Null hypothesis in statistics: an example. Null hypothesis testing
Formulation of hypotheses systematizes the assumptions of the researcher and presents them in a clear, concise manner. The decision that the researcher needs to make concerns the truth or falsity of the statistical hypothesis. There are two types of hypotheses: scientific and statistical. Scientific A hypothesis is a proposed solution to a problem (stated as a theorem). Statistical a hypothesis is simply a statement about an unknown parameter of the general population (property of a random variable or event), which is formulated to test the reliability of the relationship and which can be verified against known sample statistics (research results, available empirical data).
Statistical hypotheses subdivided into zero and alternative, directional and non-directional. Null hypothesis (H 0) this is a hypothesis about the absence of differences, the absence of the influence of a factor, the absence of an effect, etc.. This is what is supposed to be refuted if we are faced with the task of proving the significance of differences. Alternative hypothesis (H 1) it is a hypothesis about the significance of the differences. This is what is supposed to be proven, which is why it is sometimes called an experimental or working hypothesis.
herself the procedure for processing the obtained quantitative data, which consists in calculating some statistical characteristics and estimates that allow testing the null hypothesis, is called statistical analysis.
The null and alternative hypotheses can be directional or non-directional. The hypothesis is called directed if it contains an indication of the direction of the differences. Such hypotheses should be formulated, for example, in the event that in one of the groups the individual values of the subjects for any characteristic are higher, and in the other lower, or it is necessary to prove that in one of the groups under the influence of any experimental influences more pronounced changes than in the other group. The hypothesis is called non-directional, if its wording presupposes only the definition of differences or non-differences (without indicating the direction of differences). For example, if it is necessary to prove, in two different groups the forms of distribution of the trait differ.
Examples of formulating hypotheses.
The method that is used to decide on the validity of a statistical hypothesis is called hypothesis testing. The basic principle of hypothesis testing is that the null hypothesis is put forward. H 0, in order to try to refute it and thereby confirm the alternative hypothesis H 1 .
When testing any statistical hypothesis, the researcher's decision is never made with certainty, as there is always the risk of making the wrong decision.
Usually the samples used are small, and in these cases the probability of error can be significant. There is a so-called confidence level (significance level) differences. This is the probability that the differences are considered significant, but they are actually random. That is, it is the probability of deviation null hypothesis, while it is true.
When differences are stated to be significant at the 5% significance level, or at p£0.05, what is meant is that the probability that they are not significant after all is 0.05 (the lowest level). statistical significance). If a difference is stated to be significant at the 1% significance level, or at p£0.01, then it means that the probability that it is not significant after all is 0.01 (a sufficient level of statistical significance). If the differences are stated to be significant at the 0.1% significance level, or at p£0.001, then it means that the probability that they are still not significant is 0.001 ( highest level statistical significance).
The rule of rejection H 0 and acceptance H 1:
If the empirical value of the criterion equals or exceeds the critical value corresponding to p £ 0.05, then H 0 rejected, but not yet definitely accepted H 1.
If the empirical value of the criterion equals or exceeds the critical value corresponding to p £ 0.01, then H 0 rejected accepted H 1.
To visualize the decision rule, you can use the so-called "significance axis".
If the confidence level is not exceeded, then it can be considered probable that the revealed difference really reflects the state of affairs in the population. For each statistical method, this level can be found in the distribution tables of the critical values of the corresponding criteria.
T - Student's criterion
This is a parametric method used to test hypotheses about the validity of the difference in means when analyzing quantitative data in populations with a normal distribution and with the same variance. It is well applicable in the case of comparing the average random values of the measured trait in the control and experimental groups, in different sex and age groups, groups with other different characteristics.
A prerequisite for the applicability of parametric methods, including Student's t-test, to prove statistical hypotheses is the subordination empirical distribution of the characteristic under study to the law of normal distribution.
Student's method is different for independent and dependent samples.
Independent samples are obtained by studying two different groups of subjects (for example, control and experimental groups). To dependent samples include, for example, the results of the same group of subjects before and after exposure to the independent variable.
