Download Chapter 7

Statistics for Managers
5th Edition
Chapter 7
Sampling Distributions and
Sampling
Chapter Topics
• Sampling Distributions
•Sampling Distribution of the Mean
•Sampling Distribution of the Proportion
•The Situation of Finite Populations
Sampling Distribution - Definition


A sampling distribution is a probability
distribution that shows the relationship
between all of the possible values that a
sample statistic can assume and the
corresponding probabilities.
The sampling distribution of the mean shows
all of the possible values that the sample
mean can assume and the corresponding
probabilities.
Sampling Distribution of the
(X)
mean
Population = 1, 2, 3, 4
so N = 4
Sampling Distribution of X for n = 2
Samples
X
P(X)
X
P(X)
1, 2
1, 3
1, 4
2, 3
2, 4
3, 4
1.5
2.0
2.5
2.5
3.0
3.5
1/6
1/6
1/6
1/6
1/6
1/6
1.5
2.0
2.5
3.0
3.5
1/6
1/6
2/6
1/6
1/6
6/6
or
Summary Measures For
Sampling Distribution
mX_ = S X · P ( X )
= 1.5(1/6 ) + 2.0 (1/6 ) + 2.5 (2/6 ) + 3.0 (1/6 ) + 3.5 (1/6 ) = 2.5
sX_ =
=
S ( X - mX )
_ 2
·P (X)
(1.5 - 2.5)2 · 1/6 + (2.0 - 2.5)2 · 1/6 + (2.5 - 2.5)2 · 2/6+
(3.0 - 2.5)2 · 1/6 + (3.5 - 2.5)2 · 1/6
= .645
Properties of Summary
Measures




Population Mean Equal to
Sampling Mean m x = m
The Standard Error (standard deviation)
of the Sampling distribution is Less than
Population Standard Deviation
Formula:
s x
=
s
n
As n increase, s
x
decrease.
Properties of the Mean



Unbiasedness
 Mean of sampling distribution equals
population mean
Efficiency
 Sample mean comes closer to population mean
than any other unbiased estimator
Consistency
 As sample size increases, variation of sample
mean from population mean decreases
Unbiasedness
P(X)
Unbiased
m
Biased
X
Efficiency
P(X) Sampling
Distribution
of Median
Sampling
Distribution of
Mean
m
X
Consistency
Larger
sample size
P(X)
B
Smaller
sample size
A
m
X
When the Population is Normal
Population Distribution
s= 10
Central Tendency
m _ = m
x
Variation
s
_
s x =
n
Sampling with
Replacement
m = 50
X
Sampling Distributions
n=4
s X = 5
n =16
sX = 2.5
m X-X = 50
X
Central Limit Theorem
As Sample
Size Gets
Large
Enough
Sampling
Distribution
Becomes
Almost Normal
regardless of
shape of
population
X
X
Central Limit Theorem
For a population with a mean u and a
standard deviation s , the sampling
distribution of the means of all possible
samples of size n generated from the
population will be approximately normally
distributed assuming that the sample size is
sufficiently large (n ≥ 30) regardless of the
shape of the distribution of the population.
13
When The Population is
Not Normal
Central Tendency
Population Distribution
s = 10
mx = m
Variation
s
x
=
s
n
Sampling with
Replacement
m = 50
X
Sampling Distributions
n=4
s X = 5
n =30
sX = 1.8
m X = 50
X
Example:
Central Limit Theorem

The number of seconds it takes to complete a
task is normally distributed with a mean of
150 seconds and a standard deviation of 10
seconds.


What is the probability that a particular task takes
more than 152 seconds?
If a random sample of 100 tasks is drawn, what is
the probability that the sample mean number of
seconds it takes to complete them is more than
152?
Example: Central Limit Theorem
m=150 s=10
Population
Sampling Distribution of X
P ( X > 152 ) = .4207
P ( X > 152 when n = 100) = .0228
.0793
.4772
.4207
.0228
150 152
0 .20
Z=
X-m
s
=
X
Z
152 - 150
10
= .20
150 152
0 2.00
X
Z
X-m
152 - 150
Z=
=
= 2.00
10 100
s n
Population Proportions




Categorical variable (e.g., gender)
% population having a characteristic
If two outcomes, binomial distribution
Possess or don’t possess characteristic
Sample proportion (ps)
X number of successes
=
P =
n
sample size
Sampling Distribution of the
Proportion

The sampling distribution of the
proportion shows the relationship
between all of the possible values the
sample proportion can assume and the
corresponding probabilities.
Population Proportions
π = the proportion of the population
having some characteristic

Sample proportion ( p ) provides an estimate
of π:
p=
X
number of items in the sample having the attribute of interest
=
n
sample size

0≤ p≤1

p has a binomial distribution
Sampling Distribution of p

Approximated by a
normal distribution
if n ≥ 50
μp = π
where
P( p)
Sampling Distribution
.3
.2
.1
0
0
.2
.4
.6
π(1 π )
σp =
n
(where π = population proportion)
8
1
p
Sampling Distribution of Proportion
Example

40% of all patients
experience dry mouth
when taking a specific
drug. You select a
random sample of 200
patients. What is the
probability that the
sample proportion of
patients with dry mouth
would be between 40%
& 43% ?
Example: Sampling
Distribution of Proportion

Sampling
Distribution
sp = .0346
π
p
Z
=
π(1  π)
n
-
.43 - .40
.40  ( 1  .40 )
200
= .87
Standardized
Normal Distribution
s=1
..3078
mp = .40
.43
p
m = 0 .87
Z
Sampling from Finite Sample

Modify standard error if sample size (n)
is large relative to population size (N )
n  .05N or n / N  .05



Use finite population correction factor (fpc)
Standard error with FPC

sX =
sP =

S
s
n
N n
N 1
p 1  p  N  n
n
N 1
Sampling Distribution of Proportion from
Finite Population Example

40% of all 1000
patients experience dry
mouth when taking a
specific drug. You
select a random sample
of 200 patients. What
is the probability that
the sample
proportion of patients
with dry mouth would
be between 40% &
43% ?
Solution*
P(.40  P  .43)
p 
.43  .40
z
=
= .97
.40  (1  .40) 1000  200
 (1   ) N  n
200
1000  1
n
N 1
Sampling Distribution
sP = .0310
Standardized Distribution
sZ = 1
.3340
m p = 40 .43
P
m z = 0 .97
Z
Chapter Summary





Introduced sampling distributions
Described the sampling distribution of the mean
 For normal populations
 Using the Central Limit Theorem
Described the sampling distribution of a proportion
Calculated probabilities using sampling distributions
Described the use of the Finite Population Correction