If you knew that the 10% (one tailed) critical value for a z-test was 1.28 and the 20% critical value was 0.84, a rough guess at the 15% critical value would be half-way between - (1.28+0.84)/2 = 1.06 (the actual value is 1.0364), and the 12.5% value could be guessed at halfway between that and the 10% value (1.28+1.06)/2 = 1.17 (actual value 1.15+). The approaches discussed here are suitable for hand calculation when you have limited tables available, but if you're implementing a computer routine to produce p-values, there are much better approaches (if tedious when done by hand) that should be used instead. This answer is in two main parts: firstly, using linear interpolation, and secondly, using transformations for more accurate interpolation.
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