Download STA 291 Summer 2010

Lecture 11
Dustin Lueker


The larger the sample size, the smaller the
sampling variability
Increasing the sample size to 25…
10 samples
of size n=25
100 samples
of size n=25
STA 291 Summer 2010 Lecture 11
1000 samples
of size n=25
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X
Population
with mean m
and standard
deviation s
X


X
X
X
X
X
X
X
• If you repeatedly
take random
samples and
calculate the
sample mean
each time, the
distribution of the
sample mean
follows a
pattern
• This pattern is
the sampling
distribution
STA 291 Summer 2010 Lecture 11
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
As n increases, the variability decreases and
the normality (bell-shapedness) increases.
STA 291 Summer 2010 Lecture 11
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
The larger the sample size n, the
smaller the standard deviation of the
sampling distribution for the sample
sx
mean
◦ Larger sample size = better precision

As the sample size grows, the sampling
distribution of the sample mean
approaches a normal distribution

s
n
◦ Usually, for about n=30, the sampling
distribution is close to normal
◦ This is called the “Central Limit Theorem”
STA 291 Summer 2010 Lecture 11
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
If X is a random variable from a normal
population with a mean of 20, which of these
would we expect to be greater? Why?
◦ P(15<X<25)
◦ P(15< x <25)

What about these two?
◦ P(X<10)
◦ P( x <10)
STA 291 Summer 2010 Lecture 11
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
When we calculate the sample mean, x , we
do not know how close it is to the
population mean m
◦ Because m is unknown, in most cases.
 On the other hand, if n is large,
to x
m
STA 291 Summer 2010 Lecture 11
ought to be close
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
If we take random samples of size n from a
population with population mean m and
population standard deviation s , then the
sampling distribution of x
◦ has mean
E( x )  mx  m
◦ and standard error
SD( x )  s x 
s
n
 The standard deviation of the sampling distribution of
the mean is called “standard error” to distinguish it from
the population standard deviation
STA 291 Summer 2010 Lecture 11
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

The example regarding students in STA 291
For a sample of size n=4, the standard error
of x is
s
0.5
sX 

n

4
 0.25
For a sample of size n=25,
s
0.5
sX 

 0.1
n
25
STA 291 Summer 2010 Lecture 11
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
For random sampling, as the sample size n
grows, the sampling distribution of the
sample mean, x , approaches a normal
distribution
◦ Amazing: This is the case even if the population
distribution is discrete or highly skewed
 Central Limit Theorem can be proved
mathematically
◦ Usually, the sampling distribution of x is
approximately normal for n≥30
◦ We know the parameters of the sampling
distribution
E( x )  mx  m
SD( x )  s x 
STA 291 Summer 2010 Lecture 11
s
n
10


Household size in the United States (1995)
has a mean of 2.6 and a standard deviation of
1.5
For a sample of 225 homes, find the
probability that the sample mean household
size falls within 0.1 of the population mean
P(2.5  x  2.7)

Also find
P(.2  x  3.1)
STA 291 Summer 2010 Lecture 11
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p̂
Binomial
Population
with
proportion p
of successes
p̂

p̂

p̂
p̂
p̂
p̂
p̂
p̂
• If you repeatedly
take random
samples and
calculate the
sample proportion
each time, the
distribution of the
sample proportion
follows a
pattern
STA 291 Summer 2010 Lecture 11
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
As n increases, the variability decreases and
the normality (bell-shapedness) increases.
STA 291 Summer 2010 Lecture 11
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
For random sampling, as the sample size n
grows, the sampling distribution of the
sample proportion, p̂ , approaches a normal
distribution
◦ Usually, the sampling distribution of p̂ is
approximately normal for np≥5, nq≥5
◦ We know the parameters of the sampling
distribution
E ( pˆ )  m pˆ  p
SD( pˆ )  s pˆ 
p(1  p)

n
STA 291 Summer 2010 Lecture 11
p(q)
n
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

Take a SRS with n=100 from a binomial
population with p=.7, let X = number of
successes in the sample
Find
P ( pˆ  .8)
 Does this answer make sense?

Also Find
P( X  65)
 Does this answer make sense?
STA 291 Summer 2010 Lecture 11
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
Mean/center of the sampling
distribution for sample
mean/sample proportion is
always the same for all n,
and is equal to the
population mean/proportion.
E(x)  mx  m
E ( pˆ )  m pˆ  p
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

The larger the sample size n, the smaller
the variability of the sampling distribution
Standard Error
◦ Standard deviation of the sample mean or sample
proportion
◦ Standard deviation of the population divided by n
SD( x )  s x 
s
n
SD( pˆ )  s pˆ 
p(1  p)

n
STA 291 Summer 2010 Lecture 11
p(q)
n
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