Deriving the form of the best size for exponential distribution

[XeroPhobous]

Hi everyone

So in this question I am told that a supermarket claims that the mean time to serve a customer is equal to 4 mins.However, a customer claims it is greater than 4 mins. I am asked based off a sample of n=9, determine the form of the best test of which has size of a=0.05

So I know my Ho: λ=4 and my Ha: λ>4

I am just unsure on how to derive it down, I know I use the central limit theorem like in previous examples I have done but very unsure for exponential distribution !

 

[rogojel]

hi,
what is exactly the question if you already have the sample size?

regards

 

[Dason]

Clt might not be appropriate here. Have you worked with likelihood ratio tests before?

 

[GretaGarbo]

What will happen when 9 exponentially distributed variables are summed?

 

[XeroPhobous]

Clt might not be appropriate here. Have you worked with likelihood ratio tests before?

Yep I have, this is what I believe I do but again unsure:

Conduct the neyman-pearson lemma i.e. the likelihood ratios over one another. Then once I have these ratios I am basically solving down until I get something along the lines of
P(sum of xi > ....) then from here I would use my CLT

 

[XeroPhobous]

What will happen when 9 exponentially distributed variables are summed?

If these are summed it means that we would have a gamma distribution but I don't see how that would apply here

 

[GretaGarbo]

If these are summed it means that we would have a gamma distribution .....

And what would it be under the null hypothesis?