A gamma function problem

Let's define Y = the sum of the 10 observations. Under the null hypothesis each observation is assumed to be independent and identically distributed exponential with mean 2 (equavalently, with rate 1/2). The sum of 10 independent exponential random variables with parameter $\beta = 2$ follows a gamma distribution with parameters $\alpha = 10, \beta = 2$.

We can find the appropriate cutoff value for the sum of the observations, and then simply divvide it by 10 to find the cutoff value for the sample mean.

In R: qgamma(.05, 10, 1/2) yields 10.85081
In excel: GAMMA.INV(0.05, 10, 2) gives us the same result. (note that R uses the rate parameter and excel uses the mean parameter)

Divide that number by 10 and you get 1.0851. In other words, if the true lifespan of the phones is 2 years, the probability that the sample mean of 10 phones is 1.0851 or less is 0.05. 

With 1000 observations, we could use a normal distribution as an approximation because of the Central Limit Theorem, but the problem states that we want an exact test, so we should use gamma again. 

In R: qgamma(.05, 1000, 1/2) gives us 1897.12. Therefore, under the null hypothesis, if we sample 1000 random phones, the probability that the sample mean is less than or equal to 1.897 is 0.05. So an appropriate cutoff for the sample mean in this test is 1.897.