1
Chapter 7
Sampling Distributions
David Chow
Oct 2016
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Learning Objectives
In this chapter, you learn:
§The concept of the sampling distribution
§To compute probabilities related to the
sample mean and the sample proportion
§The Central Limit Theorem
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Sampling Distributions
§A sampling distribution is a distribution of all
possible values of a statistic for a given size
sample selected from a population
§Eg: Sampling distribution of mean
§Suppose 5 students are selected
with replacement from N=50
§For each sample you find a
sample mean GPA
§The sampling distribution is the
collection of all ____, one from
each possible sample
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A Simple Example
§Suppose your population consists only of four people, i.e., N=4
§Random variable, X, is age of individuals
§X = 18, 20, 22, 24 (years)
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A Simple Example
21
4
24222018
N
X
µi
=
+++
=
=
2.236
N
μ)(X
σ
2
i==
§N = 4
§X = age of individuals = 18, 20, 22, 24
§Summary statistics:
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A Simple Example
1st
Obs.
2nd Observation
18 20 22 24
18 18,18 18,20 18,22 18,24
20 20,18 29,20 20,22 20,24
22 22,18 22,20 22,22 22,24
24 24,18 24,20 24,22 24,24
Now consider all possible samples
(with replacement) of size n = 2
1st
Obs.
2nd Observation
18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
16 Sample Means
16 possible samples
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A Simple Example
Sampling Distribution of All Sample Means
1st
Obs
2nd Observation
18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
16 Sample Means
18 19 20 21 22 23 24
0
.1
.2
.3
P(X)
X
(no longer uniform)
Sampling Distribution of Mean
_
??
??
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A Simple Example
21
16
24211918
N
X
μi
X=
++++
== !
1.58
16
21)–(2421)–(1921)–(18
N
)μX(
σ
222
2
X
i
X
=
+++
=
=
!
Note that two different distributions (populations) are involved:
1. The ____ distribution
2. The ____ distribution (of all possible sample means)
Summary Measures of this Sampling Distribution:
mean
std
A Simple Example
Population (N = 4)
1.58σ 21µX==
X
2.236σ 21µ==
Sample Means Distribution
(16 observations)
0
.1
.2
.3
P(X)
0
.1
.2
.3
P(X)
_
_
Three Properties
§Suppose the original population is normally distributed with
mean = µand std = s
§The sampling distribution of mean (assuming sample size n)
has two important properties:
(1) Mean (µ`X) = µ
(2) Standard deviation*
n
σ
σX=
* Assume the sampling is conducted with replacement, or n/N < 5%
