STAT 409
Homework #10
Fall 2009
(due Thursday, November 5, by 4:00 p.m.)
1 – 2.
Bert and Ernie noticed that the
following are satisfied when
Cookie Monster eats cookies:
(a)
the number of cookies eaten during
nonoverlapping time intervals are
independent;
(b)
the probability of exactly one cookie
eaten in a sufficiently short interval
of length
h
is approximately
λ
h
;
(c)
the probability of two or more cookies eaten in a sufficiently short interval is
essentially zero.
Therefore, X
t
, the number of cookies eaten by Cookie Monster by time
t
, is a Poisson
process, and for any
t
> 0, the distribution of X
t
is Poisson (
λ
t
).
However, Bert and Ernie could not agree on the value of
λ
, the average number of cookies
that Cookie Monster eats per minute.
Bert claimed that it equals 2, but Ernie insisted that it
is greater than 2.
Thus, the two friends decided to test
H
0
:
λ
= 2
vs.
H
1
:
λ
> 2.
Bert decided to count the number of cookies Cookie Monster would eat in 5 minutes, X, and
then Reject
H
0
if X is too large.
Ernie, who was the less patient of the two, decided to note
how much time Cookie Monster needs to eats the first 10 cookies, T, and then Reject
H
0
if T
is too small.
1.
a)
Help Bert to find the best (uniformly most powerful) Rejection Region with the
significance level
α
of the test closest to 0.05.
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 Spring '11
 Stepanov
 Normal Distribution, Probability theory, Statistical hypothesis testing, Statistical power, Cumulative distribution function, rejection region

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