My impression is that most scientists feel that they don’t make very many errors in reporting their results.

My impression is that most scientists feel that they don't make very many errors in reporting their results.   My own experience from coding tells me that I make many simple mistakes.

One very simple check for mistakes in a published paper is to see if the reported statistic value matches the reported p value.  For example, if we see "t(15) = 2.3; p = 0.034” we can check if the p value for t=2.3, with 15 degrees of freedom, really is 0.034.   Obviously, this check cannot tell us about other problems that might have lead to an inflated t (etc) value.  Nevertheless, nearly 20% of reported statistics failed this test in a sample from the psychology literature:

http://dx.doi.org/10.3758/s13428-011-0089-5

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