Null and alternative hypotheses. Statistical hypotheses and criteria Null hypothesis acceptance area
STATISTICAL CHECKING OF STATISTICAL
Statistical hypothesis concept.
Types of hypotheses. Errors of the first and second kind
Hypothesis- this is an assumption about some properties of the studied phenomena. Under statistical hypothesis understand any statement about the general population that can be checked statistically, that is, based on the results of observations in a random sample. Consider two types of statistical hypotheses: hypotheses distribution laws general population and hypotheses about parameters known distributions.
So, the hypothesis that the time spent on assembling a machine unit in a group of machine shops producing products of the same name and having approximately the same technical and economic conditions of production is distributed according to the normal law is a hypothesis about the distribution law. And the hypothesis that the labor productivity of workers in two brigades performing the same work under the same conditions does not differ (while the labor productivity of workers in each brigade has a normal distribution law) is a hypothesis about the distribution parameters.
The hypothesis to be tested is called null, or basic, and denoted H 0. The null hypothesis is contrasted with competing, or alternative, hypothesis denoted H 1 . Typically the competing hypothesis H 1 is a logical negation of the main hypothesis H 0.
An example of a null hypothesis would be: the means of two normally distributed populations are equal, then a competing hypothesis may consist of the assumption that the means are not equal. This is symbolically written as follows:
H 0: M(NS) = M(Y); H 1: M(NS) M(Y) .
If the null (put forward) hypothesis is rejected, then there is a competing hypothesis.
Distinguish between simple and complex hypotheses. If the hypothesis contains only one assumption, then it is - simple hypothesis. Complex a hypothesis consists of a finite or infinite number of simple hypotheses.
For example, the hypothesis H 0: p = p 0 (unknown probability p is equal to the hypothetical probability p 0 ) is simple, and the hypothesis H 0: p < p 0 - complex, it consists of countless simple hypotheses of the form H 0: p = p i, where p i- any number, less p 0 .
The put forward statistical hypothesis can be correct or incorrect, therefore it is necessary verify based on the results of observations in a random sample; verification is carried out statistical methods, so it is called statistical.
When testing a statistical hypothesis, a specially compiled random variable is used, called statistical criterion(or statistics). The accepted conclusion about the correctness (or incorrectness) of the hypothesis is based on the study of the distribution of this random variable according to the sample data. Therefore, statistical testing of hypotheses has a probabilistic nature: there is always a risk of making a mistake when accepting (rejecting) a hypothesis. In this case, errors of two kinds are possible.
First kind error is that the null hypothesis will be rejected, although in fact it is true.
Type II error is that the null hypothesis will be accepted, although in reality the competing one is true.
In most cases, the consequences of these errors are unequal. Which is better or worse depends on the specific formulation of the problem and the content of the null hypothesis. Let's look at some examples. Let us assume that at an enterprise the quality of products is judged by the results of sampling. If the sample rate of rejects does not exceed a predetermined value p 0 , then the batch is accepted. In other words, a null hypothesis is put forward: H 0: p p 0 ... If, when testing this hypothesis, an error of the first kind is made, then we will reject suitable products. If a mistake of the second kind is made, then a defect will be sent to the consumer. Obviously, the consequences of a Type II error can be much more serious.
Another example can be cited from the field of jurisprudence. We will consider the work of judges as actions to verify the presumption of innocence of the defendant. As the main testable hypothesis, consider the hypothesis H 0 : the defendant is innocent. Then an alternative hypothesis H 1 is a hypothesis: the accused is guilty of a crime. It is obvious that the court can make mistakes of the first or second kind when sentencing the defendant. If a mistake of the first kind is made, it means that the court punished the innocent: the defendant was convicted when in fact he did not commit a crime. If the judges made a mistake of the second kind, then this means that the court has passed an acquittal, when in fact the accused is guilty of committing a crime. Obviously, the consequences of an error of the first kind for the accused will be much more serious, while for society the most dangerous are the consequences of an error of the second kind.
Probability commit a mistake first kind are called level of significance criterion and denote.
In most cases, the significance level of the criterion is taken to be 0.01 or 0.05. If, for example, the significance level is taken equal to 0.01, then this means that in one case out of a hundred there is a risk of making a type I error (that is, rejecting the correct null hypothesis).
Probability commit error of the second kind denote. Probability 
not to make a mistake of the second kind, that is, to reject the null hypothesis when it is incorrect, is called power of the criterion.
