In this paper, we propose a new strategy of estimating correlation dimensions in combination with the method of surrogate data, which is a kind of statistical control usually introduced to avoid spurious estimates of nonlinear statistics, such as fractal dimensions, Lyapunov exponents and so on. In the case of analyzing time series with the method of surrogate data, it is desirable to decide values of estimated nonlinear statistics of the original data and surrogate data sets as exactly as possible. However, when dimensional analysis is applied to possible attractors reconstructed from real time series, it is very dangerous to decide a single value as the estimated dimensions and desirable to analyze its scaling property for avoiding spurious estimates. In order to solve this defficulty, a dimension estimator algorithm and the method of surrogate data are combined by introducing Monte Carlo hypothesis testing. In order to show effectiveness of the new strategy, firstly artificial time series are analyzed, such as the Henon map with additive noise, filtered random numbers and filtered random numbers transformed by a static monotonic nonlinearity, and then experimental time series are also examined, such as wolfer's sunspot numbers and the fluctuations in a farinfrared laser data.