The tested hypothesis H 0 is that the difference between the means of the two samples is equal to zero ( = 0), in other words, this is the hypothesis about the equality of the means (). The alternative hypothesis H 1 is that this difference is non-zero ( ¹ 0) or there is a difference in the sample means ().
When independent samples to analyze the difference in the means, the formula is used: 
for n 1 , n 2 > 30
and formula 
for n 1 , n 2< 30, где
Arithmetic mean of the first sample;
Arithmetic mean of the second sample;
s 1 - standard deviation for the first sample;
s 2 - standard deviation for the second sample;
n 1 and n 2 are the number of elements in the first and second samples.
To find the critical value of t, we determine the number of degrees of freedom:
n \u003d n 1 - 1 + n 2 - 1 \u003d (n 1 + n 2) - 2 \u003d n - 2.
If |t emp | > t cr, then we discard the null hypothesis and accept the alternative one, that is, we consider the difference in the averages to be reliable. If |t emp |< t кр, то разница средних недостоверна.
When dependent samples the following formula is used to determine the reliability of the difference in the means: 
, where
d– the difference between the results in each pair (х i – y i);
å d is the sum of these partial differences;
å d2 is the sum of squared partial differences;
n is the number of data pairs.
The number of degrees of freedom in the case of dependent samples to determine the t criterion will be equal to n = n - 1.
There are other statistical criteria for testing hypotheses, both parametric and non-parametric. For example, a mathematical-statistical criterion that allows one to judge the similarities and differences in the dispersions of random variables is called the Fisher criterion.
In its most general form, the meaning of "correlation" refers to a mutual relationship. Although, speaking of correlation, the terms "correlation" and "correlation dependence" are also used, which are often used as synonyms.
Under correlation understand the coordinated changes of two or more features, i.e. the variability of one trait is in some correspondence with the variability of another.
Correlation dependence are the changes that the values of one feature make to the probability of occurrence of different values of another feature.
Thus, coordinated changes in traits and the correlation between them reflecting this may indicate not the dependence of these traits among themselves, but the dependence of both of these traits on some third trait or combination of traits not considered in the study.
Let's get acquainted with the terminology used in hypothesis testing.
But - the null hypothesis (the skeptic's hypothesis) is a hypothesis about no difference between compared samples. The skeptic believes that the differences between the sample estimates obtained from the results of the research are random.
· Н 1 – an alternative hypothesis (optimist's hypothesis) is a hypothesis about the presence of differences between the compared samples. The optimist believes that the differences between sample estimates are due to objective reasons and correspond to the differences populations
Testing of statistical hypotheses is feasible only when the elements of the compared samples can be used to compose some value(criterion), the distribution law of which is known in the case of validity H 0 . Then, for this quantity, one can specify confidence interval, in which given probability R d hits its value. This interval is called critical area. If the value of the criterion falls within the critical region, then the hypothesis H 0 is accepted. Otherwise, the hypothesis H 1 is accepted.
In medical research, P d = 0.95 or P d = 0.99 is used. These values correspond significance levels a = 0.05 or a = 0.01.
When testing statistical hypotheses significance level(a) is the probability of rejecting the null hypothesis when it is true.
Note that, at its core, the hypothesis testing procedure aimed at finding differences rather than confirming their absence. When the criterion value goes beyond the critical area, we can say “skeptic” with a pure heart - well, what else do you want?! If there were no differences, then with a probability of 95% (or 99%) the calculated value would be within the specified limits. So no!...
Well, if the value of the criterion falls into the critical region, then there is no reason to believe that the hypothesis H 0 is true. This most likely points to one of two possible causes.
a) Sample sizes are not large enough to detect differences. It is likely that continued experimentation will bring success.
b) There are differences. But they are so small that they are of no practical importance. In this case, the continuation of experiments does not make sense.
Let's move on to consider some of the statistical hypotheses used in medical research.
§ 3.6. Testing hypotheses about the equality of variances,
F - Fisher criterion
In some clinical studies, a positive effect is evidenced not so much by magnitude parameter under study, how much stabilization, reducing its fluctuations. In this case, the question arises of comparing two general variances based on the results of a sample survey. This task can be solved using Fisher's criterion.