Statistical criterion.
Critical areas
The statistical hypothesis is tested using a specially selected random variable, the exact or approximate distribution of which is known (we denote it TO). This random variable is called statistical criterion(or simply criterion).
There are various statistical criteria used in practice: U- and Z-criteria (these random variables have a normal distribution); F-criterion (the random variable is distributed according to the Fisher-Snedecor law); t-criterion (according to Student's law); -criterion (according to the "chi-square" law), etc.
The set of all possible values of the criterion can be divided into two disjoint subsets: one of them contains the values of the criterion for which the null hypothesis is accepted, and the other for which it is rejected.
The set of values of the criterion for which the null hypothesis is rejected is called critical area... We will denote the critical region by W.
The set of values of the criterion for which the null hypothesis is accepted is called area of hypothesis(or range of acceptable values of the criterion). We will denote this area as 
.
To test the validity of the null hypothesis according to the sample data, calculate observed criterion value... We will denote it TO obs.
Basic Principle of Statistical Hypothesis Testing can be formulated as follows: if the observed value of the criterion falls into the critical region (i.e. 
), then the null hypothesis is rejected; if the observed value of the criterion fell into the area of acceptance of the hypothesis (i.e. 
), then there is no reason to reject the null hypothesis.
What principles should be followed when constructing the critical area W ?
Let us assume that the hypothesis H 0
is actually correct. Then hitting the criterion 
into the critical region, due to the basic principle of testing statistical hypotheses, entails the rejection of the correct hypothesis H 0
, which means making a mistake of the first kind. Therefore, the probability of hitting 
to the region W if the hypothesis is true H 0
must be equal to the level of significance of the criterion, that is
.
Note that the probability of making an error of the first kind is chosen sufficiently small (as a rule, 
). Then hitting the criterion 
to the critical area W if the hypothesis is true H 0
can be considered an almost impossible event. If, according to the data of selective observation, the event 
nevertheless happened, then it can be considered incompatible with the hypothesis H 0
(which is rejected as a result), but consistent with the hypothesis H 1
(which is accepted as a result).
Suppose now that the hypothesis is true H 1
... Then hitting the criterion 
in the field of hypothesis acceptance 
entails acceptance of an incorrect hypothesis H 0
, which means making a mistake of the second kind. That's why 
.
Since events 
and 
are mutually opposite, then the probability of hitting the criterion 
to the critical area W will be equal to the power of the test if the hypothesis H 1
true, that is
.
Obviously, the critical area should be chosen so that at a given level of significance the power of the criterion 
was maximum. Maximizing the power of the criterion will provide a minimum probability of making a type II error.
It should be noted that no matter how small the significance level is, the criterion falling into the critical area is only an unlikely, but not absolutely impossible event. Therefore, it is possible that if the null hypothesis is correct, the value of the criterion calculated from the sample data will nevertheless be in the critical region. Rejecting in this case the hypothesis H 0 , we make an error of the first kind with probability. The smaller, the less likely it is to make a Type I error. However, with decreasing, the critical region decreases, which means that it becomes less possible for the observed value to fall into it. TO obs, even when the hypothesis H 0 is wrong. For = 0 the hypothesis H 0 will always be accepted regardless of sample results. Therefore, a decrease entails an increase in the probability of accepting an incorrect null hypothesis, that is, making a mistake of the second kind. In this sense, errors of the first and second kind are competing.
Since it is impossible to exclude errors of the first and second kind, it is necessary at least to strive in each specific case to minimize losses from these errors. Of course, it is desirable to reduce both errors at the same time, but since they are competing, then a decrease in the probability of admitting one of them entails an increase in the likelihood of admitting the other. The only way simultaneous decrease the risk of errors lies in increased sample size.
Depending on the type of competing hypothesis H 1
build unilateral (right-sided and left-sided) and bilateral critical areas. Points separating the critical area 
from the field of hypothesis acceptance 
are called critical points and denote k Crete. For finding the critical area you need to know the critical points.
Right-sided the critical region can be described by the inequality 
TO>k Crete. pr, where it is assumed that the right critical point k Crete. pr> 0. Such an area consists of points to the right of the critical point. k Crete. pr, that is, it contains many positive and sufficiently large values of the criterion TO. To find k Crete. pr set first the level of significance of the criterion. Next, the right critical point k Crete. pr find from the condition. Why exactly this requirement defines the right-sided critical region? Since the probability of an event (TO>k Crete. NS )
is small, then, due to the principle of practical impossibility of unlikely events, this event, if the null hypothesis is valid in a single test, should not occur. If, nevertheless, it did occur, that is, the observed value of the criterion calculated from the sample data 
turned out to be more k Crete. pr, then this can be explained by the fact that the null hypothesis does not agree with the observation data and therefore must be rejected. Thus, the requirement 
determines such values of the criterion at which the null hypothesis is rejected, and they constitute the right-sided critical region.
