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In inferential statistics, confidence intervals for population proportions rely upon the standard normal distribution to determine unknown parameters of a given population given a statistical sample of the population. One reason for this is that for suitable sample sizes, the standard normal distribution does an excellent job at estimating a binomial distribution. This is remarkable because although the first distribution is continuous, the second is discrete.
There are a number of issues that must be addressed when constructing confidence intervals for proportions. One of these concerns what is known as a “plus four” confidence interval, which results in a biased estimator. However, this estimator of an unknown population proportion performs better in some situations than unbiased estimators, especially those situations where there are no successes or failures in the data.
In most cases, the best attempt to estimate a population proportion is to use a corresponding sample proportion. We suppose that there is a population with an unknown proportion p of its individuals containing a certain trait, then we form a simple random sample of size n from this population. Of these n individuals, we count the number of them Y that possess the trait we are curious about. Now we estimate p by using our sample. The sample proportion Y/n is an unbiased estimator of p.
When to Use the Plus Four Confidence Interval
When we use a plus four interval, we modify the estimator of p. We do this by adding four to the total number of observations, thus explaining the phrase “plus four." We then split these four observations between two hypothetical successes and two failures, which means that we add two to the total number of successes. The end result is that we replace every instance of Y/n with (Y + 2)/(n + 4), and sometimes this fraction is denoted by p with a tilde above it.
The sample proportion typically works very well at estimating a population proportion. However, there are some situations in which we need to modify our estimator slightly. Statistical practice and mathematical theory show that the modification of the plus four interval is appropriate to accomplish this goal.
One situation that should cause us to consider a plus four interval is a lopsided sample. Many times, due to the population proportion being so small or so large, the sample proportion is also very close to 0 or very close to 1. In this type of situation, we should consider a plus four interval.
Another reason for using a plus four interval is if we have a small sample size. A plus four interval in this situation provides a better estimate for a population proportion than using the typical confidence interval for a proportion.
Rules for Using the Plus Four Confidence Interval
The plus four confidence interval is an almost magical way to calculate inferential statistics more accurately in that simply adding in four imaginary observations to any given data set, two successes and two failures, it is able to more accurately predict the proportion of a data set which fits the parameters.
However, the plus-four confidence interval isn't always applicable to every problem. It can only be used when the confidence interval of a data set is above 90% and the sample size of the population is at least 10. However, the data set can contain any number of successes and failures, though it does work better when there are either no successes or no failures in any given population's data.
Keep in mind that unlike the calculations of regular statistics, inferential statistics' calculations rely on a sampling of data to determine the most likely results within a population. Though the plus four confidence interval corrects for a larger margin of error, this margin must still be factored in to provide the most accurate statistical observation.