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I obtained two groups of data from my experiment and calculated the binomial distribution for each group, trying to see if they are significantly different from each other. However, the z sore I got was way smaller than -4, which was a little bit odd to me. Here is how I did my calculation:

The control group: # of trails n1=34; # of successes k1=2, success rate p1=2/34=5.9% The experimental group: # of trails n2=25; # of successes k2=2, success rate p2=2/25=8.0%

average p=6.8% standard error=0.0044 test statistic z=(5.9%-8%)/standard error=-4.77 It's a two-tailed test. I couldn't find a p-value corresponding to the z score I got (-4.77) on the table. Does that mean my p-value was just really small or I made mistakes here?

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If you had obtained a z-score of -4.77, then you would have observed a very significative difference However, the calculations are wrong, as (2 in 34) vs (2 in 25) should return a much closer to 1 p-value

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Your calculation of the standard error is way off. It should be closer to 0.07

In any case, the normal approximation is not applicable when there are barely any successes observed. You should be using the exact binomial distribution to do this calculation. The R function prop.test will do that test. If I'm not mistaken, you will find a p-value of 1 when you do that.

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