Parameter Estimation and Evaluation Introduction to Hypothesis Testing Introduction to Hypothesis Testing Example 4(9.4) Suppose X"is an IID N(u,o2)random sample,where u is unknown but o2 is known.We are interested in testing for Ho:u=uo versus HA:u uo,where uo is a known number. Here,o {uo}contains one parameter value uo and A contains all parameter values on the real line R except uo. Hypothesis Testing Introduction to Statistics and Econometrics July7,2020 16/110
Parameter Estimation and Evaluation Hypothesis Testing Introduction to Statistics and Econometrics July 7, 2020 16/110 Introduction to Hypothesis Testing Introduction to Hypothesis Testing Example 4 (9.4)
Parameter Estimation and Evaluation Introduction to Hypothesis Testing Introduction to Hypothesis Testing Remarks: ● To test the null hypothesis Ho u=uo,we consider the following test statistic T(X")= Xn-po where o is known. Under the null hypothesis Ho:u=wo,we have T(X)= Xn-Ho σ/元 N(0,1). Hypothesis Testing Introduction to Statistics and Econometrics Juy7,2020 17/110
Parameter Estimation and Evaluation Hypothesis Testing Introduction to Statistics and Econometrics July 7, 2020 17/110 Introduction to Hypothesis Testing Introduction to Hypothesis Testing Remarks:
Parameter Estimation and Evaluation Introduction to Hypothesis Testing Introduction to Hypothesis Testing Under HA:u丰o, o)) which diverges to infinity with probability approaching one as n→o. One can accept HA if T(X")is large.How large T(X") should be in order to be considered as "large"is deter- mined by the sampling distribution (N(0,1))of T(X") under Ho. Hypothesis Testing Introduction to Statistics and Econometrics Juy7,2020 18/110
Parameter Estimation and Evaluation Hypothesis Testing Introduction to Statistics and Econometrics July 7, 2020 18/110 Introduction to Hypothesis Testing Introduction to Hypothesis Testing
Parameter Estimation and Evaluation Introduction to Hypothesis Testing Introduction to Hypothesis Testing Specifically,by setting c=z/2,where za/2 is the upper- tailed critical value of N(0,l)at level号∈(0，l),i.e., P(Z>za/2)=g where Z~N(0,1),we can define the acceptance and rejection regions as follows: An(c)= σ/ ≤号 c={x: Hypothesis Testing Introduction to Statistics and Econometrics Juy7,2020 19/110
Parameter Estimation and Evaluation Hypothesis Testing Introduction to Statistics and Econometrics July 7, 2020 19/110 Introduction to Hypothesis Testing Introduction to Hypothesis Testing
Parameter Estimation and Evaluation Introduction to Hypothesis Testing Introduction to Hypothesis Testing The N(0,1)test decision rule: -(1)Accept Ho u=uo at the significance level a if xn∈An(c); -(2)Reject Ho:u=uo at the significance level a if xn∈Cn(c). The significance level a is the probability that the above decision rule will wrongly reject a correct null hy- pothesis.This error is called the Type I error and is unavoidable given any finite sample size n. Hypothesis Testing Introduction to Statistics and Econometrics July7,2020 20/110
Parameter Estimation and Evaluation Hypothesis Testing Introduction to Statistics and Econometrics July 7, 2020 20/110 Introduction to Hypothesis Testing Introduction to Hypothesis Testing