## Introduction

In this paper we will discuss what have we learned in mathematical statistics.

Mathematical statistics is a branch of mathematics, which is developing methods for recording, describing and analyzing the data of observations and experiments in order to construct probabilistic models of random mass phenomena. Depending on the mathematical nature of the concrete results of observations divided by the mathematical statistics Statistics numbers, multivariate statistical analysis, the functions (processes) and time series, Statistics non-numeric objects of nature.

There are three parts of mathematical statistics: descriptive statistics, estimation theory and the theory of testing hypotheses. Descriptive statistics is the collection of empirical methods used for visualization and interpretation of data (calculation of sample characteristics, tables, charts, graphs, etc.) usually do not require assumptions about the probabilistic nature of the data. Some methods involve the use of descriptive statistics capabilities of modern computers. These include, in particular, cluster analysis, aimed at the selection of groups of objects, similar to each other, and multidimensional scaling, which allows to visualize objects on the plane.

Methods of estimation and hypothesis testing based on the probabilistic model of the origin of data. These models are divided into parametric and nonparametric. In parametric models assume that the characteristics of the objects under study are described by distributions that depend on (one or more) of numerical parameters. Nonparametric models are not associated with the specification of the parametric family for the distribution of the studied characteristics. In mathematical statistics, estimate the parameters and functions of them representing important characteristics of the distributions (eg, expectation, median, standard deviation, quantiles, etc.), the density and distribution functions, etc. Use a point and interval estimates.

A large part of modern mathematical statistics - statistical sequential analysis, a fundamental contribution to the creation and development of which Wald introduced during the Second World War. Unlike traditional (non-consecutive) methods of statistical analysis based on a random sample of fixed volume, in sequential analysis allowed the formation of an array of observations by one (or, more generally, groups), with the decision on conducting the following observation (observation group) received at based on already existing array of observations. In view of this, the theory of sequential statistical analysis is closely related to the theory of optimal stopping.

In mathematical statistics, is the general theory of testing hypotheses and a large number of methods for dealing with testing specific hypotheses. Consider the hypothesis of parameter values and characteristics, the verification of homogeneity (i.e., on the coincidence of the characteristics or distribution functions in the two samples), the consent of the empirical distribution function with a given distribution function or parameter family of such functions, the symmetry of the distribution, etc.

Great importance has the section of mathematical statistics associated with sample surveys, with the properties of different types of organization of samples and construct adequate methods for estimation and hypothesis testing.

## Restoration task dependencies are actively studied for over 200 years, since the development by Gauss in 1794, the method of least squares.

Descriptive Statistics

Descriptive statistics used for simple integration of data obtained under sampling. In turn, the statistical inferences are needed to ensure that the data obtained from the sample can be extended to the entire population.

Basic methods of descriptive statistics include percentages, measures of central tendency, measures of variation and paired coupling coefficients. They allow you to compile the data available for the sample.

Measures of central tendency (mode, median and average) provide information about the typical or central value of the distribution. Fashion talks about the most common value, the median - the median value of, the arithmetic mean - the most expected value.

In the example above, fashion is the answer "There were three", as it marked the 28 people - more than any other option.

Calculated as the arithmetic mean is shown in the first chapter. In this case, the calculation of the average value is not quite correct. But if we assign Options answers specific numerical values (e.g., 1, 2, 3, 4), we can carry out the calculation of average.

Measures of variability indicate the degree of heterogeneity of distribution (eg, the magnitude of the coefficient of variation categories, standard deviation, etc.). A simple, both in terms of the received information, and in terms of counting, a scale (R). It is equal to the difference between the highest and lowest values of the distribution. For example, if the smallest increase among respondents is 147 centimeters and the largest - 198, the scale is equal to 51 cm.

For quantitative traits lower bound measures of variability equal to 0 (the researchers are interested in characterizing the observed objects do not differ). In turn, the upper limit - always open quantities defining characteristics of the studied property and observed heterogeneous distribution. For example, the standard deviation for the growth of the population of Ukraine in 2008 was equal to 9 centimeters (average is 169 cm) and for age - 17 years (the average is 46 years).

