On Wednesday, April 24, 2013 11:31:11 AM UTC+12, Paul wrote:
> from a conceptual standpoint, say H0 was some generic hypothesis > rather than residual normality. My question about whether the > significance should be small or large depending on whether I'm > interested resoundingly showing the resonableness of rejecting or not > rejecting H0 still seems to hold.
I think your basic intuition is right: Other things being equal, a _larger_ p value tends to indicate that the data are _less discrepant_ from what Ho would predict.
But this view is so oversimplified that it can easily be misleading. For example, the p value might be large because there isn't much data or because there is lots of random error, and in neither case should you conclude that the data provide good support for Ho.
When the research goal is to show that Ho is true to a good approximation, Ho testing is just not a good method. The confidence interval approach is much more informative in most such situations, because they allow conclusions of the form "Ho is not wrong by more than X units". CIs are not easily adapted to your original question of checking for normality, though, AFAIK.