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Sampling Distributions
What are they?
Why do we care?
Our Goal? To make a decisions.
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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Lesson Objectives

Know what is meant by the
“sampling distribution” of a statistic,
and the “population of all possible
X-bar values.”

Know when the population of all
possible X-bar values IS normal.

Know when the population of all
possible X-bar values IS NOT normal.
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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Statistics
Descriptive
Graphical
Numerical
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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Statistical Inference
Generalizing from a sample
to a population,
by using a statistic
to estimate
a parameter.
Goal:
 Department of ISM, University of Alabama, 1992-2003
.
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Population
Sample
Census
a guess
Statistic
True
Parameter
Statistic
Parameter
Mean:
X
estimates
m
Standard
deviation:
s
estimates
s
Proportion:
p
estimates
p
Objective of this section:
A statistic is a
.
Before we can make decisions
about parameters and control
the degree of risk, we must know:
 the
 and its
 Department of ISM, University of Alabama, 1992-2003
of the statistic
values.
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Example 1:
Original Population: 300 ST 260 students.
X = Exam 2 score
m = population mean (unknown)
s = population std deviation (unknown)
Sample
Population
Calculate:
n=4
x = mean
s = std dev
Think of X as a random variable.
Fact:
Different samples of size “n”
will produce different values
of the sample mean.
The population mean is fixed
as long as the population does
not change.
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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For samples of size n,
what is the distribution of
the statistic X?
• Shape?
(Skewed? Symmetric?)
• Center? (Mean? Median?)
• Spread? (Std. Deviation? IQR?)
• Is it one of our “special” distributions?
(Normal? Exponential?)
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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For samples of size n,
what is the distribution of ^
p,
i.e, a sample proportion?
• Shape?
(Skewed? Symmetric?)
• Center? (Mean? Median?)
• Spread? (Std. Deviation? IQR?)
• Possible values? (0/n, 1/n, 2/n, …, n/n)
• Is it one of our “special” distributions?
(normal, exponential, binomial, Poisson)
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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Example 1, continued:
From pop. of all ST 260 students,
randomly select n = 1 student.
Record exam 2 grade:
Sampled value: 76
X = 76.0
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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Example 1, continued:
From pop. of all ST 260 students,
randomly select n = 4 students.
Record exam 2 grades:
Sampled values: 64, 78, 94, 46
X = (64 + 78 + 94 + 46) / 4 = 70.5
 Department of ISM, University of Alabama, 1992-2003
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Fact:
x’s from
samples
tend to be
to
the true mean, m, than
x ’s from smaller samples.
 Department of ISM, University of Alabama, 1992-2003
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Sampling Distribution of X
is the distribution of all possible
sample means
calculated from
all possible samples
of size n.
Also called “the population
of all possible x-bars”.
 Department of ISM, University of Alabama, 1992-2003
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And so on, . . . ,
every
until we collect
possible sample of size n = 4.
How many samples
of size 4 are there from a
population of 300 members?
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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Sampling Distribution
of x for n = 4
And the shape
looks like a
Normal dist.
Based on all
samples of
size n = 4
sx
mx
 Department of ISM, University of Alabama, 1992-2003
x-axis
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Definitions, from previous slide:
mx
sx
= average of all possible X’s
(center of the sampling dist.)
= std. deviation of all pos. X’s
(spread of the sampling dist.)
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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Compare parameters of the
original population of all scores
and the parameters of the
sampling dist. of all possible x’s
Original population:
mean =
mx
sx
=
=
m
m
s
n
& std. dev. =
s
(same mean as
individual values)
(different std. dev.,
but related!)
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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Original Population: 300 ST 260 students.
X = Exam 2 score.
If
m = 75
and
s = 10,
then the population of all possible
X-values for n = 4 will have
mx = m =
s
sx =
=
n
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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Questions
What is the probability that one randomly
selected Exam2 score will be within
10 points of the population mean, 75?
s=
, X:
Z:
65 to 85
What is the probability that a sample mean
of n = 4 randomly selected Exam 2 scores
will be within 10 points of the pop. mean?
sX =
, X: 65 to 85
Z:
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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We now know the parameters
of the population of
all possible x-bar values.
?
What is the distribution
Look back at the plot exam 2 grades.
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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If original population has a
Normal dist., then the distribution
of X values is Normal also.
If X ~ N ( m,
s),
then for samples of size n,
s
X~N(m,
).
n
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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Original Population: Normal (m = 50, s = 18)
s = 18.00
0
10 20 30 40 50 60 70 80 90 100
X
 Department of ISM, University of Alabama, 1992-2003
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Original Population: Normal (m = 50, s = 18)
n = 36
n = 16
s x = 3.00
s x = 4.50
s x = 9.00
n=4
n = 2 s = 12.73
x
s = 18.00
0
10 20 30 40 50 60 70 80 90 100
X
 Department of ISM, University of Alabama, 1992-2003
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Example 2:
Bottle filling machine for soft drink.
Bottles should contain 20.00 ounces;
assume actual contents follow a
