Plan:
1. Research of methods of mathematical statistics in pedagogical research.
1. Research of methods of mathematical statistics in pedagogical research.
Recently, serious steps have been taken to introduce into pedagogy mathematical methods for assessing and measuring pedagogical phenomena and establishing quantitative relationships between them. Mathematical methods allow us to approach the solution of one of the most difficult tasks of pedagogy - the quantitative assessment of pedagogical phenomena. Only the processing of quantitative data and the resulting conclusions can objectively prove or disprove the hypothesis put forward.
In the pedagogical literature, a number of methods for statistical processing of data from a pedagogical experiment are proposed (L. B. Itelson, Yu. V. Pavlov, and others). When using the methods of mathematical statistics, it should be borne in mind that statistics itself does not reveal the essence of the phenomenon and cannot explain the reasons for the differences that arise between the individual aspects of the phenomenon. For example, an analysis of the results of the study shows that the teaching method used gave better results compared to previously recorded ones. However, these calculations cannot answer the question why the new method is better than the old one.
The most common of the mathematical methods used in pedagogy are:
1. Registration - a method of identifying the presence of a certain quality in each member of the group and a total count of the number of those who have or do not have this quality (for example, the number of children who attended classes without a pass and made passes, etc.).
2. Ranking (or ranking method) involves the arrangement of the collected data in a certain sequence, usually in ascending or descending order of any indicators and, accordingly, determining the place in this row for each of the subjects (for example, compiling a list of children depending on the number of missed classes, etc.).
3. Scaling as a quantitative research method makes it possible to introduce numerical indicators in the assessment of certain aspects of pedagogical phenomena. For this purpose, the subjects are asked questions, answering which they must indicate the degree or form of assessment chosen from among these assessments, numbered in a certain order (for example, a question about playing sports with a choice of answers: a) I am fond of, b) I do it regularly, c) do not exercise regularly, d) do not do any kind of sport).
Correlating the results with the norm (with given indicators) involves determining deviations from the norm and correlating these deviations with acceptable intervals (for example, with programmed learning, 85-90% of correct answers are often considered the norm; if there are fewer correct answers, this means that the program is too difficult if more, then it is too light).
The penetration of mathematical methods into the most diverse spheres of human activity actualizes the problem of modeling, with the help of which the correspondence of a real object to a mathematical model is established. Any model is a homomorphic image of some system in another system (homomorphism is a one-to-one correspondence between systems that preserves basic relations and basic operations). Mathematical models in relation to the simulated objects are analogues at the level of structures.
The specificity of the statistical processing of the results of psychological and pedagogical research lies in the fact that the analyzed database is characterized by a large number of indicators of various types, their high variability under the influence of uncontrolled random factors, the complexity of the correlations between the sample variables, the need to take into account objective and subjective factors that affect the diagnostic results. , especially when deciding on the representativeness of the sample and evaluating hypotheses regarding the general population. Research data can be divided into groups according to their type:
The first group is nominal variables (gender, personal data, etc.). Arithmetic operations on such quantities are meaningless, so the results of descriptive statistics (mean, variance) are not applicable to such quantities. The classic way to analyze them is to divide them into contingency classes with respect to certain nominal features and check for significant differences across classes.
The second group of data has a quantitative measurement scale, but this scale is ordinal (ordinal). In the analysis of ordinal variables, both subsampling and rank technologies are used. Parametric methods are also applicable with some limitations.
The third group - quantitative variables that reflect the severity of the measured indicator - these are Cattell's tests, academic performance and other assessment tests. When working with variables in this group, all standard types of analysis are applicable, and with a sufficient sample size, their distribution is usually close to normal. Thus, the variety of types of variables requires the use of a wide range of mathematical methods used.
The analysis procedure can be divided into the following steps:
Preparing the database for analysis. This stage includes converting the data into an electronic format, checking them for outliers, choosing a method for working with missing values.
Descriptive statistics (calculation of averages, variances, etc.). The results of descriptive statistics determine the characteristics of the parameters of the analyzed sample or subsamples specified by one partition or another.
Exploratory analysis. The task of this stage is a meaningful study of various groups of sample indicators, their relationships, identification of the main explicit and hidden (latent) factors affecting the data, tracking changes in indicators, their relationships and the significance of factors when dividing the database into groups, etc. The research tool are various methods and technologies of correlation, factor and cluster analysis. The purpose of the analysis is to formulate hypotheses concerning both the given sample and the general population.
Detailed analysis of the obtained results and statistical verification of the proposed hypotheses. At this stage, hypotheses are tested regarding the types of the distribution function of random variables, the significance of differences in means and variances in subsamples, etc. When summarizing the results of the study, the question of the representativeness of the sample is solved.
It should be noted that this sequence of actions, strictly speaking, is not chronological, with the exception of the first stage. As the results of descriptive statistics are obtained and certain patterns are identified, it becomes necessary to test emerging hypotheses and immediately proceed to their detailed analysis. But in any case, when testing hypotheses, it is recommended to analyze them by various mathematical means that adequately correspond to the model, and a hypothesis should be accepted at a particular level of significance only when it is confirmed by several different methods.
When organizing any measurement, a correlation (comparison) of the measured with the measuring instrument (standard) is always assumed. After the correlation (comparison) procedure, the measurement result is evaluated. If in technology, as a rule, material standards are used as meters, then in social measurements, including pedagogical and psychological measurements, meters can be ideal. Indeed, in order to determine whether or not a particular mental action has been formed in a child, it is necessary to compare the actual with the necessary. In this case, the necessary is the ideal model that exists in the head of the teacher.
It should be noted that only some pedagogical phenomena can be measured. The majority of pedagogical phenomena cannot be measured, since there are no standards of pedagogical phenomena, without which measurement cannot be performed.
As for such phenomena as activity, cheerfulness, passivity, fatigue, skills, habits, etc., it is not yet possible to measure them, since there are no standards of activity, passivity, vivacity, etc. Due to the extreme complexity and, for the most part, the practical impossibility of measuring pedagogical phenomena, special methods are currently used for an approximate quantitative assessment of these phenomena.