If 
fell into the range of acceptable values of the criterion 
, that is 
<
k Crete. pr, then the main hypothesis is not rejected, because it is compatible with observation data. Note that the probability of hitting the criterion 
into the range of admissible values 
if the null hypothesis is valid, it is equal to (1-) and is close to 1.
It must be remembered that the hit of the criterion values 
into the range of acceptable values is not a rigorous proof of the validity of the null hypothesis. It only indicates that there is no significant discrepancy between the hypothesis put forward and the sample results. Therefore, in such cases, it is said that the observational data agree with the null hypothesis and there is no reason to reject it.
The construction of other critical regions is carried out in a similar way.
So, leuro-sided the critical region is described by the inequality 
TO<k Crete. l where k crit.l<0.
Такая область состоит из точек, находящихся
по левую сторону от левой критической
точки k crit.l, that is, it is a set of negative, but sufficiently large in modulus values of the criterion. Critical point k crit.l is found from the condition 
(TO<k Crete. l) 
, that is, the probability that the criterion takes on a value less than k crit.l, is equal to the accepted level of significance, if the null hypothesis is correct.
Bilateral critical area 
is described by the following inequalities: ( TO<
k crit.l or TO>k Crete. pr), where it is assumed that k crit.l<0
и k Crete. pr> 0. Such an area is a set of criterion values that are sufficiently large in absolute value. Critical points are found from the requirement: the sum of the probabilities that the criterion will take a value less than k Crete. l or more k Crete. pr, should be equal to the accepted level of significance if the null hypothesis is valid, that is,
(TO<
k Crete. l )+
(TO>k Crete. NS )=
.
If the distribution of the criterion TO symmetrically about the origin, then the critical points will be located symmetrically about zero, therefore k Crete. l = - k Crete. pr. Then the two-sided critical region becomes symmetric and can be described by the following inequality: > k Crete. dv where k Crete. dv = k Crete. pr Critical point k Crete. dv can be found from the condition
P (K< -k Crete. dv ) = P (K>k Crete. dv )= .
Remark 1. For each criterion TO critical points at a given level of significance 
can be found from the condition 
only numerically. Numerical results
k crit are given in the corresponding tables (see, for example, appendices 4 - 6 in the file "Attachments").
Remark 2. The principle of testing a statistical hypothesis described above does not yet prove its truth or untruth. Acceptance of the hypothesis H 0 compared with an alternative hypothesis H 1 does not mean that we are sure of the absolute correctness of the hypothesis H 0 - just a hypothesis H 0 is consistent with our observational data, that is, it is a fairly plausible statement that does not contradict experience. It is possible that with an increase in the sample size n hypothesis H 0 will be rejected.
STATISTICAL HYPOTHESES
Sample data obtained in experiments are always limited and are largely random in nature. That is why mathematical statistics are used to analyze such data, which makes it possible to generalize the patterns obtained in the sample and extend them to the entire general population.
The data obtained as a result of the experiment on any sample serve as the basis for judging the general population. However, due to the action of random probabilistic reasons, the estimate of the parameters of the general population, made on the basis of experimental (sample) data, will always be accompanied by an error, and therefore such estimates should be considered as conjectural, and not as final statements. Such assumptions about the properties and parameters of the general population are called statistical hypotheses . According to G.V. Sukhodolsky: "A statistical hypothesis is usually understood as a formal assumption that the similarity (or difference) of some parametric or functional characteristics is accidental or, conversely, not accidental."
The essence of testing a statistical hypothesis is to establish whether the experimental data and the hypothesis put forward agree, whether it is permissible to attribute the discrepancy between the hypothesis and the result of the statistical analysis of experimental data due to random causes. Thus, a statistical hypothesis is a scientific hypothesis that allows statistical testing, and mathematical statistics is a scientific discipline whose task is to scientifically test statistical hypotheses.
Statistical hypotheses are classified into null and alternative, directed and non-directed.
