My problem statement is to identify in a healthcare organisation, which of it's doctors are lagging in providing proper care to their patients.
My Random Variable X is defined as {0 if the patient does not receive proper care, 1 if they received proper care}
I am taking all the patients that have ever visited the organization as the population. I can calculate the population mean and the standard deviation for this distribution (Is this a Bernoulli Distribution?).
I am assuming that the set of patients examined by a doctor is a sample. I can calculate the sample mean.
My question is, Can I use the Central Limit Theorem to say that the sampling distribution of these means follows a normal distribution? If so, then can I use the knowledge of normal distribution and the following hypothesis to calculate the p-value for each doctor:
Ho : Sample Mean = Population Mean
H1 : Sample Mean < Population Mean
Edit: The number of patients examined by each doctor is different.
Are you proposing an hypothesis test like this because it is strange and new to me but makes sense. You can also use proportions to evaluate the doctor as to how far or less than the mean porportion. The mean proportion is the number of properly treated patients to the total. Let this be $p_0$. Now set the hypothesis for each doctor
$H_0: p > p_0$
$H_1: p < p_0$
This type of two tailed proportion hypothesis testing and getting the p values for all your samples make sense also. You may like this to see how many of the doctors you can bucket to be below par and above par.