STA 291 Summer 2010
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STA 291Summer 2010
Lecture 11Dustin Lueker
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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
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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
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Example of Sampling Distribution of the Mean
As n increases, the variability decreases and
the normality (bell-shapedness) increases.4
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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
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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< <25)
What about these two?◦ P(X<10)◦ P( <10)
Examples
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x
x
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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
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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 )(
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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
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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 )(
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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
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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
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Example of Sampling Distributionof the Sample Proportion
As n increases, the variability decreases and
the normality (bell-shapedness) increases.13
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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)
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p̂
p̂
ppE p ˆ)ˆ( m
nqp
npppSD p
)()1()ˆ( ˆ
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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
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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
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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
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