At present, it is customary to divide all psychological and pedagogical phenomena into two large categories: objective material phenomena (phenomena that exist outside and independently of our consciousness) and subjective non-material phenomena (phenomena characteristic of a given person).
Objective material phenomena include: chemical and biological processes, movements performed by a person, sounds made by him, actions performed by him, etc.
Subjective non-material phenomena and processes include: sensations, perceptions and ideas, fantasies and thinking, feelings, desires and desires, motivation, knowledge, skills, etc.
All signs of objective material phenomena and processes are observable and can, in principle, always be measured, although modern science is sometimes unable to do this. Any property or trait can be measured directly. This means that by means of physical operations it can always be compared with some real value taken as the standard of measure of the corresponding property or attribute.
Subjective non-material phenomena cannot be measured, since there are no and cannot be material standards for them. Therefore, approximate methods for evaluating phenomena are used here - various indirect indicators.
The essence of the use of indirect indicators is that the measured property or sign of the phenomenon under study is associated with certain material properties, and the value of these material properties is taken as an indicator of the corresponding non-material phenomena. For example, the effectiveness of a new teaching method is assessed by the progress of students, the quality of a student's work - by the number of mistakes made, the difficulty of the material being studied - by the amount of time spent, the development of mental or moral traits - by the number of relevant actions or misconduct, etc.
With all the great interest that researchers usually show in the methods of quantitative analysis of experimental data and mass material obtained using different methods, the processing stage is essential - their qualitative analysis. With the help of quantitative methods, it is possible, with varying degrees of reliability, to identify the advantage of a particular method or to detect a general trend, to prove that a scientific assumption under test has been justified, etc. However, a qualitative analysis should give an answer to the question why this happened, what favored it, and what served as an obstacle, and how significant the influence of these interferences was, whether the experimental conditions were too specific for this technique to be recommended for use in other conditions, etc. At this stage, it is also important to analyze the reasons that prompted individual respondents to give a negative answer, and to identify the causes of certain typical and even random errors in the work of individual children, etc. The use of all these methods of analyzing the collected data helps to more accurately evaluate the results of the experiment, increases the reliability of the conclusions about them and provides more grounds for further theoretical generalizations.
Statistical methods in pedagogy are used only to quantify phenomena. In order to draw conclusions and conclusions, a qualitative analysis is necessary. Thus, in pedagogical research, the methods of mathematical statistics should be used carefully, taking into account the peculiarities of pedagogical phenomena.
So, most of the numerical characteristics in mathematical statistics are used when the property or phenomenon under study has a normal distribution, which is characterized by a symmetrical arrangement of the values ​​of the population elements relative to the average value. Unfortunately, in view of the insufficient study of pedagogical phenomena, the laws of distribution in relation to them, as a rule, are unknown. Further, to evaluate the results of the study, rank values ​​are often taken, which are not the results of quantitative measurements. Therefore, it is impossible to perform arithmetic operations with them, and therefore calculate numerical characteristics for them.
Each statistical series and its graphical representation is a grouped and visually presented material that should be subjected to statistical processing.
Statistical processing methods make it possible to obtain a number of numerical characteristics that make it possible to predict the development of the process of interest to us. These characteristics, in particular, make it possible to compare different series of numbers obtained in pedagogical research and draw appropriate pedagogical conclusions and recommendations.
All variation series can differ from each other in the following ways:
1. In a big way, i.e. its upper and lower limits, which are usually called limits.
2. The value of the attribute around which the majority of the variant is concentrated. This feature value reflects the central trend of the series, i.e. typical for the series.
3. Variations around the central trend of the series.
In accordance with this, all statistical indicators of the variation series are divided into two groups:
-indicators that characterize the central trend or level of the series;
-indicators characterizing the level of variation around the central trend.
The first group includes various characteristics of the mean: median, arithmetic mean, geometric mean, etc. To the second - variation range (limits), mean absolute deviation, standard deviation, variance, coefficients of asymmetry and variation. There are other indicators, but we will not consider them, because. they are not used in educational statistics.
Currently, the concept of "model" is used in various senses, the simplest of them is the designation of a sample, a standard. In this case, the model of a thing does not carry any new information and does not serve the purposes of scientific knowledge. In this sense, the term "model" in science is not used. In a broad sense, a model is understood as a mentally or practically created structure that reproduces a part of reality in a simplified and visual form. In a narrower sense, the term "model" is used to depict a certain area of ​​phenomena with the help of another, more studied, easily understood. In pedagogical sciences, this concept is used in a broad sense as a specific image of the object being studied, in which real or supposed properties, structure, etc. are displayed. Modeling is widely used in academic subjects as an analogy that can exist between systems at the following levels: the results that the compared systems give; functions that determine these results; structures that ensure the performance of these functions; elements that make up structures.
V. M. Tarabaev points out that the technique of the so-called multifactorial experiment is currently being used. In a multivariate experiment, researchers approach the problem empirically - they vary with a large number of factors on which, as they believe, the course of the process depends. This variation by various factors is carried out using modern methods of mathematical statistics.
A multivariate experiment is built on the basis of statistical analysis and using a systematic approach to the subject of research. It is assumed that the system has input and output that can be controlled, it is also assumed that this system can be controlled in order to achieve a certain result at the output. In a multifactorial experiment, the whole system is studied without an internal picture of its complex mechanism. This type of experiment opens up great opportunities for pedagogy.
Literature:
1. Zagvyazinsky, V. I. Methodology and methods of psychological and pedagogical research: textbook. allowance for students. higher ped. textbook institutions / Zagvyazinsky V.I., Atakhanov R. - M .: Academy, 2005.
2. Gadelshina, T. G. Methodology and methods of psychological research: textbook. method. allowance / Gadelshina T. G. - Tomsk, 2002.
3. Kornilova, T. V. Experimental psychology: theory and methods: a textbook for universities / Kornilova T. V. - M .: Aspect Press, 2003.
4. Kuzin, F. A. PhD thesis: writing methodology, design rules and defense procedure / Kuzin F. A. - M., 2000.