Null hypothesis(H 0) Is the hypothesis that there are no differences. If we want to prove the significance of the differences, then the null hypothesis is required refute, otherwise it is required confirm.
Alternative hypothesis (H 1) Is a hypothesis about the significance of the differences. This is what we want to prove, which is why it is sometimes called experimental hypothesis.
There are tasks when we want to prove just insignificance differences, that is, to confirm the null hypothesis. For example, if we need to make sure that different subjects receive, albeit different, but balanced in difficulty, or that the experimental and control samples do not differ in some significant characteristics. However, more often we still need to prove the significance of the differences, for they are more informative for us in our search for something new.
Null and alternative hypotheses can be directional and non-directional.
Directed hypotheses - if it is assumed that the characteristic values are higher in one group, and lower in the other:
H 0: X 1 less than X 2,
H 1: X 1 exceeds X 2.
Undirected hypotheses - if it is assumed that the forms of distribution of a characteristic in groups differ:
H 0: X 1 does not differ from X 2,
H 1: X 1 is different X 2.
If we noticed that in one of the groups the individual values of the subjects for some criterion, for example, for social activity, are higher, and in the other lower, then in order to test the significance of these differences, we need to formulate directed hypotheses.
If we want to prove that in a group A under the influence of some experimental influences, more pronounced changes occurred than in the group B, then we also need to formulate directed hypotheses.
If we want to prove that the forms of distribution of the trait in groups differ A and B, then undirected hypotheses are formulated.
Hypothesis testing is carried out using criteria for the statistical assessment of differences.
The accepted conclusion is called a statistical decision. Let us emphasize that such a solution is always probabilistic. When testing a hypothesis, experimental data may contradict the hypothesis H 0, then this hypothesis is rejected. Otherwise, i.e. if the experimental data agree with the hypothesis H 0, it does not deviate. It is often said in such cases that the hypothesis H 0 is accepted. This shows that statistical testing of hypotheses based on experimental sample data is inevitably associated with the risk (probability) of making a false decision. In this case, errors of two kinds are possible. An error of the first kind will occur when a decision is made to reject a hypothesis. H 0, although in reality it turns out to be true. An error of the second kind will occur when a decision is made not to reject the hypothesis. H 0, although in reality it will be incorrect. Obviously, correct conclusions can also be accepted in two cases. Table 7.1 summarizes the above.
Table 7.1
It is possible that the psychologist may be mistaken in his statistical decision; as we can see from Table 7.1, these errors can be of only two kinds. Since it is impossible to exclude errors when accepting statistical hypotheses, it is necessary to minimize the possible consequences, i.e. acceptance of an incorrect statistical hypothesis. In most cases, the only way to minimize errors is to increase the sample size.
STATISTICAL CRITERIA
Statistical criterion- this is a decision rule that ensures reliable behavior, that is, acceptance of a true hypothesis and rejection of a false hypothesis with a high probability.
Statistical criteria also refer to the method for calculating a certain number and the number itself.
When we say that the reliability of differences was determined by the criterion j *(the criterion is the angular Fisher transformation), then we mean that we used the method j * to calculate a specific number.
By the ratio of the empirical and critical values of the criterion, we can judge whether the null hypothesis is confirmed or refuted.
In most cases, in order for us to recognize differences as significant, it is necessary that the empirical value of the criterion exceeds the critical, although there are criteria (for example, the Mann-Whitney criterion or the sign criterion) in which we must adhere to the opposite rule.
In some cases, the calculation formula of the criterion includes the number of observations in the studied sample, denoted as n... In this case, the empirical value of the criterion is at the same time a test for testing statistical hypotheses. Using a special table, we determine what level of statistical significance of differences a given empirical value corresponds to. An example of such a criterion is the criterion j * calculated based on the angular Fisher transform.
In most cases, however, the same empirical value of the criterion may turn out to be significant or insignificant depending on the number of observations in the studied sample ( n) or on the so-called number of degrees of freedom, which is denoted as v or how df.
The number of degrees of freedom v is equal to the number of classes of the variational series minus the number of conditions under which it was formed. These conditions include the sample size ( n), means and variances.
Let's say a group of 50 people was divided into three classes according to the principle:
Knows how to work on a computer;
Knows how to perform only certain operations;
Can't work on a computer.
The first and second groups included 20 people, the third - 10.
