Let X1, X2, . . . , Xn be independent and identically distributed Poisson random variables with parameter . Consider testing H0 : = 0 versus H1 : > 0.

Suppose that n = 10 and take 0 = 1.

a. Find a suitable test statistic for this hypothesis test and identify its distribution. (Note that if Xi Pois(i), i = 1, . . . , n, are independent Poisson random variables, then the sum X1 + + Xn is also Poisson with parameter 1 + + n).

b. Find the largest critical region for which the size of the test is at most = 0.05. Considering the data

x = (1, 6, 2, 3, 5, 0, 1, 0, 0, 2),

do we have enough evidence to reject the null hypothesis?

c. Write down and plot the power function (on the interval 1 (1, 3)) of your hypothesis test, and calculate the power when 1 = 2. (Hint: Recall the result P(X > x) = 1 P(X x)).