How to do hypothesis test if we only know part of the sample Basically, I am facing a problem where I need to do a hypothesis test with the Null hypothesis being Y ~ Norm(u,q) while I only know the sample size is 'n' and the number of samples between 0 and 5 is 'k'. 
 A: You're not given any of the usual information for a test of hypothesis.
Maybe the scenario below is a clue to the approach you're expected to take.
Suppose I'm told that $u=\mu = 10, \sigma=3,$ $q=\theta=\sigma^2 = 9,$ and I know $n = 20,$ $k = 5.$
Then $P(0 < Y \le 5) = 0.0474$ with $\mu=10, \sigma=3.$ 
(Computation in R below.)
diff(pnorm(c(0,5), 10, 3))
[1] 0.04736129

If getting $0 < Y \le 5$ is a Success, should I get as many as $k=5$ Successes in $n=20$ observations?
1 - pbinom(4,20,.0474)
[1] 0.002037408

A: It's not much of an issue. A general proposition is that upon simulation a sample repeatedly (say for 1000 times), you get close to the population parameters if not juxtapose those. Let's take an example; say that you have 10K records in a sample, and don't know what the population is. And want to find out whether the sample mean (xbar) or the median is the true representation of the population mean (mu). You will have to simulate the sample say for 1000 times, taking a separate and small sample, say 10 values, each time recording sample mean for those smaller samples. One premise of the idea is that upon plotting those 1000 values, you should end up with a normal distribution. You then take the RMSE of both the given xbar and the median and take the one having lesser RMSE for the "Maximum Likelihood Estimator" (MLE) for the population parameter mu.
Having said that, I admit from the metaphorical leaves of my experience that it is always important to report and be transpicuous with the audience and readers, for in my opinion it is very difficult to always get the right answer. Nonetheless, people should always be kept informed of what they are consuming.
