I have a question which goes like this.
Two analysts are in dispute about some data they expect to arise in an experiment. In total, they will receive 20 observations. One analyst believes that these should be a random sample from an exponential distribution with mean 1. The second analyst believes instead that the data come from a normal distribution with mean 2 and standard deviation 1. They come to you for advice on how to use the data to resolve their dispute.
(you can assume that the sum of 20 independent observations from a unit exponential distribution has a Gamma(20,1) distribution)
Your first suggestion is to calculate the average of the observations. You will endorse the first analyst’s view if the average is less than 1.5, and endorse the second analyst’s view of the average is greater than this.
Calculate the probability that you will endorse the second analyst’s view if, in fact, the first analyst is correct.
Is it correct to apply the central limit theorem here since we only have 20 observations? So far I have considered standardizing the sum of the random variables and using the CLT but unsure if this was correct.
If anyone could point me in the right direction, I will be extremely thankful.