We are limited by one condition - the sample size. Therefore, even if we have lost data on how many people do not know how to work on a computer, we can determine this, knowing that in the first and second grades there are 20 subjects each. We are not free in determining the number of subjects in the third category, "freedom" extends only to the first two cells of the classification:
Since statistics as a research method deals with data in which the laws of interest to the researcher are distorted by various random factors, most statistical calculations are accompanied by testing some assumptions or hypotheses about the source of these data.
Pedagogical hypothesis (scientific hypothesis not about the advantage of one method or another) in the process of statistical analysis is translated into the language of statistical science and is formulated anew, at least in the form of two statistical hypotheses.
Two types of hypotheses are possible: the first type is descriptive hypotheses that describe the causes and possible consequences. The second type is explanatory : they explain the possible consequences of certain causes, and also characterize the conditions under which these consequences will necessarily follow, that is, they explain by virtue of what factors and conditions the given effect will be. Descriptive hypotheses lack foresight, while explanatory hypotheses do. Explanatory hypotheses lead researchers to make assumptions about the existence of certain regular relationships between phenomena, factors and conditions.
Hypotheses in educational research may suggest that one of the tools (or a group of them) will be more effective than other means. Here, hypothetically, an assumption is made about the comparative effectiveness of means, methods, methods, forms of education.
A higher level of hypothetical prediction is that the author of the study makes a hypothesis that some system of measures will not only be better than another, but also from a number of possible systems, it seems optimal from the point of view of certain criteria. Such a hypothesis needs more detailed proof.
A.P. Kulaichev Methods and tools for data analysis in the Windows environment. Ed. 3rd, rev. and add. - M: InKo, 1999, pp. 129-131
Psychological and pedagogical dictionary for teachers and heads of educational institutions. - Rostov-n / D: Phoenix, 1998, p. 92
On the basis of the data collected in statistical studies, after their processing, conclusions are drawn about the phenomena under study. These conclusions are made by putting forward and testing statistical hypotheses.
Statistical hypothesis any statement about the form or properties of the distribution of experimentally observed random variables is called. Statistical hypotheses are tested by statistical methods.
The hypothesis to be tested is called main (zero) and denoted H 0. In addition to zero, there is also alternative (competing) hypothesis H 1, denying the main . Thus, as a result of testing, one and only one of the hypotheses will be accepted. , and the second will be rejected.
Types of errors... The hypothesis put forward is tested on the basis of a study of a sample obtained from the general population. Due to the randomness of the sample, the validation does not always lead to the correct conclusion. In this case, the following situations may arise:
1. The main hypothesis is correct and accepted.
2. The main hypothesis is correct, but it is rejected.
3. The main hypothesis is incorrect and it is rejected.
4. The main hypothesis is not correct, but it is accepted.
In case 2, one speaks of error of the first kind, in the latter case we are talking about error of the second kind.
Thus, for some samples, the correct decision is made, and for others, the wrong one. The decision is made by the value of some sampling function, called statistical characteristics, statistical criterion or simply statistics... The set of values for this statistic can be divided into two disjoint subsets:
- H 0 is accepted (not rejected), called hypothesis acceptance area (feasible area);
- a subset of the statistic values for which the hypothesis H 0 is rejected (rejected) and the hypothesis is accepted H 1 is called critical area.
Conclusions:
- Criterion is a random variable K that allows you to accept or reject the null hypothesis H0.
- When testing hypotheses, errors of 2 genera can be made.
First kind error is that the hypothesis will be rejected H 0 if correct ("skip target"). The probability of making a mistake of the first kind is denoted by α and is called level of significance... Most often in practice, it is assumed that α = 0.05 or α = 0.01.
Type II error is that the hypothesis H0 is accepted if it is incorrect ("false positive"). The probability of an error of this kind is denoted by β.
Classification of hypotheses
Main hypothesis H 0 about the value of the unknown parameter q of the distribution usually looks like this:H 0: q = q 0.
Competing hypothesis H 1 can thus have the following form:
H 1: q < q 0 , H 1: q> q 0 or H 1: q ≠ q 0 .
Accordingly, it turns out left-sided, right-sided or bilateral critical areas. Boundary points of critical regions ( critical points) are determined from the distribution tables of the corresponding statistics.
When testing a hypothesis, it is prudent to reduce the likelihood of making bad decisions. Type I error tolerance usually denoted a and called level of significance... Its value is usually small ( 0,1, 0,05, 0,01, 0,001
...). But a decrease in the probability of a type I error leads to an increase in the probability of a type II error ( b), i.e. the desire to accept only correct hypotheses causes an increase in the number of rejected correct hypotheses. Therefore, the choice of the level of significance is determined by the importance of the problem posed and the severity of the consequences of an incorrectly made decision.
