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STATISTICAL HYPOTHESIS
Hypothesis
The previous sections have showed how can estimate a parameter from the data contained in a sample. It can find already it was an only number (punctual estimator) or an interval of possible values (interval of confidence). However, a lot of problems of engineering, science, and administration, require that it take a decision between accepting or refuse a proposition on some parameter. This proposition receives the name of hypothesis. This is one of the most useful appearances of the statistical inference, since a lot of types of problems of taking of decisions, proofs or experiments in the world of the engineering, can formulate like problems of proof of hypothesis.
A statistical hypothesis is a proposition or supposed on the parameters of one or more populations.
• Invalid hypothesis
The invalid hypothesis, represented byH or, is the affirmation on one or more characteristic of populations that to the start supposes some (that is to say, the "belief a priori").
• Alternative hypothesis
The alternative hypothesis, represented byH 1, is the contradictory affirmation toH or, and this is the hypothesis of the researcher.
The invalid hypothesis refuses in favour of the alternative hypothesis, only if the evidence muestral suggests that Hor is false. If the sample does not contradict decididamente toH or, continues believing in the validity of the invalid hypothesis. Then, the two possible conclusions of an analysis by proof of hypothesis are to refuse Hor or not refusing Hor.
Types of Essay
Can present three types of essay of hypothesis that are:
• Unilateral Right
• Unilateral Izquierdo
• Bilateral
Depending on the evaluation that want to do will select the type of essay.
• Unilateral Right. The researcher wishes to check the hypothesis of an increase in the parameter, in this case the level of significancia loads everything to the right side, to define the regions of acceptance and of rejection.
Essay of hypothesis:
HOr; Parameter x
H1; Parameter > x
• Unilateral Izquierdo: The researcher wishes to check the hypothesis of a decrease in the parameter, in this case the level of significancia loads everything to the left side, to define the regions of acceptance and of rejection.
Essay of hypothesis:
HOr; Parameter x
H1; Parameter < x
• Bilateral: The researcher wishes to check the hypothesis of a change in the parameter. The level of significancia divides in two and exist two regions of rejection.
Essay of hypothesis:
HOr; Parameter = x
H1; Parameter x
Nature of a random test
Nature of a deterministic test. The scientific method acts like this:
Step 1: Pose the hypotheses H0 ≡ Newtonian Theory is some vs. H1 ≡ is false
Step 2: Take the data (planetary orbits)
Step 3: Check if the data are satisfied with what predicts H0:
a) Data yes satisfied with H0 Accept H0: it is not reliable, always can have other data -no taken- that contradict it (the Science is a continuous approximation to the TRUTH).
b) b) Data no satisfied with H0 Rejection H0 Accept H1: yes it is reliable.
Nature of a random test The Statistics acts the same, but sees forced to alter the Step 3.
Example:
We suppose that we confront us to the one who advance will be called "opponent" in the following game: the opponent provides a coin that launches to the air, winning he if it goes out expensive and the reader if it goes out cross. If the opponent is not of fiar, is logical that the reader take his cautions requesting test previously the coin a number of, say n=10 times and deciding, from the results obtained, if the opponent is or no tramposo (that is to say, if the reader participates or no in the game).
Hypothesis
• H0 ≡ himself play ≡ The player is honoured ≡ p = 0,5 (p = % expensive).
• H1≡ do not play ≡ The player is fullero ≡ p>0,5 (gives too many faces).
Data
• Launch n=10 times the coin: size of sample.
• Observe the 10 results (C, C, F, …, F): random sample.
• Fix in the notable characteristic for the problem posed number of expensive x: statistician of contrast.
• With this x → B(n=10; p=unknown) and when observing it obtains x = xexp: experimental value (of the statistician of contrast).
Rule of Decision
• If xexp=10 will be able to seem rare … but is possible! (The binomial takes values of 0 to 10) always would conclude H0! It is necessary to change the rule of the deterministic case!
