USE OF MATHEMATICAL COMPETENCIES IN STUDY FOR A RESILIENT ECONOMY

. The description of the problems, solved within the statistics evaluation of the results of educational (pedagogical) experiment in resilient economy is given. Comparison of the two average samplings and hypothesis verification relative to dispersions of the two samplings are considered. It is shown the hypothesis that teaching and methodical training system of subjects of higher mathematics in technical universities, suggested on the base of the modern interactive technologies, significantly improves the quality of the base level of professional competence of future engineers. The methods for statistical processing of the obtained results, which very thoroughly research all the indexes which compose the suggested learning and methodical system for teaching higher mathematics in technical universities are used.


Introduction
Sustainable development and a resilient economy are impossible without the training of relevant personnel, because the quality of managerial decision-making depends on the professional training of specialists. Pedagogical experiments using elements of mathematical statistics are traditionally conducted in resilient economy. Mathematical statistics has a great variety of criteria, designed to verify the statistic hypothesis. The evaluation of manufacturing, medical, pedagogical and the like managerial decisions usually uses the hypothesis on numerical characteristics of the received figures. The most fully developed theory is the one for hypothesis verification relative to numerical characteristics of the normal law of distribution [1,2,10].
The second and more accurate formula: The value of kj Z , calculated by the close formulas (7) and (8), may also be used as the initial during the solution of the equation (5  Comparison problems lie behind the efficiency evaluation of managerial decisions in many spheres. The most universal are the problems for checking hypothesis on correspondence with the empirical distribution with certain theoretical model or hypothesis on significance of deviation between the empirical laws of distribution. If the parameters under investigation with the random law of distribution are transferred into parameters which are distributed against the normal law of distribution during the solution of the specific problems, then there is no need to test the hypothesis in the form of law of distribution, since the normal law may totally be determined by its mathematical expectation and dispersion. Among various approaches to the evaluation of the efficiency of managerial decisions obtained wide development the evaluation of the importance of change in mathematical expectations of the parameters under research, which reflect the trajectories of the centers of their grouping. However, even nowadays the potential of the given approach is not fully implemented.

Literature review
The definition of resilience has received considerable attention in the economics literature, for instance in publication: [3] a basis for policy discussions of resilience at the local level is presented. Resilient economy is not a very widely used concept. More often, scientists use the concept of resilience. For instance, Mustonen T., Shadrin V., Mustonen K., Vasiliev V. attempted a synthesis framework for analyzing resilience from an economics viewpoint [6,9,11]. Generally, resilience can be defined as the ability of an economy, society, organization or individual to recover effectively from an unexpected shock. This rough definition of resilience raises two related questions: what it means to "recover" and what we mean by an effective recovery [4]. Perrings C. gives a definition of resilience as "the ability of the system to withstand either market or environmental shocks without losing the capacity to allocate resources efficiently or to deliver essential services" leaves the term of analysis indeterminate [8]. Walker B., Holling C., Carpenter S., Kinzig A. have defined resilience as "the capacity of a system to absorb disturbance and reorganize while undergoing change, so as to retain essentially the same function, structure, identity and feedbacks". Hence, this definition encompasses the stability of a structure, rather than focus purely on the performance outcomes of that structure [7,10].
It is in the conditions of a resilient economy that personnel who improve its results are especially valuable. The effectiveness of the system of educational and methodical training of personnel involved in the country's economy in institutions of higher education can be evaluated using various mathematical models and pedagogical experiments. The experience of pedagogical science research draws attention to the monotony of the application of mathematical statistics methods for analyzing the results of a pedagogical experiment. Also, quite often there are cases of only a percentage assessment of the data obtained in a pedagogical experiment without accompanying statistical analysis, which would scientifically confirm or refute the significance of the obtained positive changes. Works justifying the use of statistical criteria for processing and quality assessment of data for various types of pedagogical experiments are particularly relevant. Mathematical statistics includes a significant number of criteria designed to test various statistical hypotheses. As a rule, hypotheses concern either the law of distribution of data in the sample (normal, binomial, etc.) or the numerical characteristics of the sample (mean, variance, correlation, etc.) manual, which describes the use of criteria for the normal distribution law [5,6]. IAPGOŚ 2/2023 p-ISSN 2083-0157, e-ISSN 2391-6761 As Gogol K., Brunner M., Martin R., Preckel F., Goetz T. [2] and Arens K. [1] truly state, that careful consideration is needed when selecting a specific model. Other core constructs in education research (e.g., academic anxiety, academic interest) have theoretical underpinnings that are similar to ASC as they can also be conceptualized as multidimensional and hierarchical in nature.