Statistical hypothesis testing consists of the following steps:
1) defining hypotheses H 0 and H 1 ;
2) selection of statistics and setting the level of significance;
3) determination of critical points K cr and critical area;
4) computation of the statistic value from the sample To ex;
5) comparison of the statistic value with the critical area ( K cr and To ex);
6) decision making: if the value of the statistics is not included in the critical area, then the hypothesis is accepted H 0 and the hypothesis is rejected H 1, and if it enters the critical region, then the hypothesis is rejected H 0 and the hypothesis is accepted H 1 . At the same time, the results of testing the statistical hypothesis should be interpreted as follows: if the hypothesis is accepted H 1 ,
then it can be considered proven, and if the hypothesis is accepted H 0 ,
then it was recognized that it does not contradict the results of observations. However, this property, along with H 0 may have other hypotheses as well.
Classification of hypothesis tests
Let us now consider several different statistical hypotheses and mechanisms for testing them.I) Hypothesis about the general mean of the normal distribution with unknown variance. We assume that the general population has a normal distribution, its mean and variance are unknown, but there is reason to believe that the general average is equal to a. At the significance level α, the hypothesis should be tested H 0: x = a. As an alternative, one of the three hypotheses discussed above can be used. In this case, statistics is a random variable having a Student's distribution with n- 1 degrees of freedom. The corresponding experimental (observed) value is determined t ex t cr H 1: x> a it is found according to the significance level α and the number of degrees of freedom n- 1. If t ex < t cr H 1: x ≠ a, the critical value is found according to the significance level α / 2 and the same number of degrees of freedom. The null hypothesis is accepted if | t ex |
,
having a normal distribution, and M(Z) = 0, D(Z) = 1. The corresponding experimental value is determined z ex... The critical value is found from the Laplace function table z cr... Under an alternative hypothesis H 1: x> y it is found from the condition F(z cr) = 0,5 – a... If z ex< z кр , then the null hypothesis is accepted, otherwise it is rejected. Under an alternative hypothesis H 1: x ≠ y the critical value is found from the condition F(z cr) = 0.5 × (1 - a). The null hypothesis is accepted if | z ex |< z кр .
III) The hypothesis of the equality of two mean values of normally distributed general populations, the variances of which are unknown and the same (small independent samples). At the significance level α, the main hypothesis should be tested H 0: x = y. As statistics, we use a random variable
,
having a Student's distribution with ( n x + n at- 2) degrees of freedom. The corresponding experimental value is determined t ex... From the table of critical points of the Student's distribution, the critical value is found t cr... Everything is solved similarly to hypothesis (I).
IV) Conjecture on the equality of two variances of normally distributed general populations... In this case, at the significance level a need to test the hypothesis H 0: D(NS) = D(Y). The statistics is a random variable having the Fisher - Snedecor distribution with f 1 = n b- 1 and f 2 = n m- 1 degrees of freedom (S 2 b - large variance, the volume of its sample n b). The corresponding experimental (observed) value is determined F ex... Critical value F cr under an alternative hypothesis H 1: D(NS) > D(Y) is found from the table of critical points of the Fisher - Snedecor distribution by the level of significance a and the number of degrees of freedom f 1 and f 2. The null hypothesis is accepted if F ex < F cr.
Instruction. For the calculation, you must specify the dimension of the source data.
V) The hypothesis of the equality of several variances of normally distributed general populations over samples of the same size. In this case, at the significance level a need to test the hypothesis H 0: D(NS 1) = D(NS 2) = …= D(X l). The statistics is a random variable 
with a Kochren distribution with degrees of freedom f = n- 1 and l (n - the size of each sample, l Is the number of samples). This hypothesis is tested in the same way as the previous one. A table of critical points of the Cochren distribution is used.
Vi) The hypothesis about the importance of the correlation. In this case, at the significance level a need to test the hypothesis H 0: r= 0. (If the correlation coefficient is zero, then the corresponding values are not related to each other). The statistics in this case is a random variable 
,
having a Student's distribution with f = n- 2 degrees of freedom. This hypothesis is tested in the same way as hypothesis (I).
Instruction. Indicate the amount of source data.