In the history of mathematics, two main periods can be conventionally distinguished: elementary and modern mathematics. The milestone, from which it is customary to count the era of new (sometimes they say - higher) mathematics, was the 17th century - the century of the emergence of mathematical analysis. By the end of the XVII century. I. Newton, G. Leibniz and their predecessors created the apparatus of a new differential calculus and integral calculus, which forms the basis of mathematical analysis and even, perhaps, the mathematical basis of all modern natural science.

Mathematical analysis is a vast area of ​​mathematics with a characteristic object of study (a variable), a peculiar research method (analysis by means of infinitesimals or by passing to the limit), a certain system of basic concepts (function, limit, derivative, differential, integral, series) and constantly improving and developing apparatus, which is based on differential and integral calculus.

Let's try to give an idea of ​​what kind of mathematical revolution took place in the 17th century, what characterizes the transition from elementary mathematics associated with the birth of mathematical analysis to the one that is now the subject of research in mathematical analysis, and what explains its fundamental role in the entire modern system of theoretical and applied knowledge. .

Imagine that in front of you is a beautifully executed color photograph of a stormy ocean wave running ashore: a powerful stooped back, a steep but slightly sunken chest, already tilted forward and ready to fall head with a gray mane torn by the wind. You have stopped the moment, you have managed to catch the wave, and now you can carefully study it in all its details without haste. A wave can be measured, and using the means of elementary mathematics, you will draw many important conclusions about this wave, and therefore all its oceanic sisters. But by stopping the wave, you have deprived it of movement and life. Its origin, development, run, the force with which it falls on the shore - all this turned out to be out of your field of vision, because you do not yet have either a language or a mathematical apparatus suitable for describing and studying not static, but developing, dynamic processes, variables and their interrelations.

