another way to look at this could be to compare a study done on a particular medicine, when study is on a small group of participants versus a large group the results must be shown as proportional and can only be relevant as they are calculated as how many are made well to how many do not receive any favorable results. Many times when a study is done on the effects of a substance on another it must be shown that the margin of error is proportionate to the study. As for instance the drug which conquered polio was simply a vaccine made of the polio virus and had great affects on the disease. Although there was a margin of error that the US government felt was acceptable to allow the drug to be forcibly injected in all children of the late 50's.
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The goal is to disregard the influence of sample size. When calculating Cohen's d, we use the standard deviation in teh denominator, not the standard error.
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No, more information is needed to determine the margin of error. For example, one may need to know the sample's mean, the sample size, and the standard deviations of the population and sample. Depending on the type of test one is performing, certain parameters need not be known. For example, the population standard deviation does not need to be known in a one sample T-test.