98
Instructor’s Solutions Manual for Statistics for Business & Economics, 5
th
Edition
Chapter 7: Sampling and Sampling Distributions
71
a.
Probability distribution for one die:
Die outcome
Probability
1
1/6
2
1/6
3
1/6
4
1/6
5
1/6
6
1/6
b.
Sampling distribution of the sample means from rolling a pair of dice:
Total
Sample
x
Prob. of
x
2
1 1
1
1/36
3
1 2, 2 1
1.5
2/36
4
1 3, 3 1, 2 2
2
3/36
5
1 4, 4 1, 2 3, 3 2
2.5
4/36
6
1 5, 5 1, 2 4, 4 2, 3 3
3
5/36
7
1 6, 6 1, 2 5, 5 2, 3 4, 4 3
3.5
6/36
8
2 6, 6 2, 3 5, 5 3, 4 4
4
5/36
9
3 6, 6 3, 4 5, 5 4
4.5
4/36
10
4 6, 6 4, 5 5
5
3/36
11
5 6, 6 5
5.5
2/36
12
6 6
6
1/36
7.2
The sampling distribution of the sample mean can be generated by listing out all
possible samples of size n, calculate each possible
x
, determine the probability of
each possible
x
and generate the sampling distribution.
Alternatively, the
probabilities of each
x
can be generated by use of the binomial formula.
a.
When n = 5:
Use the binomial formula for x = 0, x = 1, etc.:
X
P(X)
x
0
.07776
0
1
.25920
.2
2
.34560
.4
3
.23040
.6
4
.07680
.8
5
.01024
1.0
(
)
x
E p
= np = (5)(.4) = 2.0,
2
(1
)
p
p
p
n
σ

=
=
.4(.6)
5
=
= .048,
.2191
p
σ
=
b.
Using the result from part a:
(
)
x
E p
= np = (100)(.4) = 40,
2
(1
)
p
p
p
n
σ

=
=
.4(.6)
100
=
= .0024,
.04899
p
σ
=
73
The response should note that there will be errors in taking a census of the entire
population as well as errors in taking a sample.
Improved accuracy can be achieved via
sampling methods versus taking a complete census (see reference to Hogan, 90).
By