"Mathematical analysis is no less comprehensive than nature itself: it determines all tangible relationships, measures times, spaces, forces, temperatures." J. Fourier

Movement, variables and their relationships are all around us. Various types of motion and their regularities constitute the main object of study of specific sciences: physics, geology, biology, sociology, etc. Therefore, the exact language and appropriate mathematical methods for describing and studying variables turned out to be necessary in all areas of knowledge approximately to the same extent as numbers and arithmetic are necessary in describing quantitative relationships. So, mathematical analysis is the basis of the language and mathematical methods for describing variables and their relationships. Today, without mathematical analysis, it is impossible not only to calculate space trajectories, the operation of nuclear reactors, the running of an ocean wave and the patterns of cyclone development, but also to economically manage production, resource distribution, organization of technological processes, predict the course of chemical reactions or changes in the number of various species interconnected in nature. animals and plants, because all these are dynamic processes.

Elementary mathematics was basically the mathematics of constants, it studied mainly the relations between the elements of geometric figures, the arithmetic properties of numbers, and algebraic equations. To some extent, her attitude to reality can be compared with an attentive, even thorough and complete study of each fixed frame of a film that captures the changing, developing living world in its movement, which, however, is not visible on a separate frame and which can be observed only by looking tape as a whole. But just as cinema is unthinkable without photography, so modern mathematics is impossible without that part of it, which we conditionally call elementary, without the ideas and achievements of many outstanding scientists, sometimes separated by tens of centuries.

Mathematics is one, and its “higher” part is connected with the “elementary” one in much the same way as the next floor of a house under construction is connected with the previous one, and the width of the horizons that mathematics opens up to us in the world around us depends on which floor of this building we managed to reach. rise. Born in the 17th century mathematical analysis opened up possibilities for scientific description, quantitative and qualitative study of variables and motion in the broadest sense of the word.

What are the prerequisites for the emergence of mathematical analysis?

By the end of the XVII century. the following situation has arisen. First, within the framework of mathematics itself, over the years, certain important classes of problems of the same type have accumulated (for example, problems of measuring the areas and volumes of non-standard figures, problems of drawing tangents to curves) and methods have appeared for solving them in various special cases. Secondly, it turned out that these problems are closely related to the problems of describing an arbitrary (not necessarily uniform) mechanical motion, and in particular with the calculation of its instantaneous characteristics (velocity, acceleration at any time), as well as with finding the distance traveled for movement at a given variable speed. The solution of these problems was necessary for the development of physics, astronomy, and technology.

Finally, thirdly, by the middle of the XVII century. the works of R. Descartes and P. Fermat laid the foundations of the analytical method of coordinates (the so-called analytical geometry), which made it possible to formulate geometric and physical problems of heterogeneous origin in the general (analytical) language of numbers and numerical dependences, or, as we now say, numerical functions.

NIKOLAI NIKOLAEVICH LUZIN (1883-1950) N. N. Luzin - Soviet mathematician, founder of the Soviet school of function theory, academician (1929). Luzin was born in Tomsk, studied at the Tomsk gymnasium. The formalism of the gymnasium mathematics course alienated the talented young man, and only a capable tutor could reveal to him the beauty and grandeur of mathematical science. In 1901, Luzin entered the mathematical department of the Faculty of Physics and Mathematics of Moscow University. From the first years of study, questions related to infinity fell into the circle of his interests. At the end of the XIX century. the German scientist G. Kantor created the general theory of infinite sets, which has received numerous applications in the study of discontinuous functions. Luzin began to study this theory, but his studies were interrupted in 1905. The student, who took part in revolutionary activities, had to leave for France for a while. There he listened to lectures by the most prominent French mathematicians of that time. Upon his return to Russia, Luzin graduated from the university and was left to prepare for a professorship. Soon he again went to Paris, and then to Göttingen, where he became close to many scientists and wrote his first scientific papers. The main problem that interested the scientist was the question of whether there can be sets containing more elements than the set of natural numbers, but less than the set of points of the segment (the continuum problem). For any infinite set that could be obtained from segments using the operations of union and intersection of countable collections of sets, this hypothesis was true, and in order to solve the problem, it was necessary to find out what other ways of constructing sets were. At the same time, Luzin studied the question of whether it is possible to represent any periodic function, even if it has infinitely many discontinuity points, as the sum of a trigonometric series, i.e. sums of an infinite set of harmonic oscillations. Luzin obtained a number of significant results on these issues and in 1915 he defended his dissertation "The Integral and the Trigonometric Series", for which he was immediately awarded the degree of Doctor of Pure Mathematics, bypassing the intermediate master's degree that existed at that time. In 1917 Luzin became a professor at Moscow University. A talented teacher, he attracted the most capable students and young mathematicians. Luzin's school reached its heyday in the first post-revolutionary years. Luzin's students formed a creative team, which was jokingly called "Luzitania". Many of them received first-class scientific results during their student days. For example, P. S. Aleksandrov and M. Ya. Suslin (1894-1919) discovered a new method for constructing sets, which initiated the development of a new direction - descriptive set theory. Research in this area, conducted by Luzin and his students, showed that the usual methods of set theory are not enough to solve many of the problems that arose in it. Luzin's scientific predictions were fully confirmed in the 1960s. 20th century Many students of N. N. Luzin later became academicians and corresponding members of the Academy of Sciences of the USSR. Among them P. S. Aleksandrov. A. N. Kolmogorov. M. A. Lavrentiev, L. A. Lyusternik, D. E. Menshov, P. S. Novikov. L. G. Shnirelman and others. Modern Soviet and foreign mathematicians in their works develop the ideas of N. N. Luzin.

