STAT410 Homework #7 Complete Solution

Question: #7277

STAT 410 Homework #7 Summer 2015

1 – 2. 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 l 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 ( l t ).
However, Bert and Ernie could not agree on the value of l, 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
H 0 : l = 1.5 vs. H 1 : l < 1.5.
Bert decided to count the number of cookies Cookie Monster would eat in 7 minutes, X 7 , and
then Reject H 0 if X 7 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, T 4 , and then Reject H 0 if T 4
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. ( Hint: X 7 £ c. )
b) Find the power of the test from part (a) if l = 1.
c) Bert decided to Reject H 0 if Cookie Monster eats at most 6 cookies in 7 minutes.
Find the significance level a for this Rejection Region.
d) Suppose Cookie Monster ate 4 cookies in 7 minutes. Find the p-value of the test.
2. a) Help Ernie to find the best (uniformly most powerful) Rejection Region
with the significance level a = 0.05. ( Hint: T 4 ³ c. )
Hint: If T has a Gamma ( a , q = 1/l ) distribution, where a is an integer, then
2 T/q = 2 l T has a c
2
( 2 a ) distribution ( a chi-square distribution with
2 a degrees of freedom ).
b) Ernie decided to Reject H 0 if it takes Cookie Monster longer than 5 minutes to
eat the first 4 cookies. Find the significance level a for this Rejection Region.
Hint: If T has a Gamma ( a , q = 1/l ) distribution, where a is an integer, then
F T ( t ) = P ( T £ t ) = P ( Y ³ a ) and P ( T > t ) = P ( Y £ a – 1 ),
where Y has a Poisson ( l t ) distribution.
c) Find the power of the test from part (b) if l = 1.
d) It took Cookie Monster 6 minutes to eat the first 4 cookies. Find the p-value
of the test.
3. In the past, only 30% of the people in a large city felt that its mass transit system is
adequate. After some changes to the mass transit system were made, we wish to test
if the proportion of individuals who feel the mass transit system is adequate has
increased, that is, test H 0 : p = 0.30 vs. H 1 : p > 0.30 . Let X denote the number
of those who feel the mass transit system is adequate in a random sample of 20 persons.
a) Suppose we decided to use the rejection region “Reject H 0 if X ³ 11.”
Find the significance level a associated with this rejection region.
b) Find the Rejection Rule with the probability of Type I Error closest to 5%.
c) What is the actual value of the probability of Type I Error for the Rejection Rule
in part (b) ?
d) Suppose that 8 persons out of 20 feel the mass transit system is adequate. Find
the p-value.
4. A certain automobile manufacturer claims that at least 80% of its cars meet the tough
new standards of the Environmental Protection Agency (EPA). The EPA tests a random
sample of 25 its cars and wishes to test H 0 : p ³ 0.80 vs. H 1 : p < 0.80, where p
is the proportion of the cars that meet the new EPA standards.
a) Let X denote the number of cars in the sample that meet the new EPA standards.
Find the “best” Rejection Region with the significance level closest to 0.05.
b) What is the power of the Rejection Rule obtained in part (a) if p = 0.70 ? If p = 0.60 ?
c) Suppose that 18 out of 25 cars in our sample meet the new EPA standards. Compute
the p-value.
5. 4.5.5 ( 7th edition ) 5.5.5 ( 6th edition )
Let X 1 , X 2 be a random sample of size n = 2 from the distribution having p.d.f.
f X ( x ; q ) = θ
1
θ
e− x , 0 < x < ¥, zero elsewhere. We reject H 0 : q = 2 in favor of
H 1 : q = 1 if the observed values of X 1 , X 2 , say x 1 , x 2 , are such that
( ) ( )
( ) ( ) 2
1
; 1 ; 1
; 2 ; 2
1 2
1 2
£
×
×
f x f x
f x f x
.
Here W = { q : q = 1, 2 }. Find the significance level of the test and the power of the test
when H 0 is false.
That is, we know that the most powerful rejection region for testing
H 0 : q = 2 vs H 1 : q = 1 is
Reject H 0 if
( )
( )
( ) ( )
( ) ( )
k
f x f x
f x f x
L
L
= £
×
×
; 1 ; 1
; 2 ; 2
1
2
1 2
1 2
. Let k =
2
