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STA 291 Summer 2010. Lecture 11 Dustin Lueker. Reduce Sampling Variability. 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. 1000 samples of size n=25. Sampling Distribution. - PowerPoint PPT Presentation
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STA 291Summer 2010
Lecture 11Dustin Lueker
2
Reduce Sampling Variability The larger the sample size, the smaller the
sampling variability Increasing the sample size to 25…
10 samplesof size n=25
100 samplesof size n=25
1000 samplesof size n=25
STA 291 Summer 2010 Lecture 11
X
Population with mean m and standard deviation s
X
X
XXXXX
X
• If you repeatedly take random samples andcalculate the sample mean each time, thedistribution of the sample mean follows apattern• This pattern is the sampling distribution
Sampling Distribution
3STA 291 Summer 2010 Lecture
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Example of Sampling Distribution of the Mean
As n increases, the variability decreases and
the normality (bell-shapedness) increases.4
STA 291 Summer 2010 Lecture 11
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Effect of Sample Size The larger the sample size n, the
smaller the standard deviation of the sampling distribution for the sample mean◦ Larger sample size = better precision
As the sample size grows, the sampling distribution of the sample mean approaches a normal distribution◦ Usually, for about n=30, the sampling
distribution is close to normal◦ This is called the “Central Limit Theorem”
x nss
STA 291 Summer 2010 Lecture 11
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< <25)
What about these two?◦ P(X<10)◦ P( <10)
Examples
STA 291 Summer 2010 Lecture 11 6
x
x
7
Sampling Distribution of the Sample Mean When we calculate the sample mean, ,
we do not know how close it is to the population mean ◦ Because is unknown, in most cases.
On the other hand, if n is large, ought to be close to
mm
mx
x
STA 291 Summer 2010 Lecture 11
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Parameters of the Sampling Distribution If we take random samples of size n from a
population with population mean and population standard deviation , then the sampling distribution of
◦ has mean
◦ and standard error
The standard deviation of the sampling distribution of the mean is called “standard error” to distinguish it from the population standard deviation
ms
x
nxSD x
ss )(
mm xxE )(
STA 291 Summer 2010 Lecture 11
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Standard Error The example regarding students in STA 291 For a sample of size n=4, the standard
error of is
For a sample of size n=25,
0.5 0.254X n
ss
0.5 0.125X n
ss
x
STA 291 Summer 2010 Lecture 11
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Central Limit Theorem For random sampling, as the sample size
n grows, the sampling distribution of the sample mean, , 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 is
approximately normal for n≥30◦ We know the parameters of the sampling
distribution
x
x
mm xxE )(n
xSD xss )(
STA 291 Summer 2010 Lecture 11
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Example 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
Also find
)7.25.2( xP
)1.32(. xP
STA 291 Summer 2010 Lecture 11
Binomial Population
with proportion p of successes
• If you repeatedly take random samples andcalculate the sample proportion each time, thedistribution of the sample proportion follows apattern
p̂p̂p̂p̂p̂p̂p̂p̂
p̂
Sampling Distribution
12STA 291 Summer 2010 Lecture
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Example of Sampling Distributionof the Sample Proportion
As n increases, the variability decreases and
the normality (bell-shapedness) increases.13
STA 291 Summer 2010 Lecture 11
For random sampling, as the sample size n grows, the sampling distribution of the sample proportion, , approaches a normal distribution◦ Usually, the sampling distribution of is
approximately normal for np≥5, nq≥5◦ We know the parameters of the sampling
distribution
Central Limit Theorem (Binomial Version)
14
p̂
p̂
ppE p ˆ)ˆ( m
nqp
npppSD p
)()1()ˆ( ˆ
sSTA 291 Summer 2010 Lecture
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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
Does this answer make sense? Also Find
Does this answer make sense?
Example
15
)8.ˆ( pP
)65( XP
STA 291 Summer 2010 Lecture 11
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Mean of sampling distribution 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.
ppE
xE
p
x
ˆ)ˆ(
)(
m
mm
STA 291 Summer 2010 Lecture 11
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Reduce Sampling Variability 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
nxSD x
ss )(nqp
npppSD p
)()1()ˆ( ˆ
s
STA 291 Summer 2010 Lecture 11
