Optimal sample size in a triangular model for sensitive questions

The estimation of the fraction of a population with a stigmatizing characteristic is the issue that this study attempts to address. In this paper the nonrandomized response model proposed by Tian et al. (2007) is considered. The exact confidence interval (CI) for this fraction is constructed. The optimal sample size for obtaining the CI of a given length is also derived. In order to estimate the proportion of the population with a stigmatizing characteristic, we explore the nonrandomized response model proposed by Tian et al. (2007). The prevalent approach to constructing a CI involves applying the Central Limit Theorem. Unfortunately, such CIs fail to consistently maintain the prescribed confidence level, contradicting the Neyman (1934) definition o f C Is. I n t his p aper, w e p resent t he c onstruction o f a n e xact CIs for this proportion, ensuring adherence to the designated confidence l evel. T he l ength of the proposed CI depends on both the given probability of a positive response to a neutral question and the sample size. For these CIs, the probability of a positive response to a neutral question is established in relation to the provided limit on the privacy protection of the interviewee. Additionally, we derive the optimal sample size for obtaining a CI of a given length.