Since the calculation of the standard deviation was considered earlier, here we touch only the coefficient of variation of categories (IQV) - a single measure of variability, the use of which is justified in cases of nominal and ordinal scales. In this case, the ratio can range from 0 (no variation) and 1 (maximum variability).

## Inferential Statistics and Hypothesis Development and Testing

The purpose of inductive statistics to determine how likely it is that the two samples are from the same population, to determine whether there is enough great difference between the average of the two distributions in order to be able to explain its action of the independent variable, rather than an accident, associated with the small sample size.

## In this case there are two possible hypotheses:

1) the null hypothesis (H0), according to which the difference between the distributions is unreliable; it is assumed that the difference is significantly insufficient, and thus belong to the distribution of the same population as the independent variable had no effect;

2) alternative hypothesis (H1), which is the working hypothesis of our study. In accordance with this hypothesis, the distinction between the two distributions sufficiently significant and due to the influence of the independent variable.

The basic principle of the method of testing hypotheses is that the null hypothesis H0 extends, in order to try to refute it, and thereby confirm the alternative hypothesis H1. Indeed, if the results of the statistical test used to analyze the difference between the average, will be such that will allow discard H0, it would mean that H1 is true that extended working hypothesis is confirmed.

In the humanities, it is assumed that the null hypothesis can be rejected in favor of the alternative hypothesis, if the results of the statistical test probability of forming differences found less than 5 out of 100. If this is not achieved the level of confidence to believe that the difference can be quite random and therefore cannot reject the null hypothesis.

In order to judge whether the probability of mistake, accepting or rejecting the null hypothesis, statistical methods are used, corresponding to features of the sample.

Thus, for quantitative data when distributions are close to normal, using parametric methods based on indicators such as the mean and standard deviation. In particular, to determine the reliability of mean differences for the two-sample Student's method is used, but in order to judge the differences between three or more samples - test F, or analysis of variance.

If we are dealing with a non-quantitative data or sample size is too small to be confident that the population from which they are taken, are normally distributed, then use non-parametric methods - the criterion χ2 (chi-square) for data quality criteria and signs, ranks, Mann-Whitney, Wilcoxon and others for ordinal data.

Furthermore, statistical selection of the method depends on whether those samples, which average is compared independent (i.e., for example, taken from two different groups of subjects), or dependent (i.e. reflecting the results of the same test group before and after exposure or after two different exposures).

## Selection of Appropriate Statistical Tests

According to Institute for Digital Research and Education, the rules how to select the appropriate statistical test is given in the following table:

## Evaluating Statistical Results

Statistical hypotheses can be tested by methods of mathematical statistics. The result will be obtained by the conclusion whether to reject or accept the hypothesis put forward by it. However, these methods do not allow us to guarantee the complete accuracy of the results. Ie there is always a non-zero probability of error. It is possible error are of two types.

Error of the first kind - it is such a mistake, which resulted in the correct hypothesis is rejected. Chance make such a mistake is called the significance level. Typically, as the level of significance is accepted to use the following probabilities : 0.1, 0.05, 0.01.

Error of the second kind is to be taken the wrong hypothesis. The probability of error of the second kind indicate β.

When working with statistical hypotheses must push the main hypothesis, which is usually denoted by H0 and called the null hypothesis and the alternative hypothesis, which is usually the logical negation of the null hypothesis. Further, in these examples, as the main hypothesis H0 will be conjectured slight difference in the results. Hence, in the alternative hypothesis will be approved that results vary significantly.

Procedure informed comparison of the hypothesis with samples obtained by means of a statistical test called a statistical test of the hypothesis. The critical area of a set of values understood criteria under which the null hypothesis H0 rejected.

## Sources

R. A. Fisher (1925).Statistical Methods for Research Workers, Edinburgh: Oliver and Boyd, 1925.

Lehmann, E.L.; Romano, Joseph P. (2005). Testing Statistical Hypotheses (3E ed.). New York: Springer.

Triola, Mario (2001). Elementary statistics (8 ed.). Boston: Addison-Wesley.

Bakan, David (1966). "The test of significance in psychological research". Psychological Bulletin 66 (6): 423–437.