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# Sample size for Confidence Intervals and Hypothesis Testing for making Inferences

There are two alternatives with regards to sample size for inferences. One alternative is to take a sample size considered to be economical and practical, or as dictated by circumstances. Another is to determine sample size scientifically to meet the desired precision.

Sometimes the former approach is unavoidable, for example when conducting a survey, only a limited number of people may respond. At other times the cost of sampling may be high, dictating an economical approach. When there are time constraints and sampling and testing is time consuming sample size may be chosen on practical considerations.

The former approach should not be taken lightly and only used if it really is unavoidable. This approach usually results in sample sizes that are too small with insufficient precision, providing no benefit for decision making. It has even been shown that in the absence of the necessary precision managers tend to assume precision that is not there and make wrong decisions.

For example, a politician based on a sample size of 112 may have conclude 52% of voters are in her favour and thus believe she is leading. However, this would foolishly assume that the sample result has no error. The sample result is only a sample result and may not even remotely reflect the true proportion of voters in the electorate. In this instance beknown to the politician the opposition, assuming a two-party system, could have as high as 62% voters favouring the opposition. The survey can thus have negative consequences if the politician reads too much into the result, falsely believing she is ahead.

Another example is in General Practice. A General Medical Practioner, who can see many patients each day, is often reliant only on blood pressure measurements taken during consultation. It is not uncommon for a Practioner to base a decision on medication from 3 readings over say 3 days. Our case studies for Systolic readings has shown standard deviations around 10 on average, but as high as 20 for many cases. Such a low sample size can mean that patients are prescribed medication when they should not be taking medication, or conversely not being prescribed medication when they should take medication.

If the ad-hoc approach must be used the analyst can make use of BIS.Net Analystâ€™s sample size sensitivity feature found in the BIS.Net Sample Size App. In most instances this requires specifying the desired margin of error resulting in a sensitivity chart such as shown below. The analyst can then compare the margin of error for the practical sample size with the needed margin of error.

However, the best and scientific alternative is to determine the sample size required to satisfy the needed precision.

To determine the required sample size for confidence intervals and hypothesis testing requires specifying parameters such as the margin of error, level of significance, amount of concern and power.

For some standard text book applications, such as the confidence interval on the mean for a population, that is normally distributed, solving for n is simple because closed form equations exist that can easily be mathematically solved.

However, for some other parameters there are no closed form equations. Even to obtain a sample size for confidence intervals, or to perform a hypothesis test on the proportion, there are no closed form equations, if solutions for the most reliable confidence intervals or hypothesis tests are to be obtained. One approach typically used is to use the classical approach by using approximating functions, relying on the normal distribution. This approach is however not accurate and in some instances such as for Cp and Cpk and F Ratio cannot even be used.

The free online BIS.Net Analyst version uses classical approximations when possible or provides no solution when impossible.

Only the BIS.Net Analyst App provides solutions to previously unsolvable applications using Machine Power and Machine Learning algorithms.

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