The combination of these circumstances led to the fact that at the end of the XVII century. two scientists - I. Newton and G. Leibniz - independently managed to create a mathematical apparatus for solving these problems, summing up and generalizing individual results of their predecessors, including the ancient scientist Archimedes and contemporaries of Newton and Leibniz - B. Cavalieri, B. Pascal , D. Gregory, I. Barrow. This apparatus formed the basis of mathematical analysis - a new branch of mathematics that studies various developing processes, i.e. interrelations of variables, which in mathematics are called functional dependencies or, in other words, functions. By the way, the term “function” itself was required and naturally arose precisely in the 17th century, and by now it has acquired not only a general mathematical, but also a general scientific meaning.

Initial information about the basic concepts and the mathematical apparatus of analysis is given in the articles "Differential Calculus" and "Integral Calculus".

In conclusion, I would like to dwell on only one principle of mathematical abstraction that is common to all mathematics and characteristic of analysis, and in this connection to explain in what form mathematical analysis studies variables and what is the secret of such universality of its methods for studying all kinds of specific developing processes and their interrelations. .

Let's look at some explanatory examples and analogies.

We sometimes no longer realize that, for example, a mathematical ratio, written not for apples, chairs or elephants, but in an abstract form abstracted from specific objects, is an outstanding scientific achievement. This is a mathematical law that experience has shown to be applicable to various concrete objects. So, studying in mathematics the general properties of abstract, abstract numbers, we thereby study the quantitative relationships of the real world.

For example, it is known from a school mathematics course that, therefore, in a specific situation, you could say: “If two six-ton ​​dump trucks are not allocated to me for transporting 12 tons of soil, then you can request three four-ton dump trucks and the work will be done, and if they give only one four-ton dump truck, then she will have to make three flights. Thus, the abstract numbers and numerical regularities that are now familiar to us are connected with their concrete manifestations and applications.

Approximately in the same way, the laws of change of concrete variable quantities and developing processes of nature are connected with the abstract, abstract form-function in which they appear and are studied in mathematical analysis.

For example, an abstract ratio may be a reflection of the dependence of the box office at the cinema on the number of tickets sold, if 20 is 20 kopecks - the price of one ticket. But if we are cycling on a highway at 20 km per hour, then the same ratio can be interpreted as the relationship of the time (hours) of our bike ride and the distance covered during this time (kilometers), you can always argue that, for example, a change by several times leads to a proportional (i.e., by the same number of times) change in the value of , and if , then the opposite conclusion is also true. So, in particular, to double the box office revenue of a cinema, you have to attract twice as many viewers, and to ride a bike at the same speed twice as far, you have to ride twice as long.

Mathematics studies both the simplest dependence and other, much more complex dependences in an abstract, general, abstract form abstracted from private interpretation. The properties of a function identified in such a study or methods for studying these properties will be in the nature of general mathematical techniques, conclusions, laws and conclusions applicable to each specific phenomenon in which the function studied in an abstract form occurs, regardless of which field of knowledge this phenomenon belongs to. .

So, mathematical analysis as a branch of mathematics took shape at the end of the 17th century. The subject of study in mathematical analysis (as it appears from modern positions) are functions, or, in other words, dependencies between variables.

With the advent of mathematical analysis, it became possible for mathematics to study and reflect the developing processes of the real world; variables and motion entered mathematics.

