Exercises for Homework 14

14.6. The rests of combustive-lubricating materials in a warehouse of enterprises were studied by random way without replacement. 110 enterprises of 750 have been surveyed. The average rests have made 150 tons at the mean square deviation 42 tons. Determine with confidential probability 0,95 borders in which there will be average rests of combustive-lubricating materials per one enterprise and the general rests of combustive-lubricating materials.

(rests – остатки; combustive-lubricating materials – горюче-смазочные материалы; warehouse – склад).

14.7. To define the humidity of a grain 25 tests have been taken by random way. The average percent of humidity of a grain has made 16 %, and the sample mean square deviation – 2,5 %. Determine: a) the unbiased estimations of the mathematical expectation, dispersion and mean square deviation; b) an interval which will cover the mathematical expectation with confidential probability 0,95.

(humidity – влажность; test – проба).

14.8. Find by the method of moments the pointwise estimations of the parameters  and  of the scale-distribution:

Direction: Make the substitution и, using the scale-function find firstly and equate

(scale-distribution – гамма-распределение; scale-function – гамма-функция; to equate – приравнять).

14.9. Find by the method of the greatest plausibility on sample x₁, x₂, …, x_n the pointwise estimation of the unknown parameter  of scale-distribution if the parameter  is known.

Direction: Use density of the scale-distribution presented in Exercise 14.8.

L E C T U R E 15

Testing of statistical hypotheses

The principle of practical confidence: If the probability of an event A in a given trial is very small, one can be confident at a single realization of the trial that the event A will not take place, and in practical activities one should behave as if the event A is impossible in general.

Testing of statistical hypotheses is closely connected to the theory of statistical estimation of parameters. It is used every time when it is necessary the substantiated conclusion on advantages of this or that way of investments, measurements, shooting, technological process, about efficiency of a new method of training, management, on benefit of brought fertilizer, medicine, on a level of profitableness of securities, on importance of a mathematical model, etc.

Any assumption on a kind or parameters of an unknown law of distribution is said to be a statistical hypothesis.

One distinguishes simple and complex statistical hypotheses. A simple hypothesis in contrast to complex completely determines the theoretical function of distribution of a random variable. For example, hypotheses «the probability of occurrence of the event in Bernoulli circuit is equal to 1/2», «the law of distribution of a random variable is normal with parameters a = 0,  ² = 1» are simple, and hypotheses «the probability of occurrence of the event in Bernoulli circuit is enclosed between 0,3 and 0,6», «the law of distribution is not normal» are complex.

Tested hypothesis is usually called null (or basic) hypothesis and is denoted by H₀. Alongside with a null hypothesis H₀ one considers alternative or competing hypothesis being logic denying of H₀. Null and alternative hypotheses represent two opportunities of the choice which is carried out in problems of testing of statistical hypotheses.

The essence of testing of a statistical hypothesis is that a specially composed sampling characteristic (statistics) received on sample X₁, X₂, …, X_n is used, of which exact or approximate distribution is known. Then on this sampling distribution a critical value _cr is determined – such that if the hypothesis H₀ is true, the probability is small; so in according to the principle of practical confidence in conditions of the given investigation the event can be considered (with some risk) as practically impossible. Therefore if in the given concrete case the deviation holds then the hypothesis H₀ is rejected, and if appearance of the value is considered compatible with the hypothesis H₀ then the hypothesis H₀ is accepted (more precisely, is not rejected). The rule, by which the hypothesis H₀ is rejected or accepted, is said to be a statistical criterion or statistical test.

Thus, the set of possible values of statistics of criterion (critical statistics) is partitioned into two disjoint subsets: critical area (area of rejection of the hypothesis) W and area of allowable values (area of acceptance of the hypothesis) . If actually observable value of statistics of criterion gets in critical area W, the hypothesis H₀ is rejected. Thus, four cases are possible:

The hypothesis H0	It is accepted	It is rejected
True	The correct decision	Mistake of the 1st sort
False	Mistake of the 2nd sort	The correct decision

Probability  to make a mistake of the 1st sort, i.e. to reject the hypothesis H₀ when it is true, is said to be a significance level (or a size) of criterion.

