In nearly all scientific disciplines, the statistical inference about the population relies on the quality of the obtained sampled data. For the statistical inference, the observed sample observations are often recorded as precise numerical values. From the centuries, continuous measurements are recorded as precise numbers, and statistics offer robust techniques and models for translating the sampled raw observations into valuable information. However, the modern metrology sciences recommend that getting precise measurements of continuous phenomena is not very realistic, and measurements inherently entail another form of uncertainty known as fuzziness. Therefore, for optimal inference, it is indispensable to integrate the potential uncertainties in the estimation through recording the continuous measurements by up-to-date fuzzy numbers. The objective of this study is to formulate Bayesian parameter estimators for the Normal distribution, aiming to address both uncertainties i.e. fuzziness and random variation. To completely employ all obtainable uncertainties, the proposed estimators are developed that are based on the informative fuzzy priors and fuzzy observations.