I know that if the number of successes and failures are all greater than 10, the sampling distribution would be approximately normal. However, if one of the failures are not equal to ten, but it is stated that the population is normal, can I still use the 2 proportion z-test?
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Normal approximation to binomial distribution was very useful when there was no computer or computer was slow. Fisher's exact test was developed long time ago, but because we did not have tools for calculation, therefore we need the normal approximation. Currently, it is easy and fast to perform the Fisher's exact test for 2x2 table (as your situation). So normal approximation for 2x2 table should go to statistical museum.
So my suggestion is using Fisher's exact test which has no requirement on sample size because it is exact method. Some webpages provide this calculation.
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