normal distribution with a
mean of 20.18 oz. and a
standard deviation of 0.12 oz.
X = contents of one randomly selected bottle
X ~ N( m = 20.18, s = 0.12)
 Department of ISM, University of Alabama, 1992-2003
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a. Find the proportion of
individual bottles contain
less than 20.00 oz?
X = content of one bottle.
X ~ N(m = 20.18, s =
)
-4.0
P( X < 20.00) =
= P( Z <
=
=
)
-3.0
Z =
20.0 20.18
0
-2.0
-1.0
0.0
1.0
2.0
X-axis
Z-axis
3.0
4.0
=
of the bottles will
contain less than 20.00 ounces.
Is this a problem?
 Department of ISM, University of Alabama, 1992-2003
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a. Find the proportion of
six-packs whose mean
content is less than 20.00 oz?
Is population of x-bars Normal?
Yes; because original pop. is Normal.
X = mean of six-pack.
X ~ N(m X = 20.18, sx =
sx =
)
P( X < 20.00) =
= P( Z <
=
=
)
-4.0
-3.0
Z =
)
20.0 20.18
0
-2.0
-1.0
0.0
1.0
2.0
X-axis
Z-axis
3.0
4.0
=
Only ________% of the six-packs
will contain an average
less than 20.00 ounces.
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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 Department of ISM, University of Alabama, 1992-2003
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New situation
But what if the
original population
not
is
normally
distributed?
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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Demonstration
of the
Central Limit Theorem
Page 289
 Department of ISM, University of Alabama, 1992-2003
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Original Population: Exponential (m = 4)
s=4
0.6
0.5
0.4
0.3
0.2
0.1
n=1
0.0
0
44
8
12
16
Original Population: Exponential (m = 4)
s=4
0.6
s x = 0.730
n = 30
0.5
0.4
n = 15
0.3
0.2
n=5
n=2
0.1
0.0
0
44
8
s x = 1.033
s x = 1.789
s x = 2.828
12
16
Sampling distribution for X
If original population does
NOT have a Normal dist.,
the X values are approximately
Normal IF n is large.
If X ~ NOT Normal, then
for large samples of size n,
s
X~N(m,
), approximately.
n
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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This same phenomena will happen
for ANY non-normal distribution,
IF “n” is BIG!
How big is
BIG?
Bigger is better, but
is enough!
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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Example 3 (C.L.T.)
“Investment opportunity”
Earnings: x -80
0 +60
P(X=x) .40 .10 .50
P(player looses) = .40
Expected value:
m=
-80 (.40) + 0 (.10) + 60 (.50)
= -2.00.
 Department of ISM, University of Alabama, 1992-2003
Also, s = 66.0
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After 36 plays, what
is the probability that
the average earnings
is negative?
X = earning for one play
X ~ NOT Normal
X = Avg. earnings, 36 plays
s x=
-4.0
-3.0
-2.0
-1.0
-2.0
0
0.0
1.0
2.0
X-axis
Z-axis
3.0
4.0
X-bar pop. is Normal because n is BIG.
X ~ N(m = -2.0, sx =
)
P( X < 0.0) = ?
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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After 6400 plays, what
is the probability that
the average earnings
is negative?
Same as previous,
BUT . . . .
s x=
X ~ N(m = -2.0, sx =
-4.0
-3.0
-2.0
-1.0
-2.0
0.0
1.0
2.0
X-axis
Z-axis
3.0
4.0
)
P( X < 0.0) = ?
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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Summary different values of”n”
Number
of plays
1
36
100
6400
12,000
P( you lose)
.4000
.5714
.6179
.9922
.9995
 Department of ISM, University of Alabama, 1992-2003
Expected Total
Amount of
earnings
-2
-72
-200
-12,800
-24,000
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 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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Example 4
(C.L.T.)
On-the-job accidents in a company.
X = number of accidents in one week
X ~ Poisson ( l = 2.2 acc/wk )
a. Find the probability of having
two or fewer accidents in one
randomly selected week.
P(X < 2) =
,
from Table A.4.
This is a Chapter 6 problem.
The probability of being two or less is greater than .5,
but the mean is 2.2! How is this possible?
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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Example 4
continued
On-the-job accidents in a company.
X = number of accidents in one week
X ~ Poisson ( l = 2.2 acc/wk )
Poisson, mean = 2.2
0.300
0.250
0.200
0.150
0.100
0.050
0.000
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
Number of Accidents
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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Example 4
continued
On-the-job accidents in a company.
X = number of accidents in one week
X ~ Poisson ( l = 2.2 acc/wk )
b. What is the probability that the
average number of accidents for
next 52 weeks will be 2.0 or less?
X = mean for 52 weeks; n = 52.
What is the sampling distribution?
Ori. pop. is definitely NOT normal; BUT n is large!
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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Example 4
continued
On-the-job accidents in a company.
X = number of accidents in one week
X ~ Poisson ( l = 2.2 acc/wk )
What is the sampling distribution of X ?
By the C.L.T., it is approximately Normal.
Recall: for Poisson the mean is l, the
standard deviation is the square root of l.
X ~
N (m
X
= 2.2 ,
s X = 1.483/
52 )
= 0.2057
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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Example 4 cont.
b. What is the probability that
the average number of
accidents for next 52 weeks
will be 2.0 or less?
X = mean of accidents.
X ~ N(m x= 2.2, sx = _______)
-4.0
P( X < 2.0) =
-3.0
-2.0
-1.0
2.2
0
0.0
1.0
2.0
X- axis
Z-axis
3.0
4.0
It is much less likely that the
average number of accidents
per week will be two or less,
than any one specific week.
 Department of ISM, University of Alabama, 1992-2003
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Reminder
When is the population of
all possible X values Normal?
 Anytime the original pop.
is Normal (true for any n).
 Anytime the original pop.
is not Normal, but
n is BIG (n > 30).
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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Remember
When is the population of
all possible X values NOT Normal?

Anytime the original population
is not Normal AND
n is NOT BIG.
 Department of ISM, University of Alabama, 1992-2003
M31- Dist of X-bars
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