1
.
( That is, reject H 0 if it is more than twice as likely to observe a data set like ours under
the assumption that H 1 is true than under the assumption that H 0 is true. )
Find (i) the significance level of the test and (ii) the power of the test.
6 – 7. Let X 1 , X 2 , … , X n be a random sample from the distribution with
probability density function
( ) ( ) ( )θ
fX x ; θ = θ +1 × 1− x , 0 < x < 1, q > – 1.
Recall: Y = Σ ( )
=
− −
n
i
i
1
ln 1 X is a sufficient statistic for q
and Y has a Gamma ( a = n, “usual q” =
1
1
θ +
) distribution.
( “usual l” = q + 1 )
We wish to test H 0 : q = 4 vs. H 1 : q < 4. Suppose n = 10.
6. a) Find the form of the uniformly most powerful rejection region.
b) Find the uniformly most powerful rejection region with a = 0.01.
c) Find the power of the test in part (b) at q = 3.18.
Hint: If T has a Gamma ( a , b = 1/l ) distribution, where a is an integer, then
2 T/b = 2 l T has a c
2
( 2 a ) distribution ( a chi-square distribution with
2 a degrees of freedom ).
7. Consider the rejection region “Reject H 0 if Σ ( )
=
− −
10
1
ln 1
i
xi ³ 3”.
d) Find the significance level a of this rejection region.
e) Find the power of this rejection region at q = 3, at q = 2, and at q = 1.
f) Suppose Σ ( )
=
− −
10
1
ln 1
i
xi = 2.7. Find the p-value.
Hint: If T has a Gamma ( a , 1/l ) distribution, where a is an integer, then
P ( T > t ) = P ( Y £ a – 1 ), where Y has a Poisson ( l t ) distribution.
8 – 9. Let l > 0, and let X 1 , X 2 , … , X n be a random sample from the distribution
with the probability density function
( )
2 3 5 λ
; λ λ
x f x x e− = , x > 0.
We wish to test H 0 : l =
5
1
vs. H 1 : l =
3
1
.
Recall: Y = Σ
=
n
i
i
1
2
X is a sufficient statistic for l. ( Homework 6 )
Y = Σ
=
n
i
i
1
2
X has Gamma ( a = 3 n, q =
λ
1
) distribution. ( Homework 4 )
8. a) If n = 4, find the most powerful rejection region with a = 0.10.
b) Suppose n = 4, and
x 1 = 4, x 2 = 2, x 3 = 4, x 4 = 3.
Find the p-value of the test.
9. Consider the rejection region “Reject H 0 if Σ
=
4
1
2
i
i x ≤ 30”.
c) Find the significance level of this test.
d) Find the power of this test.
10. Let X 1 , X 2 , … , X n be a random sample of size n from a N ( 0 , s
2
) distribution.
We are interested in testing H 0 : s = 2 vs. H 1 : s = 5.
a) Use the likelihood ratio to show that the best rejection region is
C = { ( x 1 , x 2 , … , x n ) : Σ =
n
i i x
1
2
> c }.
b) If n = 10, find the value of c such that a = 0.10.
Hint:
( )
2
2
σ
X μ
Σ i −
has a c
2
( n ) distribution; here μ = 0.
c) If n = 10 and c is from part (b), find the probability of Type II Error.
11. Suppose the lifetime of a particular brand of light bulbs is normally distributed.
A random sample of 27 light bulbs yields an average lifetime of 410 hours and a
sample standard deviation of 67 hours.
a) Find the p-value of the test H 0 : μ = 432 hours vs. H 1 : μ ¹ 432 hours.
b) Test H 0 : s = 56 hours vs. H 1 : s > 56 hours at a 5% level of significance.
c) Find the power of the test in part (b) at s = 84 hours.
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If you are registered for 4 credit hours:
( please put the solution to these problems on the last page of your homework paper )
12. Consider
f 1 ( x ) = sin x, 0 < x < p/2 , zero elsewhere,
f 2 ( x ) = cos x, 0 < x < p/2 , zero elsewhere.
You will have just a single observation of X on which to base your choice between
H 0 : X has p.d.f. f 1 ( x ) vs. H 1 : X has p.d.f. f 2 ( x ).
Use the likelihood ratio to find the best rejection region with the significance level
a = 0.10 and find the power of this test.
13. You will have just a single observation of X on which to base your choice between
H 0 : X has a Normal distribution with mean μ = 5 and standard deviation s = 2
vs.
H 1 : X has a Binomial distribution with n = 25 and p = 0.20. Consider the rejection rule “Reject H 0 if X is an integer”. Find a = P ( Type I Error ) and b = P ( Type II Error ). Justify your answer.