I'd like to start a discussion related to Confidence Intervals (CI) and Hypothesis Testing (HT). I've read many books including the topic but I still sometimes find books and readings which establish a relationship between CI and HT.
They say that HT is based on CI to come to a conclusion. I don't agree.
Let's say, suppose we want to work out a problem like this: 30-data sample, IID Normal distributed variables, variance not know and hypothesis:
If our data show this numbers:
We would use a T-test and would write that under null hypothesis, we would have:
And this is a probability, not a CI, which have to be compared to the we were given as the decision's rule. Under the same logic, if we were going to check our test through statistics, we would just write:
(where I wrote "?" to mean a comparison to resolve the test)
So, I come to the conclusion that CI are not related to HT because we compared the probability related to the estimator and our Type-I error maximum accepted probability.
On the other hand, CI is just a tool to "presume", without any good reason, what the value of the parameter is. Despite it use all the sample, there are not a probability argument behind it. We just do a free-of-all-meaning calculation and asign some number as "confidence".
Using the same data as above, the 0.95 confidence intervel for the population mean would be:
In this case, we do NOT work with probability. So, again, neither are CI and HT related through probabiltiy nor through same argument.
Am I wrong?
All the best,