(Solved by Humans)-Bert and Ernie noticed that the following are satisfied when

Question

Bert and Ernie noticed that the following are satisfied when Cookie Monster eats cookies:

(a) the number of cookies eaten during non-overlapping time intervals are independent;

(b) the probability of exactly one cookie eaten in a sufficiently short interval of length h is approximately ? h;

(c) the probability of two or more cookies eaten in a sufficiently short interval is essentially zero. Therefore, Xt, the number of cookies eaten by Cookie Monster by time t, is a Poisson process, and for any t > 0, the distribution of Xt is Poisson ( ? t ). However, Bert and Ernie could not agree on the value of ?, the average number of cookies that Cookie Monster eats per minute. Bert claimed that it equals 1.5, but Ernie insisted that it has been less than 1.5 ever since Cookie Monster was forced to eat broccoli and carrots. Thus, the two friends decided to test H0 : ? = 1.5 vs. H1 : ? <1.5

Bert decided to count the number of cookies Cookie Monster would eat in 7 minutes, X7 , and then Reject H0 if X7 is too small. Ernie, who was the less patient of the two, decided to note how much time Cookie Monster needs to eat the first 4 cookies, T4 , and then Reject H0 if T4 is too large.

1 a) Help Bert to find the best (uniformly most powerful) rejection region with the significance level a closest to 0.05