Tiêu đề Hypothesis Large sample.pdf ✅

1 Statistical Hypothesis Statistical hypothesis is an assertion are conjecture concerning one are more population. Now we take a random sample of the population and use the data contained in this sample to provide evidence that either support or does not support the hypothesis. Evidence from the sample that is inconsistent with the stated hypothesis leads to the rejection of the hypothesis where as evidence supporting the hypothesis leads to its acceptance. Null Hypothesis Any hypothesis we wish to test and is denoted by H0 (hypothesis of no difference and is tested for possible rejection under the assumption that it is true). Alternative Hypothesis Any hypothesis which is complementary to null hypothesis is called and alternative hypothesis, denoted by H1 and rejection of null hypothesis H0 leads to the acceptance of H1 . A null hypothesis concerning a population parameter will always be stated so as to specify an exact value of the parameter, where as H1 allows the possibility of several values. e.g.; If H0: p = 0.5 for a binomial population, then H1 maybe any one of the following i. p > 0.5 , H1 is known as right tailed alternative. ii. p < 0.5 , H1 is known as left tailed alternative . iii. p ≠ 0.5 , H1 is known as two tailed alternative . Test of Significance and Test of Hypothesis During testing of hypothesis we calculate the deviation between observed sample statistic and hypothetical parameter value, or deviation between two independent sample statistic. And the magnitude of significance of deviation enable us to accept or reject the hypothesis. This procedure is called the test of significance
2 Error in Hypothesis Decision Type 1 error: Reject H0 when it is true, i.e.; we reject a hypothesis when it should be accepted. If we write P{reject H0 when it is true} = P{reject H0 |H0 } = α Type 2 error: Accept H0 when it is wrong. i.e.; accept H0 when H1 is true P{accept H0 when it is wrong} = P{accept H0 |H1 } = β α & β are called size of type-1 and type 2 error respectively. Regions of Acceptance and Rejection(critical) A region (corresponding to a statistic ‘t’) in the sample space S which amounts to the rejection of H0 is known as critical region of rejection. If ω is the critical region P(t ∈ ω|H0 ) = α ...(A) Critica l region Acceptanc e region Critica l region Acceptanc e region Critica l region Critica l region Acceptanc e region Two tailed Right tailed Left tailed The complementary region of ω with respect to S is the region of acceptance, i.e.; ω ∪ ω̅ = S & ω ∩ ω̅ = φ P(t ∈ ω|H1 ) = β ...(B) Level of Significance In testing a given hypothesis the maximum probability with which we would be willing to take risk of Type 1 error is called level of significance, i.e.; ′α′ as given in question (A). The level of significance usually employed in testing of hypothesis are 5% and 1% . The level of significance is always fixed in advance before collecting the sample information. Critical values are Significant values We know that large sample can be approximated as a normal distribution. And by property of area of Normal Distribution curve we have only (for two tailed reg.)

4 Procedure for Testing of Hypothesis i. Setup H0 & H1 ii. Choose the appropriate level of significance of depending upon the permissible risk i.e.; zα . iii. Computer test statistic z under H0 . iv. If |z| < zα , we say it is not significant, by this we mean that difference of t − E(t) is just due to fluctuation of sampling and the sample data does not give sufficient evidence against the null hypothesis, which therefore may be accepted. If |z| > zα then H0 is rejected for the chosen level of significance . For Test of Significance of large sample i. Compute z under H0 . ii. If |z| > 3 , H0 is always rejected . iii. If |z| ≤ 3 we test its significance by comparing values with critical value (depending upon the test). iv. If the difference between the observed and expected value is not significant then H0 is accepted.