Homework 10 Solution
1. Provided that the two sample sizes are large, a CI for
μ
1

μ
2
with a conﬁdence level of
approximately 1

α
is
¯
x

¯
y
±
z
α/
2
r
s
2
1
m
+
s
2
2
n
Plug in the data, we get the following 95% conﬁdence interval,
10
.
4

9
.
26
±
1
.
96
×
r
4
.
83
2
97
+
4
.
68
2
148
,i.e.(0.08,2.36).
2. (a). Use the paired t test
¯
d
±
t
α/
2
,n

1
s
D
/
√
n
, we get the following 95% conﬁdence interval,
(2.67,17.17).
(b).If we mistakenly thought that these were independent, then we use the 2sample test
¯
X

¯
Y

(
μ
1

μ
2
)
r
S
2
1
m
+
S
2
2
n
(4.09,18.59).
(c). The correlation between the U and A samples is
P
x
i
y
i
σ
x
σ
y
= 0
.
366.
3. Use F test
S
2
1
σ
2
1
S
2
2
/σ
2
2
to get the conﬁdence interval of the two variances.
P
(
F
1

α/
2
,m

1
,n

1
< F < F
α/
2
,m

1
,n

1
) = 1

α
, plug in the data, we get the 90% conﬁdence interval for the ratio
σ
2
2
/σ
2
1
(0.717,15.63), and the
CI for the ratio of the two standard deviations is (0.85,3.95).(The CI for
σ
2
1
/σ
2
2
is(0.064,1.39) and
the CI for
σ
1
/σ
2
is (0.253,1.18)).
4. (a). The picture is on the last page.
(b).
A
∼
(
μ
1
,σ
2
1
),
B
∼
N
(
μ
2
,σ
2
2
).
H
0
:
σ
2
1
/σ
2
2
= 1
, H
1
:
σ
2
1
/σ
2
2
6
= 1.
s
2
1
= (11
.
2735)
2
,s
2
2
= (11
.
8022)
2
.
The test statistic value is:
f
=
s
2
1
s
2
2
= 0
.
9124
We reject
H
0
if
f
≥
F
0
.
05
,
10
,
10
= 2
.
9782 or
f
≤
F
0
.
95
,
10
,
10
= 0
.
3358
.
Since
f
is not in the rejection
region, we fail to reject
H
0
and conclude that there is no signiﬁcant diﬀerence between between 2
standard deviations. The 90% CI of
σ
2
1
/σ
2
2
is given by
[
1
F
0
.
05
,
10
,
10
s
2
1
s
2
2
,
1
F
0
.
95
,
10