The linear cross correlation to quantify statistical connections or relationships finds wide applications in various geophysical and geodetic fields. However, there remains a dearth of comprehensive discourse regarding the significance testing for cross correlation, which would serve to differentiate statistically meaningful outcomes from those merely stemming from pure randomness. This study aims to develop a significance testing method for cross correlation in both white noise and red noise based on the -distribution within a rigorous statistical framework, which enables the establishment of significance tests for cross correlation as function of relative time shift within specified ranges, as demonstrated by Monte Carlo experiments that we perform. Via these results we critically examine the previously-claimed significant correlations between ENSO (El Niño Southern Oscillation ) and the global terrestrial water storage variations derived from the GRACE (Gravity Recovery and Climate Experiment ) satellite mission.