Mathematical Methods of Operations Research

regression analysis model programmatic

Introduction

Description of the subject area and statement of the research problem

Practical part

Conclusion

Bibliography

Introduction

In economics, the basis of almost any activity is forecasting. Already on the basis of the forecast, a plan of action and measures is drawn up. Thus, we can say that the forecast of macroeconomic variables is a fundamental component of the plans of all economic entities. Forecasting can be carried out both on the basis of qualitative (expert) and quantitative methods. The latter by themselves can do nothing without a qualitative analysis, just as expert assessments must be supported by sound calculations.

Now forecasts, even at the macroeconomic level, are of a scenario nature and are developed according to the following principle: what if… , - and are often a preliminary stage and justification for major national economic programs. Macroeconomic forecasts are usually made with a lead time of one year. The modern practice of the functioning of the economy requires short-term forecasts (half a year, a month, a decade, a week). Designed for the tasks of providing advanced information to individual participants in the economy.

With changes in the objects and tasks of forecasting, the list of forecasting methods has changed. Adaptive methods of short-term forecasting have received rapid development.

Modern economic forecasting requires developers to have versatile specialization, knowledge from various fields of science and practice. The tasks of a forecaster include knowledge of the scientific (usually mathematical) apparatus of forecasting, the theoretical foundations of the forecasting process, information flows, software, interpretation of forecasting results.

The main function of the forecast is to substantiate the possible state of the object in the future or to determine alternative paths.

The importance of gasoline as the main type of fuel today is difficult to overestimate. And it is just as difficult to overestimate the impact of its price on the economy of any country. The nature of the development of the country's economy as a whole depends on the dynamics of fuel prices. An increase in gasoline prices causes an increase in prices for industrial goods, leads to an increase in inflationary costs in the economy and a decrease in the profitability of energy-intensive industries. The cost of petroleum products is one of the components of the prices of goods in the consumer market, and transportation costs affect the price structure of all consumer goods and services without exception.

Of particular importance is the issue of the cost of gasoline in the developing Ukrainian economy, where any change in prices causes an immediate reaction in all its sectors. However, the influence of this factor is not limited to the sphere of the economy; many political and social processes can also be attributed to the consequences of its fluctuations.

Thus, the study and forecasting of the dynamics of this indicator is of particular importance.

The purpose of this work is to forecast fuel prices for the near future.

1. Description of the subject area and statement of the research problem

The Ukrainian gasoline market can hardly be called constant or predictable. And there are many reasons for this, starting with the fact that the raw material for the production of fuel is oil, the prices and volume of production of which are determined not only by supply and demand in the domestic and foreign markets, but also by state policy, as well as special agreements between manufacturing companies. In conditions of strong dependence of the Ukrainian economy, it is dependent on the export of steel and chemicals, and the prices for these products are constantly changing. And speaking of gasoline prices, one cannot fail to note their upward trend. Despite the restraining policy pursued by the state, their growth is habitual for the majority of consumers. Prices for petroleum products in Ukraine today change daily. They mainly depend on the cost of oil on the world market ($ / barrel) and the level of the tax burden.

The study of gasoline prices is very relevant at the present time, since the prices of other goods and services depend on these prices.

In this paper, we will consider the dependence of gasoline prices on time and such factors as:

ü oil prices, US dollar per barrel

ü official exchange rate of the dollar (NBU), hryvnia per US dollar

ü consumer price index

The price of gasoline, which is a product of oil refining, is directly related to the price of the specified natural resource and the volume of its production. The dollar exchange rate has a significant impact on the entire Ukrainian economy, in particular on the formation of prices in its domestic markets. The direct connection of this parameter with gasoline prices directly depends on the US dollar exchange rate. The CPI reflects the general change in prices within the country, and since it is economically proven that a change in the prices of some goods in the vast majority of cases (in conditions of free competition) leads to an increase in the prices of other goods, it is reasonable to assume that a change in the prices of goods across the country affects the studied indicator at work.

Description of the mathematical apparatus used in the calculations

Regression analysis

Regression analysis is a method of modeling measured data and studying their properties. The data consists of pairs of values ​​of the dependent variable (the response variable) and the independent variable (the explanatory variable). Regression model<#"19" src="doc_zip1.jpg" />. Regression analysis is the search for a function that describes this relationship. Regression can be represented as a sum of non-random and random components. where is the regression dependence function, and is an additive random variable with zero mat expectation. The assumption about the nature of the distribution of this quantity is called the data generation hypothesis<#"8" src="doc_zip6.jpg" />has a Gaussian distribution<#"20" src="doc_zip7.jpg" />.

