When looking at goodness of fit statistics, what you are determining is the probability of whether the data does not fit the specific distribution. In Charles Annis page on AndersonDarling, his Note 1 does a good job of explaining what you are really looking at. You really are not supporting that you have a normal distribution, you are supporting the probability that you do not, and for your data using that one calculation you are not rejecting that you do not have a normal distribution  a much different term than concluding you do.
The process I use is the "Distribution Analyzer" , which calculates the goodness of fit for a series of common distributions, and gives you the result of the distribution with the highest pvalue. (I believe this is similar to the Pearson analysis.) The logic then behind that is the highest pvalue curve has a lowest probability of being rejected as the specific curve. So, it looks at several distributions, rather than making a judgment toward just one.
That is why I prefer it. As you can see, the tool also permits me to show the data and curves for any of the distributions  which is handy. If the best curve p value is very close to the normal curve's pvalues, I may assume that it is "normal enough' to suit my statistical intentions.

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Goodness of Fit: Is it normal? Probably...or probably not  it's all probability
It's not what you think!!!