4.
(a) The random variable,
Y
is the number of
tigers with the bacteria present.
A rea
sonable model (given the available infor
mation) is
Y
∼
B
(15
, p
).
The hypothe
ses can be written:
H
0
:
p
= 0
.
05 vs
H
A
:
p >
0
.
05. Evidence against the null
is obtained for ’large’ observed values of
Y
. Using the basic definition of pvalues,
p

val
=
P
(
Y
≥
2). This can be written
as 1

(
P
(
Y
= 0) +
P
(
Y
= 1)). Using the
binomial formula results in
p

val
= 0
.
171.
This means that there is no meaningful ev
idence against the claim that the rate of
occurrence of this bacteria is 0.05 or less.
(b) The random variable
Y
is the number of
cats with bacteria present.
A reasonable
model is
Y
∼
B
(60
, p
).
H
0
:
p
= 0
.
15 vs
H
A
:
p
≥
0
.
15. Since
np
(= 9) and
n
(1

p
)(= 51) are both larger than 5, the nor
mal approx may be used.
Let ˆ
p
=
Y/
60.
Then, under
H
0
,
ˆ
p
NA
∼
N
(
.
15
,
(
.
0461)
2
).
α
=
P
(ˆ
p
≥
0
.
25) =
P
(
Z
≥
2
.
17) =
.
015.
5.
(a) The general form for a CI for the difference
between two proportions is: ( ˆ
p
A

ˆ
p
B
)
±
Z
α/
2
*
p
A
*
(1

p
A
)
n
A
+
p
B
*
(1

p
B
)
n
B
.
Because
p
A
= 0
.
4 and
p
B
= 0
.
6, both expressions
of the form
p
(1

p
) are the same.
Since
Z
α/
2
affects all CIs equally (given the same
1

α
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 Fall '08
 Staff
 Normal Distribution, Standard Deviation, Null hypothesis, Trigraph, binomial formula results

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