How to I prove a rate is at least x% higher than the other?

Suppose I have option 1 and option 2. Option 1 yields a success rate of 80%, option 2 yields a success rate of 70%. The samples are independent. A z test can reject the H0 hypothesis and I can conclude that option 1 has a higher success rate.

My question is: if I use a lowered success rate 77% for option 1, the z test can still reject the H0 hypothesis. Can I conclude that Option 1 can yield a success rate at least 3% higher than option 2? The reasoning is that 77% is proved to be significant, my real data is 80%, so my data is 3% higher than the proved improvement.

I request you to tell me whether this right or wrong. If it turns out to be wrong, what statistical method can give a conclusion that option 1 is at least x% higher than option 2? Thank you in advance.


TS Contributor
the p test, for instance, gives you a confidence interval of the difference in the output. Isnt this what you need?