We aim to study the practical statistical inference of the online second-order Newton method for general unconstrained stochastic optimization problems under the fixed dimension setting. We consider the adaptive inexact stochastic Newton method, which is reduced from an existing stochastic sequential programming (StoSQP) method to the unconstrained setting. Based on the asymptotic normality of the last iteration, we propose a weighted sample covariance matrix, which is a consistent covariance matrix estimator. With this estimator, we are able to conduct statistical inference on the solution of the stochastic optimization problem in practice. The update of the estimator is entirely online and efficient in computation and memory. We demonstrate the empirical performance through numerical experiments on linear regression models.