Specific Hypothesis I have one sample group. (I know my test statistics is nonparametric.)
My null hypothesis is a CI  and alternative is ~ . 
What is your idea for testing this hypothesis?

 A: If I understand your question correctly, You should measure the variable in your sample population and if its mean is between 1.5 and 3 then you should accept the null hypothesis; however if the measured mean is outside the given range then you should discard the null hypothesis and accept the alternative. 
A: Null hypothesis stated here is not a CI. It is just a given interval, confidence interval has a specific meaning in context of estimation. It is simply meaningless to call any interval as CI. And also the interval should be expressed as $[1.5,3]$.
You can approach this problem like this: do 2 hypothesis tests, first with $H_0=1.5$ $H_A<1.5$, $\alpha=.025$ and second with $H_0=3$, $H_A>3$, $\alpha=.025$. If both tests fail to reject $H_0$, you will fail to reject $H_0=[1.5,3]$, if one of them rejects $H_0$, you will reject $H_0=[1.5,3]$ with $\alpha=.05$ .
A: This sounds like an equivalence test would do the trick. You are testing whether the mean is equivalent to 2.25 and you have your equivalence interval. Usually in equivalence testing you have two treatments (or whatever) to compare, but here you have one. It seems to me you could just use two one sample tests.
But I don't see a way to do it with one test. In a sense, the two tests above are one test - they are a test of a single hypothesis. 