Formulation of the problem
normal law distribution. Sample sizes n 1 and n 2 , and sample variances are equal respectively. Needs to be compared general variances.
Tested hypotheses:
H 0– general dispersions are the same;
H 1 - general variances different.
Shown if samples are drawn from populations with normal law distribution, then if the hypothesis H 0 is true, the ratio of sample variances obeys the Fisher distribution. Therefore, as a criterion for checking the validity of H 0, the value F, calculated by the formula
where are the sample variances.
This ratio obeys the Fisher distribution with the number of degrees of freedom of the numerator n 1 = n 1 -1, and the number of degrees of freedom of the denominator n 2 = n 2-1. The boundaries of the critical region are found using Fisher distribution tables or using the computer function FDISP.
For the example presented in Table. 3.4, we get: n 1 \u003d n 2 \u003d 20 - 1 \u003d 19; F = 2.16/4.05 = 0.53. At a = 0.05, the boundaries of the critical region are equal, respectively: F left = 0.40, F right = 2.53.
The value of the criterion fell into the critical region, so the hypothesis H 0 is accepted: the general variances of the samples are the same.
§ 3.7. Testing hypotheses regarding the equality of means,
t-Student's test
Comparison problem medium two general populations arises when it is the magnitude the trait under study. For example, when comparing the duration of treatment with two different methods, or the number of complications that occur when using them. In this case, Student's t-test can be used.
Formulation of the problem.
Two samples (X 1 ) and (X 2 ) are obtained from populations with normal law distribution and equal variances. Sample sizes n 1 and n 2 , sample means are equal, and sample variances- , respectively. Needs to be compared general averages.
Tested hypotheses:
H 0– general averages are the same;
H 1 - general averages different.
It is shown that in the case of the validity of the hypothesis H 0, the value t, calculated by the formula
, (3.10)
distributed according to Student's law with the number of degrees of freedom n= n 1 + n 2 - 2.
Here where n 1 = n 1 - 1 - the number of degrees of freedom for the first sample; n 2 = n 2 – 1 is the number of degrees of freedom for the second sample.
The boundaries of the critical region are found from the tables t-distribution or with the help of the computer function STUDRASP. The Student's distribution is symmetrical about zero, so the left and right boundaries of the critical region are the same in absolute value and opposite in sign: - t gr and t gr.
For the example presented in Table. 3.4, we get: n 1 \u003d n 2 \u003d 20 - 1 \u003d 19; t= –2.51, n= 38. At a = 0.05 tgr = 2.02.
The value of the criterion goes beyond the left border of the critical region, so we accept the hypothesis H 1: general averages different. At the same time, the average of the general population first sample less.
STATISTICAL CHECK OF STATISTICAL
The concept of statistical hypothesis.
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 verified statistically, that is, based on the results of observations in a random sample. Two types of statistical hypotheses are considered: hypotheses about the laws of distribution general population and hypotheses about parameters known distributions.
Thus, the hypothesis that the time spent on assembling a machine assembly in a group of machine shops that produce products of the same name and have approximately the same technical and economic production conditions is distributed according to the normal law is a hypothesis about the law of distribution. And the hypothesis that the productivity of workers in two teams performing the same work under the same conditions does not differ (while the productivity of workers in each team 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 opposed competing or alternative hypothesis, which is H one . As a rule, the competing hypothesis H 1 is a logical negation of the main hypothesis H 0.
An example of a null hypothesis would be that the means of two normally distributed populations are equal, then the competing hypothesis might consist of the assumption that the means are not equal. Symbolically it is written like this:
H 0: M(X) = M(Y); H 1: M(X) M(Y) .
If the null (proposed) hypothesis is rejected, then there is a competing hypothesis.
There are simple and complex hypotheses. If a 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 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 than p 0 .
The proposed statistical hypothesis may be correct or incorrect, so it is necessary to 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 composed 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 is probabilistic in nature: there is always a risk of making a mistake when accepting (rejecting) a hypothesis. In this case, errors of two kinds are possible.
