Binomial distribution confidence intervals

Using the normal approximation to a binomial distribution, confidence intervals (CIs) for p(y =j X) can be established for a specific significance level, a ... 

In the introduction to this section, two differences between "classical" and Bayes statistics were mentioned. One of these was the Bayes treatment of failure rate and demand probttbility as random variables. This subsection provides a simple illustration of a Bayes treatment for calculating the confidence interval for demand probability. The direct approach taken here uses the binomial distribution (equation 2.4-7) for the probability density function (pdf). If p is the probability of failure on demand, then the confidence nr that p is less than p is given by equation 2.6-30. 

The split-sample method is often used with so few samples in the test set, however, that the validation is almost meaningless. One can evaluate the adequacy of the size of the test set by computing the statistical significance of the classification error rate on the test set or by computing a confidence interval for the test set error rate. Because the test set is separate from the training set, the number of errors on the test set has a binomial distribution. 

This is also a confidence interval for the parameter p, probability of success, of the binomial distribution. The use of the Z distribution for this interval is made possible because of the Central Limit Theorem. Consider the random variable X taking on values of 0 or 1, such that the sampling distribution of the sample mean (the proportion) is approximately normally distributed. A table of the most commonly encountered values of the standard normal distribution is provided in Table 8.3 for quick reference. Others are provided in Appendix 1. 

The confidence interval is nonparametric since the constants f and u depend only on k and q. Determining t and u is difficult when the number of repUcations, k, is large (and k should be large for quantile estimation), but large k allows a normal approximation to the binomial distribution. The approximation leads to setting... 

The arguments leading to the determination of the lower and upper endpoints of the confidence interval are analogous to those of the preceding section. Instead of summing over a Poisson distribution this has to be done over a binomial distribution according to Eq. (9.35) for a number failures which are smaller, respectively larger than the observed number x. If the relationship between sums over binomial distributions and Fisher s F-distribution is used [24], we obtain for the lower endpoint with a level of confidence of y... 

The "Total Number" is the number of test insects while the "Pairs Mating" is the observed number of insect pairs mating within the three hrs of the test. "Confidence Interval" determined by Binomial Distribution. Values significantly different at P < 0.05. 

Pairs Mating" indicates the number of insect pairs observed mating within the three hr of the study. Confidence intervals determined with Binomial Distribution. C.I. followed by same letter not significantly different at P < 0.05. 

For example, if we set as the number of positive prediction estimates A that the predicted compound C belongs to class a of active compounds. Within the framework of the conservative strategy, compound C is considered to be active if 6>21, and defined as inactive if 0<17 (95% is the confidence interval for the median in binomial distribution [41]). If the 17 <6 <27 prediction is discontinuous, compound C can be considered conditionally active if 22 <6 <21, and conditionally inactive if 1<6<22. 

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