• Check if the result is unlikely under H0 (p=50%): to) Resulted no unlikely under H0 (like xexp=5) accept H0: it is not reliable, as a wrong coin (p>50%) can give 5 faces. b) Resulted yes unlikely under H0 (like xexp=10) rejection H0 accept H1: it is not reliable, as a correct coin (p=50%) can that it give 10 faces.
• In Statistics the two conclusions can be erroneous: the notable is that these errors occur with little probability (error α if it concludes H1; error β if it concludes H0).
• New Step 3). Check if the data are unlikely or no under H0. If the data are unlikely, refuses H0 and accepts H1. If the data are not unlikely, accepts H0
In general, to all number that, obtained from the observations of a sample (such is x), serves to decide by H0 or by H1, calls him statistical of contrast. In our case the statistician of contrast will be X = "number of faces obtained when launching 10 times the coin" Observe that X is a v.To. With distribution B(n=10;p)
Types of errors
Any one was the decision taken from a proof of hypothesis, already was of acceptance of the Ho or of Has it, can incur in error:
A error type I presents if the invalid hypothesis Ho is refused when it is true and had to be accepted. The probability to commit an error type I designates with the letter alpha α
A error type II, denotes with the Greek letter β presents if the invalid hypothesis is accepted when in fact it is false and had to be refused.
In any one of the two cases commits an error when taking a mistaken decision.
In the following table show the decisions that can take the researcher and the possible consequences.
So that any essay of hypothesis was well, has to design so that it minimise the errors of decision. In the practice a type of error can have more importance that the another, and like this has to achieve put a limitation to the error of greater importance. The only form to reduce both types of errors is to increase the size of the sample, which can be or not being possible.
The probability to commit an error of type II denoted with the Greek letter beta β, depends on the difference between the supposed and real values of the parameter of the population. As it is easier to find big differences, if the difference between the statistics of sample and the corresponding parameter of population is big, the probability to commit an error of type II, probably was small.
The study and the conclusions that obtain for a population any one, will have supported exclusively in the analysis of a part of this. Of the probability with which are had to assume these errors, will depend, for example, the size of the sample required. The contrastaciones support in that the data of game follow a normal distribution
It exists a reverse relation between the magnitude of the errors α and β: according to increase, β it diminishes. This forces to establish with care the value of to for the statistical proofs. The ideal would be to establish α and β.In the practice establishes the level α and to diminish the Error β increases the number of observations in the sample, as like this they shorten limit them of confidence with regard to the hypothesis posed .It put it of the statistical proofs is to refuse the hypothesis posed. In other words it is desirable to increase when this is true, or was, increase what calls power of the proof (1- β) The acceptance of the hypothesis posed has to interpret like that the random information of the available sample does not allow to detect the falsity of this hypothesis
Intervals of confidence
In statistics, calls interval of confidence to a pair or several pairs of numbers between which estimates that it will be some unknown value with a determinate probability of tarpaulin. Formally, these numbers determine a interval, that calculates from data ofa sample, and the unknown value is a populational parameter. The probability of success in the estimate represents with 1 - α and designates level of confidence. In these circumstances, α is the called random error or level of significance, this is, a measure of the possibilities to fail in the estimate by means of such interval.1
The level of confidence and the amplitude of the interval vary jointly, so that a wider interval will have more probability of tarpaulin (greater level of confidence), whereas for a smaller interval, that offers a more precise estimate, increases his probability of error.
For the construction of a determinate interval of confidence is necessary to know the theoretical distribution that follows the parameter to estimate, θ.2 it Is usual that the present parametera normal distribution. Also they can build intervals of confidence withthe inequality of Chebyshev.
In definite, an interval of confidence to the 1 - α percent for the estimate of a populational parameter θ that follows a determinate distribution of probability, is an expression of the type [θ1, θ2] such that P[θ1 ≤ θ ≤ θ2] = 1 - α, where P is thefunction of distribution of probability of θ
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