Type I error is that the null hypothesis will be rejected even though it is in fact true.
Type II error is that the null hypothesis will be accepted, although the competing one is in fact true.
In most cases, the consequences of these errors are unequal. What is better or worse depends on the specific formulation of the problem and the content of the null hypothesis. Consider examples. Assume that at the enterprise the quality of products is judged by the results of selective control. If the sample fraction of marriage does not exceed a predetermined value p 0 , then the batch is accepted. In other words, the null hypothesis is put forward: H 0: p p 0 . If a Type I error is made in testing this hypothesis, we will reject the good product. If an error of the second kind is made, then the reject will be sent to the consumer. Obviously, the consequences of a Type II error can be much more serious.
Another example can be given from the field of jurisprudence. We will consider the work of judges as actions to verify the presumption of innocence of the defendant. The main hypothesis to be tested is the hypothesis H 0 : the defendant is innocent. Then the alternative hypothesis H 1 is the hypothesis: the accused is guilty of a crime. It is obvious that the court may make errors of the first or second kind in sentencing the defendant. If a mistake of the first kind is made, then this 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 delivered a verdict of not guilty, when in fact the accused is guilty of a crime. Obviously, the consequences of an error of the first kind for the accused will be much more serious, while for society the consequences of an error of the second kind are the most dangerous.
Probability commit mistake first kind called significance level criteria and denote .
In most cases, the significance level of the criterion is taken equal to 0.01 or 0.05. If, for example, the significance level is taken equal to 0.01, then this means that in one case in a hundred there is a risk of making a type I error (that is, rejecting the correct null hypothesis).
Probability commit type II error denote . Probability 
not making a Type II error, that is, rejecting the null hypothesis when it is false, is called the power of the criterion.
Statistical criterion.
Critical areas
A 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 ( random value distributed according to the Fisher-Snedekor 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 non-overlapping subsets: one of them contains the values of the criterion under which the null hypothesis is accepted, and the other - under which it is rejected.
The set of test values under which the null hypothesis is rejected is called critical area. We will denote the critical region by W.
The set of criterion values under which the null hypothesis is accepted is called hypothesis acceptance area(or range of acceptable values of the criterion). We will refer to this area as 
.
To test the validity of the null hypothesis, according to the sample data, we calculate observed criterion value. We will denote it To obs.
The basic principle of testing statistical hypotheses can be formulated as follows: if the observed value of the criterion fell into the critical region (that is, 
), then the null hypothesis is rejected; if the observed value of the criterion fell into the area of accepting the hypothesis (that is, 
), then there is no reason to reject the null hypothesis.
What principles should be followed when constructing a critical region W ?
Let's assume that the hypothesis H 0
is actually true. Then hitting the criterion 
into the critical region, by virtue of the basic principle of testing statistical hypotheses, entails the rejection of the correct hypothesis H 0
, which means making a Type I error. Therefore, the probability of hitting 
to the region W if the hypothesis is true H 0
should be equal to the significance level of the criterion, i.e.
.
Note that the probability of making a Type I error is chosen to be 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 sampling data, the event 
nevertheless occurred, then it can be considered incompatible with the hypothesis H 0
(which as a result is rejected), but compatible with the hypothesis H 1
(which is eventually accepted).
Let us now assume that the hypothesis is true H 1
. Then hitting the criterion 
into the area of acceptance of the hypothesis 
leads to the adoption of an incorrect hypothesis H 0
which means committing a Type II error. That's why 
.
Since the 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 criterion if the hypothesis H 1
true, that is
.
Obviously, the critical region should be chosen so that, at a given level of significance, the power of the criterion 
was maximum. Maximizing the power of the test will provide a minimum probability of making a Type II error.
