STAT 420
Fall 2007
Homework #1
(due Friday, August 31, by 3:00 p.m.)
1.
We would like to test the effect of drugs 1 and 2 on a physiological measure X.
Consider the model:
X
1
1
, X
1
2
, … , X
1
n
are
i.i.d.
N
(
μ
1
,
σ
2
)
X
2
1
, X
2
2
, … , X
2
n
are
i.i.d.
N
(
μ
2
,
σ
2
)
Assume that
μ
1
= 6,
μ
2
= 5,
σ
2
= 4.
a)
X
1
1
~
N
(
μ
1
,
σ
2
)
=
N
(
6
, 4
)
2
6
X
11

= Z ~
N
(
0
, 1
)
i)
Find P
(
3 < X
1
1
< 9
)
.
P
(
3 < X
1
1
< 9
)
= P
(
–
1.50 < Z < 1.50
)
=
0.8664
.
ii)
Find
ε
so that P
(

X
1
1
– 6

<
ε
)
= 0.95.
2
°
= 1.96.
ε
=
3.92
.
b)
Let
1
X
=
°
=
n
i
i
n
1
1
X
1
.
i)
Suppose
n
= 25. Find P
(
5 <
1
X
< 7
)
.
1
X
~
N
(
μ
1
,
n
2
±
)
=
N
(
6
,
25
4
)
25
2
6
X
1

= Z ~
N
(
0
, 1
)
P
(
5 <
1
X
< 7
)
= P
(
–
2.50 < Z < 2.50
)
=
0.9876
.
ii)
Suppose
n
= 25. Find
ε
so that P
(

1
X
– 6

<
ε
)
= 0.95.
25
2
°
= 1.96.
ε
=
0.784
.
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iii)
Find the smallest value of
n
so that P
(

1
X
– 6

< 0.1
)
> 0.95.
n
2
0.1
> 1.96.
n
> 39.2.
n
> 1536.64.
n
≥
1537
.
c)
Let
2
X
=
°
=
n
i
i
n
1
2
X
1
, D =
1
X
–
2
X
. Suppose
n
= 25. Find P
(
0 < D < 2
)
.
D =
1
X
–
2
X
~
N
(
μ
1
–
μ
2
,
n
n
2
2
±
±
+
)
=
N
(
6 – 5
,
25
4
25
4
+
)
D ~
N
(
1
, 0.32
)
32
.
0
1
D

= Z ~
N
(
0
, 1
)
P
(
0 < D < 2
)
≈
P
(
–
1.77 < Z < 1.77
)
=
0.9232
.
2.
The salary of junior executives in a large retailing firm is normally distributed with
standard deviation
σ
= $1,500. If a random sample of 25 junior executives yields
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 Spring '07
 STEPANOV
 Statistics, Normal Distribution, Standard Deviation, Statistical hypothesis testing, µ, leastsquares regression line

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