A neutrosophic number linear programming (NN-LP) method reveals a useful tool for solving optimal production planning problems in an indeterminate environment. The optimal feasible solutions of the decision variables and the objective function in the NN-LP method are obtained only depending on the subjectively specified uncertain range of neutrosophic numbers without considering some distribution and confidence level of product sample data. Due to the lack of some probability distribution and confidence level/interval in indeterminate situations, existing indeterminate optimization methods are difficult to guarantee the reliability and credibility of the optimal interval feasible solutions. To solve these problems, this study proposes confidence neutrosophic number linear programming methods based on some probability distributions to objectively determine the confidence level/interval of a sample dataset/multivalued set in the NN-LP problems and strengthen the reliability and credibility of the optimal interval feasible solutions. Therefore, this article first introduces a transformation method from product sample datasets (multivalued sets) with normal and log-normal distributions to confidence neutrosophic numbers (CNNs) (confidence intervals) from a probabilistic point of view. Then, CNN linear programming (CNN-LP) models and solution methods are proposed according to the normal and log-normal distributions and confidence levels of product sample datasets to obtain the optimal interval feasible solutions. Furthermore, the proposed CNN-LP methods are applied to two real cases in a manufacturing company of Shaoxing City in China to perform the production planning problems under normal and log-normal distributions and 95% CNN/confidence interval of the product sample datasets. Compared with existing related linear programming methods, the proposed CNN-LP methods can make the optimal interval feasible solutions more reasonable and credible/reliable than the existing linear programming methods and overcome the defects of the existing linear programming methods in indeterminate situations.