Chapter 8:
Confidence
Intervals
Read Chapters 8 and 9
Sampling Distributions
A sample statistic is a random variable
!
This
means that it has its own distribution.
Def 1: The probability distribution of a sample
statistic is called the
sampling distribution
of
that statistic.
Def 2: The
sampling distribution
of a statistic is
the population of all possible values for that
statistic under repeated sampling with a fixed
sample size.
3
If random samples, each consisting of
n
measurements, are repeatedly drawn from the
same population having true mean
μ
and standard
deviation
σ
, then when
n
is large, the relative
frequency histogram for the sample means
(calculated from the repeated samples) will be
approximately normal with mean
and standard
deviation
, that is,
Central Limit Theorem
~
approx Normal( ,
)
X
n
n
Point Estimate and Interval Estimate
±
A
point estimate
is a
single number
that is our “best guess” for the parameter
±
An
interval estimate
is an
interval of
numbers
within
which the parameter
value is believed to fall.
Point Estimate vs. Interval Estimate
±
A
point estimate
doesn’t tell us how close
the estimate is likely to be to the
parameter
±
An
interval estimate
is more useful
²
It incorporates a margin of error which helps
us to gauge the accuracy of the point estimate
Properties of Point Estimators
±
Property 1:
A good estimator has a sampling
distribution that is centered at the parameter
²
An estimator with this property is
unbiased
±
The sample mean is an unbiased estimator of
the population mean
±
The sample proportion is an unbiased estimator
of the population proportion
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View Full DocumentProperties of Point Estimators
±
Property 2:
A good estimator has a
small
standard error
compared to other
estimators
²
This means it tends to fall closer than other
estimates to the parameter
±
The sample mean has a smaller standard error
than the sample median when estimating the
population mean of a normal distribution
Confidence Interval
±
A
confidence interval
is an interval
containing the most believable values for a
parameter
±
The probability that this method produces
an interval that contains the parameter is
called the
confidence level (1
α
)x100%
²
This is a number chosen to be close to 1,
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 '07
 VELLEMANP
 Normal Distribution

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