more appropriate. Cumulative distribution function Let Xi, i = 1, 2, ..., n be independent observations taken from a normal distribution with unknown mean μ and known variance σ2. Consider two hypotheses, a null hypothesis: and an alternative hypothesis: for some 'smallest significant difference' μ* > 0. This is the smallest value for which we care about observing a difference. Now, if we wish to (1) reject H0 with a probability of at least 1 − β when Ha is true (i.e. a power of 1 − β), and (2) reject H0 with probability α when H0 is true, then we need the following: If zα is the upper α percentage point of the standard normal distribution, then and so 'Reject H0 if our sample average () is more than ' is a decision rule which satisfies (2). (This is a 1-tailed test.) Now we wish for this to happen with a probability at least 1 − β when Ha is true. In this case, our sample average will come from Normal distribution with mean μ*. Therefore, we require Through careful manipulation, this can be shown (see Statistical power Example) to happen when where is the normal cumulative distribution function. Stratified sample size With more complicated sampling techniques, such as stratified sampling, the sample can often be split up into sub-samples. Typically, if there are H such sub-samples (from H different strata) then each of them will have a sample size nh, h = 1, 2, ..., H. These nh must conform to the rule that n1 + n2 + ... + nH = n (i.e., that the total sample size is given by the sum of the sub-sample sizes). Selecting these nh optimally can be done in various ways, using (for example) Neyman's optimal allocation. There are many reasons to use stratified sampling: to decrease variances of sample estimates, to use partly non-random methods, or to study strata individually. A useful, partly non-random method would be to sample individuals where easily accessible, but, where not, sample clusters to save travel costs. In general, for H strata, a weighted sample mean is with The weights, , frequently, but not always, represent the proportions of the population elements in the strata, and . For a fixed sample size, that is , which can be made a minimum if the sampling rate within each stratum is made proportional to the standard deviation