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Sampling Distributions
Introduction
Learning objectives for this lesson
Upon completion of this lesson, you should be able to:
determine the standard error for the sample proportion
determine the standard error for the sample mean
apply the Central Limit Theorem properly to a set of continuous data
Sampling Distributions of Sample Statistics
Two common statistics are the sample proportion,
, (read as “phat”) and sample mean,
, (read as “xbar”). Sample statistics are random variables and
therefore vary from sample to sample. For instance, consider taking two random samples, each sample consisting of 5 students, from a class and calculating the
mean height of the students in each sample. Would you expect both sample means to be exactly the same? As a result, sample statistics also have a distribution
called the
sampling distribution
. These sampling distributions, similar to distributions discussed previuosly, have a mean and standard deviation. However, we
refer to the standard deviation of a sampling distribution as the
standard error
. Thus, the standard error is simply the standard deviation of a sampling
distribituion. Often times people will interchange these two terms. This is okay as long as you understand the distinction between the two: standard error refers to
sampling
distributions and standard deviation refes to
probability
distributions.
Sampling Distributions for Sample Proportion, phat
If numerous repetitions of samples are taken, the distribution of
is said to approximate a normal curve distribution. Alternatively, this can be assumed if BOTH
n
*p and
n
*(1  p) are
at least
10. [
SPECIAL NOTE:
Some textbooks use 15 instead of 10 believing that 10 is to liberal. We will use 10 for our discussions.]
Using this, we can estimate the true population proportion, p, by
and the true standard deviation of p by s.e.(
) =
, where s.e.(
) is interpreted as
the
standard error of
Probabilities about the number X of successes in a binomial situation are the same as probabilities about corresponding proportions.
In general, if
np
>= 10 and
n
(1
p
) >= 10, the sampling distribution of
is about normal with mean of p and standard error SE(
) =
.
Example.
Suppose the proportion of all college students who have used marijuana in the past 6 months is
p
= .40. For a class of size
N
= 200, representative of all
college students on use of marijuana, what is the chance that the proportion of students who have used mj in the past 6 months is less than .32 (or 32%)?
Sampling Distributions
http://onlinecourses.science.psu.edu/stat200/book/export/html/42
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9/27/2010 11:48 PM
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View Full Document Solution. The mean of the sample proportion
is p and the standard error of
is SE(
)
=
.
For this marijuana example, we are given that p = .4. We
then determine
SE(
) =
=
=
= 0.0346
So, the sample proportion
is about normal with mean p = .40 and SE(
) = 0.0346.
The zscore for .32 is z = (.32  .40) / 0.0346 = 2.31. Then using
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This note was uploaded on 10/08/2010 for the course STAT 200 taught by Professor Barroso,joaor during the Fall '08 term at Pennsylvania State University, University Park.
 Fall '08
 BARROSO,JOAOR
 Statistics, Standard Error

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