TECHNOLOGY OVERVIEW

# Hypothesis testing for a binomial probability and ppm

If the sample size is small a popular option is to calculate a confidence interval using sample information. If the confidence interval covers the reference value, then the null hypothesis is accepted. If it does not include the reference value the null hypothesis is rejected.

If the sample size is not too small then the following steps, which rely on a Normal distribution can be used. Randomly sample n items from population.

• Count the number of occurrences (successes) of the variable in question, e.g. number of red marbles, number of defectives, number of voters voting for your party. (For ppm n is equal to 1 million and hence the count must be converted to count per million. E.g. If 1 defective were found in a 1000 items sampled then the count per 1 million is 1*1000000/1000= 1000 (ppm)).
• Specify a level of significance
• Specify an equality (<. > or <>)
• Specify a reference value for the proportion (or ppm) to test against.
• Determine the test statistic with the following expression: (r-np)/sqrt(npq) - where r is number of successes (ppm for ppm applications), n the sample size (1 million for ppm), p the reference value and q (1-reference value)
• Determine the critical region for the test statistic. The critical region depends on the equality i.e. < or > or <>.
• Compare the test statistic with the Critical region and conclude significance if the test statistic falls outside the critical region.

Alternatively, a p-value can be calculated and if the p-value falls below the specified level of significance the Null Hypothesis is rejected, and the alternative hypothesis is accepted.

BIS.Net Analyst and the BIS.Net Inferences app both use p-values for hypothesis testing as these are more flexible than imposing a predefined level of significance on users. To place perspective on the p value a P Curve is provided for the analyst.

The online BIS.Net Analyst.com service uses mainstream technology only which is out-of-date. The BIS.Net Inferences APP uses advanced machine power algorithms to more reliably calculate p values. These are derived from score confidence intervals (directly related to hypothesis testing) which have been proven to be more reliable than the Wald interval, the basis of the mainstream technology tests.

P-Curve example from BIS.Net Analyst (Inferences APP)

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