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Chapter 7
Sampling Distributions
McGraw-Hill/Irwin
Copyright © 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved.
Sample from Population
Population
Random Sample of size n
Sample x
Parameter mean µ
inference
7-2
Sampling Distribution of the Sample
Mean
The sampling distribution of the
sample mean x of size n is
the
probability distribution of the population of
the sample means obtainable from all
possible samples of size n from a
population.
7-3
General Conclusions
•
The mean of all possible sample means
equals the population mean
–
•
That is, m = mx
The variance s2x of the sampling distribution
of x is
s 
2
x
•
Continued
s
2
n
The standard deviation sx of the sampling
distribution of x is
s
sx 
n
7-4
Sample from Population
Population
Random Sample of size n
Parameter mean µ and
standard deviation σ.
x has mean mx=m, and standard
deviation s  s
x
n
7-5
General Conclusions
1. If the population of individual items is
normal, then the population of all sample
means, x‘s, is also2 normal with mean mx=m
s
2
s

and variance x n
or equivalently
sx 
s
n
2. Suppose the population has mean m and
standard deviation σ. Even if the population
of individual items is not normal, but the
sample size is big enough (n ≥30), the
distribution of x is approximately normal
s
with mx=m and s x 
(Central Limit Theorem)
n
7-6
Example: Effect of Sample Size
The larger the sample
size, the more nearly
normally distributed is
the population of all
possible sample means
Also, as the sample
size increases, the
spread of the sampling
distribution decreases
7-7
Properties of the Sampling
Distribution of the Sample Mean #3
• The standard deviation sx of the
sampling distribution of x is
sx 
s
n
• That is, the standard deviation of the
sampling distribution of x is
– Directly proportional to the standard
deviation of the population
– Inversely proportional to the square root of
the sample size
7-8
How Large?
• How large is “large enough?”
• If the sample size is at least 30, then for
most sampled populations, the sampling
distribution of sample means is
approximately normal
– Here, if n is at least 30, it will be assumed
that the sampling distribution of x is
approximately normal
• If the population is normal, then the
sampling distribution of x is normal
regardless of the sample size
7-9
Example 7.2: Car Mileage Statistical
Inference
• Suppose for the Chapter 3 mileage
example, x = 31.56 mpg for a sample of
size n=50 and suppose the population
standard deviation σ = 0.8
• If the population mean µ is exactly 31
and standard deviation is 0.8, what is
the probability of observing a sample
mean that is greater than or equal to
31.56?
7-10
Example 7.2: Car Mileage Statistical
Inference #2
• Calculate the probability of observing a sample mean
that is greater than or equal to 31.6 mpg if µ = 31
mpg
– Want P(x > 31.56 if µ = 31 σ=0.8 for the
population)
• Since the condition for Central Limit Theorem are
satisfied, we can use normal distribution to
approximate the probability. To find out the normal
distribution, we need to decide the mean and
standard deviation:
mx=m31
sx 
s
0.8

 0.113
n
50
7-11
Example 7.2: Car Mileage Statistical
Inference #3
• Then

31.56  m x 

Px  31.56 if m  31  P z 
sx


31.56  31 

 P z 

0
.
113


 Pz  4.96
 1  0.99997  0.00003
• The distribution of x bar has mean 31
and standard deviation 0.113
• But z = 4.96 is off the standard normal
table, 3.99, P(Z<4.96)=0.99997
• The largest z value in the table is 3.99,
which has a right hand tail area of
7-12
Central Limit Theorem
• Now consider sampling a non-normal
population
• Still have: mx=m and sx=s/n
– Exactly correct if infinite population
– Approximately correct if population size N finite
but much larger than sample size n
• But if population is non-normal, what is the
shape of the sampling distribution of the
sample mean?
– The sampling distribution is approximately normal
if the sample is large enough, even if the
population is non-normal (Central Limit Theorem)
7-13
The Central Limit Theorem #2
• No matter what is the probability distribution
that describes the population, if the sample
size n is large enough, then the population of
all possible sample means is approximately
normal with mean mx=m and standard
deviation sx=s/n
• Further, the larger the sample size n, the
closer the sampling distribution of the sample
mean is to being normal
– In other words, the larger n, the better the
approximation
– norecord
7-14
The Central Limit Theorem #3
Random Sample (x1, x2, …, xn)
x
X
as n  large
Population Distribution
(m, s)
(right-skewed)
m
Sampling
Distribution of
Sample Mean
x
 m,s x  s
n

(nearly normal)
7-15