If the variance is known and equal to , the critical point can be replaced by of the standard normal distribution.
For example, in order for vaccine to be approved for widespread use, it must be established that the probability of serious adverse reaction must be less than . In this case we should set `` versus '' to see whether we can reject in favor of .
Let be the frequency of the specified type in categorical data of size . For example, in order for vaccine to be approved for widespread use, we must estimate the probability of serious adverse reaction. In this example will be the number of participants who suffered adverse reaction among participants. Then the two different formulas
are available for the confidence interval with level . Although the first formula is easier to calculate, the second is known to be more accurate and widely used.