I have been working on this question and I am little confused about how to solve it.
To evaluate the prevalence of periodontal diseases in a population, suppose that $x_i$, $i=1,\ldots,n$ are the outcomes of $n$ iid observations of $X$ regarding the status of participants recruited in a clinical trial. $X_i= 0$ (if the patient does not have the periodontal disease) or $1$ (if the patient has the disease) in the population with the unkown survival rate $p= P(X=0).$
a) If it is known that $0.3\leq P \leq 0.8,$ find the MLE of $P$
b) if $ 0<p<1 $ and $n \geq 4$, derive the UMVUE of $g(p)= p^4 +3p^2-p^3 + 0.7$
With question a, I do not understand how the range of $p$ would affect the MLE. Since the MLE depends on our X values. My initial approach was to find the MLE of bernoulli (p ) which is the mean but I do not know how the range of $P$ as given affect my answer.
For question b, I was thinking of using the rao blackwell theorem but I sincerely do not know how to set it up.
I have an exams in a few days and an understanding of this is very important.Looking forward to a response soon.