It should be noted that no matter how small the value of the significance level , the criterion falling into the critical region is only an unlikely, but not absolutely impossible event. Therefore, it is possible that with a true null hypothesis, the value of the criterion calculated from the sample data will still be in the critical region. Rejecting the hypothesis in this case H 0 , we make a Type I error with probability . The smaller , the less likely it is to make a Type I error. However, with a decrease, 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. At =0 hypothesis H 0 will always be accepted regardless of the sample results. Therefore, a decrease entails an increase in the probability of accepting an incorrect null hypothesis, that is, making a Type II error. 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 the losses from these errors. Of course, it is desirable to reduce both errors simultaneously, but since they are competing, a decrease in the probability of making one of them leads to an increase in the probability of making the other. The only way simultaneous decrease the risk of error lies in increasing the sample size.
Depending on the type of competing hypothesis H 1
are building one-sided (right-sided and left-sided) and two-sided critical regions. Points separating the critical region 
from the area of acceptance of the hypothesis 
, called critical points and denote k Crete. For finding the critical region you need to know the critical points.
right hand 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 a region consists of points located on the right side of the critical point k Crete. pr, that is, it contains a set of positive and sufficiently large values of the criterion TO. For finding k Crete. pr set first the significance level of the criterion . Next, the right critical point k Crete. pr is found from the condition . Why exactly this requirement defines a right-handed critical region? Since the probability of an event (TO>k Crete. etc )
is small, then, due to the principle of the practical impossibility of unlikely events, this event should not occur if the null hypothesis is true in a single test. If, nevertheless, it has come, that is, the observed value of the criterion calculated from the data of the samples 
turned out to be more k Crete. pr, this can be explained by the fact that the null hypothesis is not consistent with the observational data and therefore should be rejected. Thus the requirement 
determines such values of the criterion under which the null hypothesis is rejected, and they constitute the right-hand 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 the observational data. Note that the probability of hitting the criterion 
into the range of acceptable values 
if the null hypothesis is true, it is equal to (1-) and close to 1.
It must be remembered that the hit of the criteria 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 proposed hypothesis and the results of the sample. Therefore, in such cases, we say that the observational data are consistent with the null hypothesis and there is no reason to reject it.
Other critical regions are constructed similarly.
So, lleft-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 modulo 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 a value less than k crit.l, is equal to the accepted level of significance if the null hypothesis is true.
bilateral critical region 
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 sufficiently large modulo values of the criterion. 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 true, that is
(TO<
k Crete. l )+
(TO>k Crete. etc )=
.
If the distribution of the criterion To symmetrical about the origin, then the critical points will be located symmetrically about zero, so k Crete. l = - k Crete. etc. Then the two-sided critical region becomes symmetric and can be described by the following inequality: > k Crete. dw, where k Crete. dw = k Crete. pr Critical point k Crete. dw 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. Results of numerical calculations
k crit are given in the corresponding tables (see, for example, appendix 4 - 6 in the file "Appendices").
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 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 agrees with the observational data that we have, 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.
5. Main problems of applied statistics - data description, estimation and testing of hypotheses
Key Concepts Used in Hypothesis Testing
Statistical hypothesis - any assumption concerning the unknown distribution of random variables (elements). Here are the formulations of several statistical hypotheses:
1. The results of observations have normal distribution with zero mathematical expectation.
2. The results of observations have a distribution function N(0,1).
3. The results of observations have a normal distribution.
4. The results of observations in two independent samples have the same normal distribution.
5. The results of observations in two independent samples have the same distribution.
There are null and alternative hypotheses. The null hypothesis is the hypothesis to be tested. An alternative hypothesis is every valid hypothesis other than the null hypothesis. The null hypothesis is H 0 , alternative - H 1(from Hypothesis - “hypothesis” (English)).
The choice of one or another null or alternative hypotheses is determined by the applied tasks facing the manager, economist, engineer, researcher. Consider examples.
Example 11. Let the null hypothesis be hypothesis 2 from the list above, and the alternative hypothesis be hypothesis 1. This means that the real situation is described by a probabilistic model, according to which the results of observations are considered as realizations of independent identically distributed random variables with a distribution function N(0,σ), where the parameter σ is unknown to the statistician. In this model, the null hypothesis is written as follows:
H 0: σ = 1,
and an alternative like this:
H 1: σ ≠ 1.