Vii) Hypothesis about the significance of the probability of the occurrence of an event. A fairly large number of n independent trials in which the event A occurred m once. There is reason to believe that the probability of this event occurring in one test is p 0... Required at significance level a test the hypothesis that the probability of an event A is equal to the hypothetical probability p 0... (Since the probability is estimated by the relative frequency, the hypothesis being tested can be formulated in another way: whether the observed relative frequency and the hypothetical probability differ significantly or not).
The number of trials is large enough, so the relative frequency of the event A distributed according to the normal law. If the null hypothesis is true, then its mathematical expectation is p 0, and the variance. In accordance with this, as a statistic, we choose a random variable 
,
which is distributed approximately according to the normal law with zero mathematical expectation and unit variance. This hypothesis is tested in exactly the same way as in case (I).
Instruction. For the calculation, you must fill in the initial data.
Statistics is a complex science of measuring and analyzing various data. As with many other disciplines, there is a concept of hypothesis in this industry. So, a hypothesis in statistics is any position that needs to be accepted or rejected. Moreover, in this industry there are several types of such assumptions, which are similar by definition, but differ in practice. The null hypothesis is the subject of our study today.
From general to specific: hypotheses in statistics
Another, no less important, departs from the basic definition of assumptions - a statistical hypothesis is the study of a general set of objects important for science, about which scientists draw conclusions. It can be checked using a sample (part of the population). Here are some examples of statistical hypotheses:
1. Overall grade performance may be affected by the educational level of each student.
2. The elementary course of mathematics is equally assimilated by both children who came to school at the age of 6 and by children who came at 7.
A simple hypothesis in statistics is an assumption that unambiguously characterizes a certain parameter of a value taken by a scientist.
A complex one consists of several or an infinite number of simple ones. Is a certain area indicated or there is no exact answer.
It is useful to understand several definitions of hypotheses in statistics so as not to confuse them in practice.
The null hypothesis concept
The null hypothesis is a theory that there are some two sets that do not differ from each other. However, at the scientific level, there is no concept "do not differ", but there is "their similarity is equal to zero." From this definition, the concept was formed. In statistics, the null hypothesis is denoted as H0. Moreover, the extreme value of the impossible (unlikely) is considered from 0.01 to 0.05 or less.
It is better to understand what the null hypothesis is, an example from life will help. The teacher at the university suggested that the different level of preparation of students of the two groups for test work is caused by insignificant parameters, random reasons that do not affect the general level of education (the difference in the preparation of the two groups of students is zero).
However, on the contrary, it is worth giving an example of an alternative hypothesis - an assumption that refutes the assertion of the zero theory (H1). For example: the director of the university suggested that the different level of preparation for the test work among the students of the two groups was caused by the use of different teaching methods by the teachers (the difference in the preparation of the two groups is significant and there is an explanation for that).
Now you can immediately see the difference between the concepts of "null hypothesis" and "alternative hypothesis". Examples illustrate these concepts.
Testing the null hypothesis
Making a guess is half the trouble. Testing the null hypothesis is considered a real challenge for newbies. This is where difficulties await many.
Using the method of alternative hypothesis, which asserts something opposite of the null theory, you can compare both options and choose the correct one. This is how statistics work.
Let the null hypothesis be H0, and the alternative hypothesis H1, then:
Н0: c = c0;
Н1: c ≠ c0.
Here c is a certain average value of the general population to be found, and c0 is the initially given value against which the hypothesis is tested. There is also a certain number X - the average value of the sample, which is used to determine c0.
So, the check consists in comparing X and c0, if X = c0, then the null hypothesis is accepted. If X ≠ c0, then by hypothesis the alternative is considered valid.
"Trusted" method of verification
There is the most powerful way by which the null statistical hypothesis is easily tested in practice. It consists in plotting a range of values up to 95% accuracy.
First, you need to know the formula for calculating the confidence interval:
X - t * Sx ≤ c ≤ X + t * Sx,
where X is the initially given number based on an alternative hypothesis;
t - tabular values (Student's coefficient);
Sx is the standard mean error, which is calculated as Sx = σ / √n, where the numerator is the standard deviation and the denominator is the sample size.
So let's assume a situation. Before the repair, the conveyor produced 32.1 kg of final products per day, and after the repair, according to the entrepreneur, the efficiency increased, and the conveyor, according to a weekly check, began to produce 39.6 kg on average.
The null hypothesis will state that the repair did not affect the conveyor efficiency in any way. An alternative hypothesis would say that the repair drastically changed the efficiency of the conveyor, therefore, its productivity increased.
