Transcript STA 291-021 Summer 2007 - University of Kentucky

Lecture 12
Dustin Lueker

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.
STA 291 Spring 2008 Lecture 12
E(x)  x  
E ( pˆ )   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 )   x 

n
SD( pˆ )   pˆ 
STA 291 Spring 2008 Lecture 12
p(1  p)

n
p(q)
n
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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 )   x  
STA 291 Spring 2008 Lecture 12
SD( x )   x 

n
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For random sampling, as the sample size n
grows, the sampling distribution of the
sample proportion, pˆ , approaches a normal
distribution
◦ Usually, the sampling distribution of pˆ is
approximately normal for np≥5, nq≥5
◦ We know the parameters of the sampling
distribution
E( pˆ )   pˆ  p
SD( pˆ )   pˆ 
p(1  p)

n
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p(q)
n
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Inferential statistical methods provide
predictions about characteristics of a
population, based on information in a sample
from that population
◦ Quantitative variables
 Usually estimate the population mean
 Mean household income
◦ For qualitative variables
 Usually estimate population proportions
 Proportion of people voting for candidate A
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Point Estimate
◦ A single number that is the best guess for the
parameter
 Sample mean is usually at good guess for the
population mean
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Interval Estimate
◦ Point estimator with error bound
 A range of numbers around the point estimate
 Gives an idea about the precision of the estimator
 The proportion of people voting for A is between 67%
and 73%
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A point estimator of a parameter is a sample
statistic that predicts the value of that parameter
A good estimator is
◦ Unbiased
 Centered around the true parameter
◦ Consistent
 Gets closer to the true parameter as the sample size gets
larger
◦ Efficient
 Has a standard error that is as small as possible (made use
of all available information)
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An estimator is unbiased if its sampling
distribution is centered around the true
parameter
◦ For example, we know that the mean of the
sampling distribution of x equals μ, which is the
true population mean
 Thus, x is an unbiased estimator of μ
 Note: For any particular sample, the sample mean
be smaller or greater than the population mean
 Unbiased means that there is no systematic
underestimation or overestimation
STA 291 Spring 2008 Lecture 12
x
may
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A biased estimator systematically
underestimates or overestimates the
population parameter
◦ In the definition of sample variance and sample
standard deviation uses n-1 instead of n, because
this makes the estimator unbiased
◦ With n in the denominator, it would systematically
underestimate the variance
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An estimator is efficient if its standard error is
small compared to other estimators
◦ Such an estimator has high precision
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A good estimator has small standard error
and small bias (or no bias at all)
◦ The following pictures represent different
estimators with different bias and efficiency
◦ Assume that the true population parameter is the
point (0,0) in the middle of the picture
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Note that even an
unbiased and efficient
estimator does not
always hit exactly the
population parameter.
But in the long run,
it is the best estimator.
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Sample mean is unbiased, consistent, and
(often) relatively efficient for estimating μ
Sample standard deviation is almost unbiased
for estimating population standard deviation
◦ No easy unbiased estimator exists
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Both are consistent
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Suppose we want to estimate the proportion of
UK students voting for candidate A
We take a random sample of size n=100
The sample is denoted X1, X2,…, Xn, where Xi=1
if the ith student in the sample votes for A, Xi=0
otherwise
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Estimator1 = the sample mean (sample
proportion)
Estimator2 = the answer from the first
student in the sample (X1)
Estimator3 = 0.3
Which estimator is unbiased?
Which estimator is consistent?
Which estimator is efficient?
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