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Lecture, On, TEST OF SIGNIFICANCE, ➢Hypothesis:-, , Population, , ➢Critical region :➢Level of significance:➢Decision error :➢Degree of freedom:-, , sample

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Introduction :, The main problems in statistical inference can be broadly classified in, to two areas:, 1. The area of estimation of population parameter., 2. Tests of statistical hypothesis ., Hypothesis : A hypothesis is a claim (assumption) about one or more, population parameters., Statistical hypothesis: a statistical hypothesis is some statement or, assertion about a population or equivalently about the probability, distribution characterizing a population, which we want to verify on the, basis of information available from a sample

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Statistical hypothesis are classified in two types i.e. simple and, composite, When a hypothesis specifies all the parameters of a probability, distribution is kwon as simple hypothesis., If the hypothesis specifies only some of the parameters of a, probability distribution is kwon as composite hypothesis, Test of a statistical hypothesis: a Test of a statistical hypothesis is a, two action decision problem after the experimental sample values have, been obtained, the two actions being the acceptance or rejection of the, hypothesis under consideration.

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Critical region:, The region of rejection of 𝑯𝟎 when 𝑯𝟎 is True is that region of the, outcome set where 𝑯𝟎 is rejected if the sample point false in that region, and is called critical region.

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Decision error: The decision to accept or reject the null hypothesis is, made on the basis of the information supplied by the observed sample, observations. The conclusion drawn on the basis of a particular sample, may not always be true in the respect of the population. The four, possible situations that arise in any test procedure are given in the, following table, , Decision from sample, 𝑯𝟎 true, 𝑯𝟎 false, 𝑯𝟏 true, , Reject 𝑯𝟎, Wrong, Type I error, 𝛼, Correct, 𝟏−𝛽, , Accept 𝑯𝟎, Correct, 𝟏−𝛼, Wrong, Type II error, 𝛽

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From the above table it is obvious that in an testing problem we are, liable two type of errors type I error and type II error, type I error: the error of rejecting 𝐻0 when 𝐻0 is true is called type I, error. The probability of type I error is denoted b 𝛼, 𝛼 = probability of type I error., = probability of rejecting 𝐻0 when 𝐻0 is true., , type II error the error of accepting 𝐻0 when 𝐻0 is false is called type II, error. The probability of type II error is denoted b 𝛽, 𝛽 = probability of type II error., = probability of accepting 𝐻0 when 𝐻0 is false

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Level of significance:, The probability that the statistic will fall in the critical region is, 𝛼, 𝛼, + = 𝛼. this 𝛼 is nothing but the probability of committing type I, 2, , 2, , error. Technically, the probability of committing type I error is known as, level of significance. It is also called the size of the critical region.

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Power of the test : 𝟏 − 𝛽 is called the power function of the test, hypothesis 𝐻0 against the alternative hypothesis 𝐻𝟏 . The value of the, power function at a parameter point is called the power of the test at that, point ., Degree of freedom : the number of degree of freedom is the number of, observation that are free to vary after certain restriction have been, placed on the data . if there are n observations in the sample, for each, restriction imposed upon the original observations the number of degree, of freedom is reduced by one i.e. n- 1

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Steps in testing of hypothesis :, The process of testing a hypothesis involves many steps. These are, 1., 2., 3., 4., , Formulation of null and alternative hypothesis, Specification of the level of significance, Selection of test statistic and its computation, Finding out the critical value from tables using the level of, significance , sampling distribution and its degree of freedom, 5. Determination of the significance of the test statistic, 6. Decision about the null hypothesis based on the significance of the, test statistic, 7. Writing the conclusion in such a way that it answers the question, on hand

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