example (approximate confidence interval for the meanlet xn be an iid sample from distribution with mean and variance (both assumed to be unknownby the central limit theorem and the law of large numberstx approx ( )sn for large nwhere is the sample standard deviation rearranging the approximate equality [| - / awhere - / is the / quantile of the standard normal distributionyields - / - / an so that - / - / abbreviated as + - / is an approximate stochastic ( aconfidence interval for ( since ( is an asymptotic result onlycare should be taken when applying it to cases where the sample size is small or moderate and the sampling distribution is heavily skewed hypothesis testing null hypothesis alternative hypothesis test statistic hypothesis testing suppose the model for the data is described by family of probability distributions that depend on parameter th th the aim of hypothesis testing is to decideon the basis of the observed data twhich of two competing hypotheses holds truethese being the null hypothesish th th and the alternative hypothesish th th in classical statistics the null hypothesis and alternative hypothesis do not play equivalent roles contains the "status quostatement and is only rejected if the observed data are very unlikely to have happened under the decision whether to accept or reject is dependent on the outcome of test statistic ( for simplicitywe discuss only the one-dimensional case two (relatedtypes of decision rules are generally used decision rule reject if falls in the critical region critical region here the critical region is any appropriately chosen region in in practice critical region is one of the followingleft one-sided(- ]right one-sided[ )two-sided(- [ critical values for examplefor right one-sided testh is rejected if the outcome of the test statistic is too large the endpoints cc and of the critical regions are called critical values