According to the table, we find n = 7, t = 2.447, from which the formula will take the following form:
39.6 - 2.447 * 4.2 ≤ s ≤ 39.6 + 2.447 * 4.2;
29.3 ≤ s ≤ 49.9.
It turns out that the value 32.1 is included in the range, and therefore the value proposed by the alternative - 39.6 - is not automatically accepted. Remember that the null hypothesis is tested first for correctness, and then the opposite.
Varieties of denial
Prior to this, such a variant of constructing a hypothesis was considered, where H0 asserts something, and H1 refutes it. Where could such a system be compiled from:
H0: c = c0;
Н1: с ≠ с0.
But there are two more related ways of refutation. For example, the null hypothesis states that the average grade for a class is greater than 4.54, while the alternative hypothesis will then say that the average grade for the same class is less than 4.54. And it will look like a system like this:
Н0: s ⩾ 4.54;
H1: s< 4.54.
Note that the null hypothesis states that the value is greater than or equal to, and the statistical hypothesis that it is strictly less. The severity of the inequality sign is of great importance!
Statistical check
Statistical testing of null hypotheses involves the use of a statistical test. Such criteria are subject to various distribution laws.
For example, there is an F-test that is calculated using the Fisher distribution. There is a T-test, most often used in practice, which depends on the Student distribution. Pearson's square goodness test, etc.
Null Hypothesis Acceptance Area
In algebra, there is the concept of "range of admissible values". This is such a segment or point on the X-axis, on which there is a set of statistic values for which the null hypothesis is true. The extreme points of the line segment are critical values. The rays on the right and left sides of the segment are critical areas. If the found value is included in them, then the zero theory is refuted and an alternative one is accepted.
Refutation of the null hypothesis
The null hypothesis in statistics is a very dodgy concept at times. When checking it, you can make two types of errors:
1. Rejection of the correct null hypothesis. Let's denote the first type as a = 1.
2. Acceptance of a false null hypothesis. The second type is designated as a = 2.
It should be understood that these are not the same parameters, the outcomes of errors can differ significantly from each other and have different samples.
An example of two types of errors
Complex concepts are easier to understand with an example.
During the production of a certain drug, scientists need to be extremely careful, since exceeding the dose of one of the components provokes a high level of toxicity of the finished drug, from which patients taking it can die. However, it is impossible to detect an overdose at the chemical level.
Because of this, before releasing a drug on the market, a small dose of it is tested on rats or rabbits, injecting them with the drug. If most of the subjects die, then the drug is not allowed on sale, if the test subjects are alive, then the drug is allowed to be sold in pharmacies.
The first case: in fact, the drug was not toxic, but during the experiment, an oversight was made and the drug was classified as toxic and was not allowed for sale. A = 1.
The second case: during another experiment, when testing another batch of a drug, it was decided that the drug was not toxic, and it was allowed for sale, although in fact the drug was poisonous. A = 2.
The first option will entail large financial costs for the supplier-entrepreneur, since you will have to destroy the entire batch of medicine and start from scratch.
The second situation will provoke the death of patients who bought and used this medicine.
Probability theory
Not only zero, but all hypotheses in statistics and economics are divided according to the level of significance.
Significance level - the percentage of occurrence of errors of the first kind (rejection of the correct null hypothesis).
The first level is 5% or 0.05, that is, the probability of a mistake is 5 to 100 or 1 to 20.
the second level is 1% or 0.01, that is, the probability is 1 in 100.
the third level is 0.1% or 0.001, the probability is 1 in 1000.
Hypothesis Testing Criteria
If the scientist has already concluded that the null hypothesis is correct, then it must be tested. This is necessary to rule out an error. There is a basic criterion for testing the null hypothesis, which consists of several stages:
1. Take the admissible error probability P = 0.05.
2. Selected statistics for criterion 1.
3. According to a well-known method, the range of admissible values is found.
4. Now the value of the T statistic is calculated.
5. If T (statistics) belongs to the area of acceptance of the null hypothesis (as in the "confidence" method), then the assumptions are considered true, and therefore the null hypothesis itself remains true.
This is how statistics work. The null hypothesis, if properly tested, will be accepted or rejected.
It is worth noting that for ordinary entrepreneurs and users, the first three stages can be very difficult to complete correctly, so they are trusted by professional mathematicians. On the other hand, stages 4 and 5 can be performed by any person who is sufficiently knowledgeable of statistical methods of verification.
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