Example 12. Let the null hypothesis be still hypothesis 2 from the above list, and the alternative hypothesis be hypothesis 3 from the same list. Then, in a probabilistic model of a managerial, economic, or production situation, it is assumed that the results of observations form a sample from a normal distribution N(m, σ) for some values m and σ. Hypotheses are written like this:
H 0: m= 0, σ = 1
(both parameters take fixed values);
H 1: m≠ 0 and/or σ ≠ 1
(i.e. either m≠ 0, or σ ≠ 1, or both m≠ 0, and σ ≠ 1).
Example 13 Let H 0 is hypothesis 1 from the above list, and H 1 - hypothesis 3 from the same list. Then the probabilistic model is the same as in example 12,
H 0: m= 0, σ is arbitrary;
H 1: m≠ 0, σ is arbitrary.
Example 14 Let H 0 is hypothesis 2 from the above list, and according to H 1 observational results have a distribution function F(x), not matching the standard normal distribution function F(x). Then
H 0: F(x) = F(x) for all X(written as F(x) ≡ F(x));
H 1: F(x 0) ≠ F (x 0) at some x 0(i.e. it is not true that F(x) ≡ F(x)).
Note. Here ≡ is the sign of the identical coincidence of functions (i.e., coincidence for all possible values of the argument X).
Example 15 Let H 0 is hypothesis 3 from the above list, and according to H 1 observational results have a distribution function F(x), not being normal. Then
For some m, σ;
H 1: for any m, σ there is x 0 = x 0(m, σ) such that 
.
Example 16 Let H 0 - hypothesis 4 from the above list, according to the probabilistic model, two samples are drawn from populations with distribution functions F(x) and G(x), which are normal with parameters m 1 , σ 1 and m 2 , σ 2 respectively, and H 1 - negation H 0 . Then
H 0: m 1 = m 2 , σ 1 = σ 2 , and m 1 and σ 1 are arbitrary;
H 1: m 1 ≠ m 2 and/or σ 1 ≠ σ 2 .
Example 17. Let, under the conditions of Example 16, it is additionally known that σ 1 = σ 2 . Then
H 0: m 1 = m 2 , σ > 0, and m 1 and σ are arbitrary;
H 1: m 1 ≠ m 2 , σ > 0.
Example 18. Let H 0 - hypothesis 5 from the above list, according to the probabilistic model, two samples are drawn from populations with distribution functions F(x) and G(x) respectively, and H 1 - negation H 0 . Then
H 0: F(x) ≡ G(x) , where F(x)
H 1: F(x) and G(x) are arbitrary distribution functions, and
F(x) ≠ G(x) with some X.
Example 19. Let, in the conditions of Example 17, it is additionally assumed that the distribution functions F(x) and G(x) differ only in the shift, i.e. G(x) = F(x- a) at some a. Then
H 0: F(x) ≡ G(x) ,
where F(x) is an arbitrary distribution function;
H 1: G(x) = F(x- a), a ≠ 0,
where F(x) is an arbitrary distribution function.
Example 20. Let, in the conditions of Example 14, it is additionally known that according to the probabilistic model of the situation F(x) is a normal distribution function with unit variance, i.e. has the form N(m, one). Then
H 0: m = 0 (those. F(x) = F(x)
for all X); (written as F(x) ≡ F(x));
H 1: m ≠ 0
(i.e. it is not true that F(x) ≡ F(x)).
Example 21. In the statistical regulation of technological, economic, managerial or other processes, consider a sample drawn from a population with a normal distribution and known variance, and hypotheses
H 0: m = m 0 ,
H 1: m= m 1 ,
where parameter value m = m 0 corresponds to the established course of the process, and the transition to m= m 1 indicates a breakdown.
Example 22. With statistical acceptance control, the number of defective product units in the sample obeys a hypergeometric distribution, the unknown parameter is p = D/ N is the defect level, where N- the volume of the batch of products, D – total number defective items in a batch. Used in regulatory, technical and commercial documentation (standards, supply contracts, etc.), control plans are often aimed at testing a hypothesis
H 0: p < AQL
H 1: p > LQ,
where AQL – acceptance level of defectiveness, LQ is the defectiveness level of defects (obviously, AQL < LQ).