Material and methods
The aim of the research is the solution of problems of the statistics evaluation of the results of pedagogical experiment.
Objectives of the research: 1. Formulate and briefly describe the problems, solved by means of mathematical statistics for the assessing the efficiency of managerial decisions. 2. According to the suggested method and the presented algorithm check the efficiency of the teaching and methodical training system of subjects of higher mathematics in technical universities.

Comparison of the two average samplings
One of the main tasks during the evaluation of the efficiency of managerial decisions the evaluation of the importance of change in mathematical expectations of the parameters under research, which reflect the trajectories of the centers of their grouping. In the simplest case there shall be done the comparison of the averages in the two samplings, which correspond to the two types of managerial decisions. The task for comparing is formed as follows. There shall be researched the random quantities Upon the results of the researches we receive the independent samplings   So the task of comparing the averages of the two samplings may be formed as follows.

Hypothesis verification relative to dispersions of the two samplings
Dispersion characterizes the stability of any process, including the learning one. Any managerial decisions are directed to the decreases in dispersion. Hence, the test of the hypothesis about the dispersion change significance is an important task which is formulated as follows: there are two normally   and is used for the determination of reliable boundaries of rootmean-square deviation.
According to the suggested method [10] and the presented algorithm we set up a hypothesis and test it 0 H : teaching and methodical training system of subjects of higher mathematics in technical universities, suggested on the base of the modern interactive technologies, significantly improves the quality of the base level of professional competence of future engineers [10].

Formation of motivational competence
For the statistical analysis of the indexes of the level of awareness of motivational competence we have composed a table of significance of their disagreements the students experience when they study parts of higher mathematics.
The results obtained allow to make a decision that the index of attitude to the chosen speciality remains stable. That is, the deviation percentage of the index of the respond to the question: "If you had to enter the university again, would you choose: a) same university, same speciality; b) same university, but the other speciality; c) other university?" in table 1  experiments, in which training followed the suggested methodical system (an adjusting experiment disclosed the drawbacks, which were corrected prior to the formative experiment) are significant This allows to state, that the suggested teaching and methodical training system of subjects of higher mathematics in technical universities significantly increases the level of students' motivational competence during the first training year.
The improvement of teaching methods caused positive changes in indexes of conceptions as for the application of the knowledge obtained in writing course and bachelor papers. The difference in statistical averages of the formative and adjusting experiments is significant (column 1), but in dispersions (column 4)it is not. It means that the index remains stable and its deviation from the statistical variable is insignificant.
Conceptions as for the chosen speciality

Improvement of the formation level of components of cognitive and creative competencelevel of the knowledge and skills obtained in higher mathematics
Results of comparison of the theoretical knowledge and skills in solution of usual and applied tasks, making decisions on all the topics following the results of the test papers and exams are presented in table 2.
Analysis of the results of statistical research of significances of differences in average and dispersions between the groups show that the suggested system for learning mathematical subjects in technical universities greatly increases the level of application of the obtained knowledge and skills for the solution of tasks of any content. This is shown by the data in columns 2 and 5.
Index "1" points to the significant difference in students' knowledge of all the topics towards the favor of the suggested teaching and methodical system. Really, the statistical average from all the topics is higher in the experimental group of the formative experiment. After the statistical processing of the results of the adjustable stage of an experiment we see that despite the higher test results and better examination answers in the experimental group "E" in comparison with the screening group "K", the difference of indexes was not that significantly stable (columns 3 and 6).
This means that the suggested part of the teaching system does not influence the results we would like to obtain. The methods of lecture-wise and practical classes in vector algebra, analytical geometry, multiple integral, theory of series, function of complex variety and operational calculus had been improved.
The results of statistical analysis after the formational experiment proved the positive shift in the results of knowledge obtained after the correction of the teaching methods, which is proved by data in columns 1 and 4 as well as 2 and 6. This may be explained by the fact that the study of these sections following the curricula requires more hours. To change the situation, we issued study books aimed at organization of independent learning some sections of higher mathematics.