The problem of finding a regression model of several free variables is posed as follows. A sample is given<#"24" src="doc_zip8.jpg" />values ​​of free variables and the set of corresponding values ​​of the dependent variable. These sets are denoted as the set of initial data.

A regression model is given - a parametric family of functions depending on parameters and free variables. It is required to find the most probable parameters:

The probability function depends on the data generation hypothesis and is given by Bayesian inference<#"justify">Least square method

The method of least squares is a method of finding the optimal parameters of linear regression, such that the sum of squared errors (regression residuals) is minimal. The method consists in minimizing the Euclidean distance between two vectors - the vector of recovered values ​​of the dependent variable and the vector of actual values ​​of the dependent variable.

The task of the least squares method is to choose a vector to minimize the error. This error is the distance from vector to vector. The vector lies in the column space of the matrix, since there is a linear combination of the columns of this matrix with coefficients. Finding a solution using the least squares method is equivalent to the problem of finding a point that lies closest to and is located in the column space of the matrix.

Thus, the vector must be a projection onto the column space, and the residual vector must be orthogonal to this space. Orthogonality is that each vector in the column space is a linear combination of columns with some coefficients, that is, it is a vector. For everything in space, these vectors must be perpendicular to the residual:

Since this equality must be true for an arbitrary vector, then

The least squares solution of an inconsistent system consisting of equations with unknowns is the equation

which is called the normal equation. If the columns of a matrix are linearly independent, then the matrix is ​​invertible and the only solution

The projection of a vector onto the column space of a matrix has the form

The matrix is ​​called the projection matrix of the vector onto the column space of the matrix. This matrix has two main properties: it is idempotent, and it is symmetric, . The converse is also true: a matrix with these two properties is a projection matrix onto its column space.

Let we have statistical data about the parameter y depending on x. We present these data in the form

xx1 X2 …..Xi…..Xny *y 1*y 2*......y i* …..y n *

The least squares method allows for a given type of dependence y= ?(x) choose its numerical parameters so that the curve y= ?(x) displayed the experimental data in the best way according to the given criterion. Consider the justification from the point of view of probability theory for the mathematical definition of the parameters included in ? (x).

Suppose that the true dependence of y on x is exactly expressed by the formula y= ?(x). The experimental points presented in Table 2 deviate from this dependence due to measurement errors. The measurement errors obey the normal law according to Lyapunov's theorem. Consider some value of the argument x i . The result of the experiment is a random variable y i , distributed according to the normal law with mathematical expectation ?(x i ) and with standard deviation ?i characterizing the measurement error. Let the measurement accuracy at all points x=(x 1, X 2, …, X n ) is the same, i.e. ?1=?2=…=?n =?. Then the normal distribution law Yi looks like:

As a result of a series of measurements, the following event occurred: random variables (y 1*,y 2*, …, yn *).

Description of the selected software product

Mathcad - computer algebra system from the class of computer-aided design systems<#"justify">4. Practical part

The task of the study is to forecast gasoline prices. The initial information is a 36-week time series - from May 2012 to December 2012.

Statistics data (36 weeks) are presented in the Y matrix. Next, we will create the H matrix, which will be needed to find the vector A.

Let's present the initial data and the values ​​calculated using the model:

To assess the quality of the model, we use the coefficient of determination.

First, let's find the average value of Xs:

The part of the variance, which is due to regression, in the total variance of the indicator Y characterizes the coefficient of determination R2.

Determination coefficient, takes values ​​from -1 to +1. The closer its value of the coefficient modulo to 1, the closer the relationship of the effective feature Y with the studied factors X.

The value of the coefficient of determination serves as an important criterion for assessing the quality of linear and nonlinear models. The greater the share of the explained variation, the less the role of other factors, which means that the regression model approximates the initial data well and such a regression model can be used to predict the values ​​of the effective indicator. We obtained the coefficient of determination R2 = 0.78, therefore, the regression equation explains 78% of the variance of the effective feature, and 22% of its variance (i.e., residual variance) falls to the share of other factors.

Therefore, we conclude that the model is adequate.

Based on the data obtained, it is possible to make a forecast of fuel prices for the 37th week of 2013. The formula for the calculation is as follows:

The calculated forecast using this model: the price of gasoline is UAH 10.434.

Conclusion

In this paper, we have shown the possibility of conducting a regression analysis to predict gasoline prices for future periods. The purpose of the course work was to consolidate knowledge in the course "Mathematical Methods of Operations Research" and gain skills in developing software that allows you to automate operations research in a given subject area.

