Equivalence of Test Hypothesis Thinking about Test Hypothesis I came up with a question related to them. When you make the test with a $90\%$ confidence interval, you have $10\%$ probability of committing a type I error. Analogously, for $99\%$, you have $1\%$ probability of committing a type I error. I wondered the following: Making a $99\%$ confidence interval test, is equivalent to making two $90\%$ confidence interval tests and reject only the null hypothesis if both let you reject it?

If you make a $90\%$ confidence interval test, the probability to make a type I error is 10%. Then you repeat it. The probability to make a type I error is $10\%$ again. The probability to make a type I error in both of the tests is $10\% \cdot 10\%=1\%$.

The probability to make a type I error in one $99\%$ confidence interval test is $1\%$.

Therefore in both cases the probability to make a type I error is $1\%$.