Formed components of self-education and communicative competence learning higher mathematics
The results of analysis of statistical indexes of self-instruction skills levels, strivings to an advanced learning of the subject, selfeducation, gaining skills of public speaking, research work are presented in table 3.
Results of comparing the indexes of acquiring skills for selfdependent work and strivings to self-education prove our hypothesis on the reliability level of 0.95, this means that the suggested learning and methodical system significantly increases the level of self-dependent work (rows 1-6), striving to self-education in comparison with traditional training methods. This is shown in columns 2 and 5, as well as 3 and 6. Throughout the indexes the difference of the statistical average and dispersions of the coefficient of availability of the knowledge of selfdependent work, strivings to an advanced learning of material, to self-education is significant.
The data of the columns 1 and 4 show that the correction of the methods for giving lectures and the additional developments of interactive methods for practical classes aimed at improvement of the research indexes after the adjusting experiment was a success. The Indicators of differences of the statistical averages and dispersions of parameters, which determine the skills for pedagogical and research activities show that the suggested learning methods for higher mathematics significantly increases the level of skills in the experimental groups Ф and Е in comparison with the screening group К (columns 2 and 5, 3 and 6). Improvement of methods after the adjusting experiment made great positive changes in indicators of public speech skills, research activities and partiallyabilities in self-education work with textbooks and independent solution of applied tasks in experimental groups of formative and adjusting experiments (columns 1 and 4).
For the statistical analysis of the indexes of level of formation of communicative competence we have considered the questionnaire data as well as quantitive evaluation of levels of awareness of this competence. The results of comparing indexes of levels of awareness of communicative competence are presented in item 11 of table 3 and item 3 of table 4.
Analyses of the results of significance in differences of the indexes of numerical characteristics of awareness show that there is a significant difference between the suggested teaching and methodical system and a traditional one. Differences between the values in the screening group of the formative (Ф) and adjusting (Е) experiments show that the decision to test it by the adjusting experiment with further formative one was correct.
So, the results of statistical analysis of comparison of the averages and dispersions proved our hypothesis 0 H : the suggested teaching and methodical system increases the formation of the basic level of professional competence of future specialists with technical education.
To process the obtained results in pedagogical experiment we used some methods for analysis. The quantitative assessment of the level of knowledge was the known index of progress. For obviousness and convenience of numerical indexes there had been built the bar graphs and diagrams, but the percentage analysis may not always answer the questions of efficiency of the suggested methods, therefore we used the methods for statistical processing of the obtained results, which very thoroughly researches all the indexes which compose the suggested learning and methodical system for teaching higher mathematics in technical Universities. IAPGOŚ 2/2023 p-ISSN 2083-0157, e-ISSN 2391-6761 Table 3. Significance of differences in indexes of numerical characteristics of development of components of self-educational and communicative competence ("1" -significant deviation, "0"insignificant deviation)

Conclusions
In the conditions of a resilient economy that personnel who improve its results are especially valuable. The effectiveness of the system of educational and methodical training of personnel involved in the country's economy in institutions of higher education can be evaluated using various mathematical models and pedagogical experiments. The experience shows that students of technical universities find it difficult to learn mathematical subjects. Academic course of parts and some subjects of higher mathematics is not always useful for future engineers. The development and the introduction of the new interactive training methods which rest on tasks with technical content considering the speciality, in particular RPG's and business games, project method, simulation, «brain storming colloquium», «scientific and technical seminar», «scientific and technical conference», «optimal project» etc. is a complex process for the fundamental subjects in technical universities, which requires interdisciplinary approach and teachers' wish to give students knowledge and form the basic level of professional competence. However, future graduates of environmental specialties must possess mathematical apparatus, in particular comparative analysis of the obtained statistical data, which is the basis for evaluating the effectiveness of management decisions. In our opinion, the gaps in the knowledge of the use of mathematical apparatus are due to the uniformity of the application of teaching methods to sections of higher mathematics and the filling of mathematical textbooks with general examples. As our long-term experience shows, the application of modern innovative teaching methods and the use of textbooks for sections of higher mathematics with tasks of applied content significantly increase not only knowledge from them, but also the ability of future ecologists to make the right management decisions.
The advantage of the suggested algorithm for the statistical analysis of the results of pedagogical experiment lies in the fact that after receiving the differences between the parameters under p-ISSN 2083-0157, e-ISSN 2391-6761 IAPGOŚ 2/2023 85 research it is possible to make a decision on significance of difference of numerical characteristics of formation of the indexes received. The difference in percentage does not always respond to the efficient results. Correlation analysis, Pierson's criteria and other methods for the evaluation of the results of pedagogical experiment do not present such a strict, significant difference as the correlation of the statistical average and dispersion of the two samplings.