Probability 1–  not to make a mistake of the 1st sort, i.e. to accept the hypothesis H₀ when it is true, sometimes name the operative characteristic of criterion.

Probability to make a mistake of the 2^nd sort, i.e. to accept the hypothesis H₀ when it is false, is usually denoted by .

Probability 1 –  not to make a mistake of the 2^nd sort, i.e. to reject the hypothesis H₀ when it is false, is said to be power (or power function) of criterion.

Using terminology of statistical control of production quality, it is possible to tell that probability  represents «risk of the supplier» connected with rejection by results of sampling control of products of all party satisfying the standard, and probability  – «risk of the consumer» connected to acceptance under the analysis of sample of the party which are not satisfying the standard.

Applying the juridical terminology,  – probability of giving out by court of a verdict of guilty when actually an accused is innocent,  – probability of giving out by court of a verdict of "not guilty" when actually an accused is guilty in fulfillment of a crime. In a number of applied researches a mistake of the first sort  means probability of that a signal intending to the observer will not be accepted, and a mistake of the second sort  – probability of that the observer will accept a false signal.

Probabilities of mistakes of the 1st and 2nd sort ( and ) are uniquely determined by a choice of critical area. Obviously, it is desirable to make somehow small  and . However it is inconsistent requirements: at the fixed volume of sample it is possible to make somehow small only one of sizes –  or  that is connected to inevitable increase of another. It is possible only at increase of volume of sample a simultaneous reduction of probabilities  and .

What principles should to be guided by at construction of critical area W?

The requirements to critical area can be written down:

i.e. the critical area W should be chosen so that the probability of hit of statistics of criterion in this area would be minimal and equal to  if the hypothesis H₀ is true, and would be maximal if otherwise. Other words, the critical area must be such that at a given significance level  criterion power 1 –  would be maximal. The problem of construction of such a critical area W (or, say, construction of the most powerful criterion) for simple hypotheses is solved by the following theorem:

Theorem (lemma) of Neyman-Pearson. Among all criteria of a given significance level , testing the simple hypothesis H₀, the criterion of relation of plausibility is the most powerful.

Let's explain sense of the theorem, assuming a random variable of X is continuous.

If a simple hypothesis Н₀ is true, density of probability (x) is uniquely determined and a function of plausibility L₀(x) expressing density of probability of joint occurrence of results of sample (x₁, x₂, …, x_n) has the following form:

Let's remind that the function is a measure of plausibility of reception of sampling observations x₁, x₂, …, x_n.

Similarly, if an alternative hypothesis H₁ is true, function of plausibility has the following form:

The relation of plausibility L₁/L₀ (for L₀  0) is considered at Neyman-Pearson theorem: the more plausible a sample in conditions of the hypothesis H₁, the greater the relation L₁/L₀ or its logarithm ln (L₁/L₀). And criterion of this relation by the conclusion of theorem is the most powerful among other possible criteria.

Using this criterion, one can find such a constant C (or ln C = c) that

By means of the obtained constant C (or c) a critical area W of criterion and its power are determined.

The principle of testing of a statistical hypothesis does not give the logic proof of its fidelity or incorrectness. Acceptance of the hypothesis H₀ in comparison with the alternative hypothesis H₁ does not mean that we are sure in absolute correctness of H₀ or that the statement stated in the hypothesis H₀ is the best and unique suitable; simply the hypothesis H₀ does not contradict the sampling data available at us, and other hypotheses can possess the same property alongside with H₀. Thus, an acceptance of the hypothesis H₀ should be regarded not just and for ever established absolutely true fact, and only as plausible enough statement not contradicting to experience.

Under the applied maintenance statistical hypotheses can be subdivided into some basic types:

• on equality of numerical characteristics of parent populations;

• on numerical values of parameters;

• on the law of distribution;

• on homogeneity of samples (i.e. belonging of their same parent population);

• on stochastic independence of sample units.