Why is my t-test intuition so far from reality? Interpretation of t-test value (2 tailed): ~20% chance that these results could have been obtained randomly.
My intuition: MEAN_2 - MEAN_1 = ~3% of MIN(STD_1, STD2). Calculate area under normal normal PDF between -3% and +3% of std from mean which is ~2.5%. 
Why is the t-test result so far off from my intuition???

COUNT_1 = 3572
MEAN_1 = 702
STD_1 = 2926

COUNT_2 = 4596
MEAN_2 = 621
STD_2 = 2730

t = 1.26
df = 7403
 A: The test statistic in the ordinary two sample t-test is a difference in means divided by the standard deviation of that difference in means (the standard deviation of the distribution of some statistic, such as a difference in means is called a standard error). 
The standard error of the difference in means decreases as the sample sizes increase (because the individual sample means become more precise).
You seem to be using (as @whuber suggests) the standard deviations of the samples to obtain the intuition of 'how different' the samples are. This ignores the fact that sample means have smaller standard deviations than the samples from which they're computed.
For example, if I have 100 observations with mean 40 and standard deviation 15, then the standard error of their mean is one tenth as large as the standard deviation of the data.
If the sample sizes were only 4, the standard error of the means would be half as large as the standard deviations of the data - five times as big as when the sample sizes were 100.