Example 23. As indicators of the stability of a technological, economic, managerial or other process, a number of characteristics of the distributions of controlled indicators are used, in particular, the coefficient of variation v = σ/ M(X). Need to test the null hypothesis
H 0: v < v 0
under the alternative hypothesis
H 1: v > v 0 ,
where v 0 is some predetermined boundary value.
Example 24. Let the probabilistic model of two samples be the same as in Example 18, let us denote the mathematical expectations of the results of observations in the first and second samples M(X) and M(At) respectively. In some situations, the null hypothesis is tested
H 0: M(X) = M(Y)
against the alternative hypothesis
H 1: M(X) ≠ M(Y).
Example 25. It was noted above great importance in mathematical statistics distribution functions that are symmetric with respect to 0. When checking the symmetry
H 0: F(- x) = 1 – F(x) for all x, otherwise F arbitrary;
H 1: F(- x 0 ) ≠ 1 – F(x 0 ) at some x 0 , otherwise F arbitrary.
In probabilistic-statistical decision-making methods, many other formulations of problems for testing statistical hypotheses are also used. Some of them are discussed below.
The specific task of testing a statistical hypothesis is fully described if the null and alternative hypotheses are given. The choice of a method for testing a statistical hypothesis, the properties and characteristics of the methods are determined by both the null and alternative hypotheses. To test the same null hypothesis under different alternative hypotheses, generally speaking, different methods should be used. So, in examples 14 and 20, the null hypothesis is the same, while the alternative ones are different. Therefore, in the conditions of example 14, methods based on fit criteria with a parametric family (Kolmogorov type or omega-square type) should be used, and in the conditions of example 20, methods based on Student's test or Cramer-Welch test. If, in the conditions of example 14, the Student's criterion is used, then it will not solve the tasks set. If, in the conditions of Example 20, we use a Kolmogorov-type goodness-of-fit criterion, then, on the contrary, it will solve the tasks set, although, perhaps, worse than the Student's criterion specially adapted for this case.
When processing real data, the correct choice of hypotheses is of great importance. H 0 and H one . The assumptions made, such as the normality of the distribution, must be carefully justified, in particular, statistical methods. Note that in the vast majority of specific applied settings, the distribution of observation results is different from normal.
A situation often arises when the form of the null hypothesis follows from the formulation of the applied problem, and the form of the alternative hypothesis is not clear. In such cases, an alternative hypothesis should be considered. general view and use methods that solve the problem for all possible H one . In particular, when testing hypothesis 2 (from the list above) as null, one should use as an alternative hypothesis H 1 from example 14, and not from example 20, if there are no special justifications for the normality of the distribution of the results of observations under the alternative hypothesis.
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Based on the collected statistical studies data after their processing, conclusions are drawn about the studied phenomena. These conclusions are made by putting forward and testing statistical hypotheses.
Statistical hypothesis any statement about the form or properties of the distribution of random variables observed in the experiment 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 , negating the main . Thus, as a result of the test, one and only one of the hypotheses will be accepted , and the second one will be rejected.
Error types. The put forward hypothesis is tested on the basis of a study of a sample obtained from the general population. Due to the randomness of the sample, the test does not always draw the correct conclusion. In this case, the following situations may occur:
1. The main hypothesis is true and it is accepted.
2. The main hypothesis is true, but it is rejected.
3. The main hypothesis is not true and it is rejected.
4. The main hypothesis is not true, but it is accepted.
In case 2, one speaks of error of the first kind, in the latter case it is error of the second kind.
Thus, for some samples, the correct decision is made, and for others, the wrong one. The decision is made according to the value of some sampling function, called statistical characteristic
, statistical criterion or simply statistics. The set of values of this statistic can be divided into two non-overlapping subsets:
- H 0 is accepted (not rejected), called hypothesis acceptance area (allowable area);
- subset of 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 called a random variable K , which allows you to accept or reject the null hypothesis H0 .
- When testing hypotheses, errors of 2 kinds can be made.
Type I error is to reject the hypothesis H 0 if it is true ("skip target"). The probability of making a Type I error is denoted by α and is called significance level. 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 false ("false positive"). The probability of this kind of error is denoted by β.
Hypothesis classification
Main hypothesis H 0 about the value of the unknown parameter q of the distribution usually looks like this:H 0: q \u003d q 0.
Competing hypothesis H 1 may look like this:
H 1: q < q 0 , H 1:q> q 0 or H 1: q ≠ q 0 .
Accordingly, it turns out left side, right side or bilateral critical areas. Boundary points of critical regions ( critical points) is determined from the distribution tables of the relevant statistics.
When testing a hypothesis, it is reasonable to reduce the likelihood of making wrong decisions. Permissible Type I Error Probability usually denoted a and called significance level. Its value is usually small ( 0,1, 0,05, 0,01, 0,001
...). But a decrease in the probability of a type 1 error leads to an increase in the probability of a type 2 error ( b), i.e. the desire to accept only true hypotheses causes an increase in the number of rejected correct hypotheses. Therefore, the choice of significance level is determined by the importance of the problem posed and the severity of the consequences of an incorrect decision.
Testing a statistical hypothesis consists of the following steps:
1) definition of hypotheses H 0 and H 1 ;
2) selection of statistics and assignment of significance level;
3) definition critical points K cr and critical area;
4) calculation of the value of statistics from the sample K ex;
5) comparison of the statistics value with the critical region ( K cr and K ex);
6) decision making: if the value of the statistic is not included in the critical region, then the hypothesis is accepted H 0 and reject the hypothesis H 1 , and if it enters the critical region, then the hypothesis is rejected H 0 and the hypothesis is accepted H one . At the same time, the results of testing the statistical hypothesis should be interpreted as follows: if the hypothesis is accepted H 1 ,
then we can consider it proven, and if we accept the hypothesis 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.
Hypothesis Test Classification
Let us further consider several different statistical hypotheses and mechanisms for testing them.I) Hypothesis of 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 a significance level of α, it is necessary to test the hypothesis H 0: x=a. As an alternative, one of the three hypotheses discussed above can be used. In this case, the statistic is a random variable , which has 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 by 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 from 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. From the table of the Laplace function, the critical value is found z cr. Under the alternative hypothesis H 1: x >y it is found from the condition F(z cr) = 0,5 – a. If a z ex< z кр , then the null hypothesis is accepted, otherwise it is rejected. Under the 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 means of normally distributed general populations, the variances of which are unknown and the same (small independent samples). At a significance level of α, it is necessary to test the main hypothesis H 0: x=y . As a statistic, we use a random variable
,
which has a Student distribution with ( n x + n– 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) The hypothesis of the equality of two variances of normally distributed populations. In this case, at the significance level a need to test the hypothesis H 0: D(X) = D(Y). The statistic is a random variable , which has 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 the alternative hypothesis H 1: D(X) > D(Y) is found from the table of critical points of the Fisher-Snedecor distribution by significance level 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 populations over samples of the same size. In this case, at the significance level a need to test the hypothesis H 0: D(X 1) = D(X 2) = …= D(Xl). The statistic is a random variable 
, which has the Cochran 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. The table of critical points of the Cochran distribution is used.
vi) Hypothesis about the significance 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 equal to zero, then the corresponding quantities are not related to each other). In this case, the statistic is a random variable 
,
having a Student's distribution with f = n– 2 degrees of freedom. The verification of this hypothesis is carried out similarly to the verification of hypothesis (I).
Instruction. Specify the amount of source data.
VII) Hypothesis about the value of the probability of occurrence of an event. Sufficiently large number of n independent trials in which the event BUT happened m once. There is reason to believe that the probability of this event occurring in one trial is equal to p 0. Required at significance level a test the hypothesis that the probability of an event BUT equal to the hypothetical probability p 0. (Because the probability is estimated by the relative frequency, the tested hypothesis can be formulated in another way: the observed relative frequency and the hypothetical probability differ significantly or not).
The number of trials is quite large, so the relative frequency of the event BUT distributed according to the normal law. If the null hypothesis is true, then its expected value 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.