The forecast for the future price of gasoline, of course, is not unambiguous, which is due to the peculiarities of the initial data and the developed models. However, based on the information received, it is reasonable to assume that, of course, gasoline prices will not fall in the near future, but most likely will remain at the same level or will grow slightly. Of course, factors related to consumer expectations, customs policy and many other factors are not taken into account here, but I would like to note that they are largely mutually repayable . And it would be quite reasonable to note that a sharp jump in gasoline prices at the moment is really extremely doubtful, which, first of all, is connected with the policy pursued by the government.

Bibliography

1.Buyul A., Zöfel P. SPSS: the art of information processing. Analysis of statistical data and restoration of hidden patterns. - St. Petersburg: OOO "DiaSoftUP", 2001. - 608 p.

2. Internet resources http://www.ukrstat.gov.ua/

3. Internet resources http://index.minfin.com.ua/

Internet resources http://fx-commodities.ru/category/oil/

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The project method, which has enormous potential for the formation of non-versal educational activities, is becoming more and more widespread in the school education system. But it is rather difficult to "fit" the project method into the classroom system. I include mini-studies in a regular lesson. This form of work opens up great opportunities for the formation of cognitive activity and ensures that the individual characteristics of students are taken into account, paves the way for the development of skills on large projects.

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“If a student at school has not learned to create anything himself, then in life he will only imitate, copy, since there are few who, having learned to copy, would be able to make an independent application of this information.” L.N. Tolstoy.

A characteristic feature of modern education is a sharp increase in the amount of information that students need to learn. And the degree of development of the student is measured and evaluated by his ability to independently acquire new knowledge and use them in educational and practical activities. The modern pedagogical process requires the use of innovative technologies in teaching.

The Federal State Educational Standard of the new generation requires the use of activity-type technologies in the educational process, the methods of design and research activities are defined as one of the conditions for the implementation of the main educational program.

A special role is given to such activities in mathematics lessons, and this is not accidental. Mathematics is the key to understanding the world, the basis of scientific and technological progress and an important component of personality development. It is designed to instill in a person the ability to understand the meaning of the task assigned to him, the ability to reason logically, to learn the skills of algorithmic thinking.

It is rather difficult to fit the project method into the class-lesson system. I try to intelligently combine the traditional and student-centered system by incorporating research elements into a regular lesson. I will give a number of examples.

So, when studying the topic “Circle”, we conduct the following study with students.

Mathematical study "Circle".

1. Think about how to build a circle, what tools are needed for this. Circle designation.
2. In order to define a circle, let's see what properties this geometric figure has. Let's connect the center of the circle with a point belonging to the circle. Let's measure the length of this segment. Let's repeat the experiment three times. Let's make a conclusion.
3. The line segment connecting the center of the circle with any point on it is called the radius of the circle. This is the definition of a radius. Radius notation. Using this definition, construct a circle with a radius of 2cm5mm.
4. Construct a circle of arbitrary radius. Build a radius, measure it. Record the measurement results. Build three more different radii. How many radii can be drawn in a circle.
5. Let's try, knowing the property of the points of the circle, to give its definition.
6. Construct a circle of arbitrary radius. Connect two points of the circle so that this segment passes through the center of the circle. This segment is called the diameter. Let's define the diameter. Diameter designation. Build three more diameters. How many diameters does a circle have.
7. Construct a circle of arbitrary radius. Measure the diameter and radius. Compare them. Repeat the experiment three more times with different circles. Make a conclusion.
8. Connect any two points on the circle. The resulting segment is called a chord. Let's define a chord. Build three more chords. How many chords does the circle have.
9. Is the radius a chord. Prove it.
10. Is the diameter a chord. Prove it.

Research works can be propaedeutic in nature. Having examined the circle, one can consider a number of interesting properties that students can formulate at the level of a hypothesis, and then prove this hypothesis. For example, the following study:

"Mathematical Research"

1. Construct a circle with a radius of 3 cm and draw its diameter. Connect the ends of the diameter to an arbitrary point on the circle and measure the angle formed by the chords. Carry out the same constructions for two more circles. What do you notice.
2. Repeat the experiment for a circle of arbitrary radius and formulate a hypothesis. Can it be considered proven with the help of the constructions and measurements carried out.

When studying the topic “Mutual arrangement of lines on a plane”, a mathematical study is carried out in groups.

Tasks for groups:

1. Group.

1. In one coordinate system, plot the graphs of the function

Y=2x, y=2x+7, y=2x+3, y=2x-4, y=2x-6.

2. Answer the questions by filling out the table: