Problem Solutions for Required Homework #9
from Chapter 7, Section 7.5
Required homework:
Problems 29 and 45 (problem 45 is from the Supplementary
Exercises” on page 289).
29.
Given:
μ
= $2.34 per gallon
σ
= $0.20 per gallon
The sampling distribution of the sample mean:
1. E(xbar) =
μ
= $2.34
2. Standard error of the mean =
σ
/√n
For n = 30, standard error of the mean = $0.20/√
30 = $0.0365
For n = 50, standard error of the mean = $0.20/√
50 = $0.0283
For n = 100, standard error of the mean = $0.20/√
100 = $0.0150
For n = 50, standard error of the mean = $0.20/√
100 = $0.0200
3. Shape is normal, by the CLT, because all the sample sizes are
30
≥
a.
With
n
= 30, want
Prob
(
μ
– $0.03 ≤ xbar ≤
μ
+ $0.03)
=
Prob(
$2.14 ≤ xbar ≤ $2.54)
82
.
0
0365
.
0
$
03
.
0
$

=
=
=
x
x
z
σ
μ
Prob
($2.14
≤
x
≤
$2.54)
=
Prob
(0.82
≤
z
≤
0.82)
=
Prob
(
z
≤
0.82) –
Prob
(
z
≤
0.82)
= 0.7939 – 0.2061
= 0.5878
b. With
n
= 30, want
Prob
(
μ
– $0.03 ≤ xbar ≤
μ
+ $0.03)
=
Prob(
$2.14 ≤ xbar ≤ $2.54)
06
.
1
0283
.
0
$
03
.
0
$

=
=
=
x
x
z
σ
μ
ASW 5e Req HW #8 from Chap 7
1
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Prob
($2.14
≤
x
≤
$2.54)
=
Prob
(1.06
≤
z
≤
1.06)
=
Prob
(
z
≤
1.06) –
Prob
(
z
≤
1.06)
= 0.8554 – 0.1446
= 0.7108
c.
With
n
= 100, want
Prob
(
μ
– $0.03 ≤ xbar ≤
μ
+ $0.03)
=
Prob(
$2.14 ≤ xbar ≤ $2.54)
50
.
1
02
.
0
$
03
.
0
$

=
=
=
x
x
z
σ
μ
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 Spring '08
 SINGER
 Standard Error, PROB

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