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Chapter 4
Probability and Sampling Distributions
Chapter 4
1
Preliminary Terminology
•
Random behavior is unpredictable in the short term but has
a regular pattern in the long run that follows the laws of
probability.
•
The outcomes of
independent
events are not inﬂuenced by
any others. We aim to have observations on different
individuals be independent.
•
A
random variable
is a variable whose value is a numerical
outcome of a random event.
•
A statistic (such as the mean) calculated from a randomly chosen
sample is an example of a random variable.
•
Random variables typically have a
distribution
, which is the set of
values the random variable can take and how often it takes each
value.
Chapter 4
2
Section 4.4
The Sampling Distribution of a Sample Mean
Section 4.4
3
Sampling Variability
•
In section 3.3, we saw that a statistic from a
random sample will take different values if we
take more samples from the same population.
•
The statistic is a random variable and varies
according to its
sampling distribution
.
•
In this section we will examine the sampling
distribution of the sample mean
¯
x
.
Section 4.4
4
Sampling Distribution Deﬁned
•
The
sampling distribution
of a statistic is the set
of values that the statistic could take under all
possible samples of size
n
from the population
and how frequently the statistic takes each value.
Section 4.4
5
Estimation
•
Why do we compute
¯
x
for a sample in the ﬁrst
place?
•
If we would like to know something about the
population mean
μ
, it seems
¯
x
would be a
reasonable estimate of
μ
.
•
A simple random sample (SRS) should “fairly”
represent the population, so the mean
¯
x
should
(on average) be near the mean
μ
of the
population.
•
By “fairly”, we mean that a SRS reduces the chance of bias.
•
The mean
¯
x
from a SRS is an
unbiased estimate
of
μ
.
Section 4.4
6
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View Full DocumentSampling Variability
•
Will
¯
x
be exactly equal to
μ
?
•
We do not know
μ
, so there is no way to know for sure.
•
Most likely,
¯
x
is not exactly equal to
μ
.
•
Recall that
¯
x
varies from sample to sample.
•
If we choose another SRS, we will probably get a
different value of
¯
x
.
•
With all of this variability, why is
¯
x
a reasonable
estimate of
μ
?
Section 4.4
7
Sampling Variability
•
¯
x
is a reasonable estimate of
μ
because of the
characteristics we saw in section 3.3.
•
As we include more and more individuals in our sample,
probability theory assures that the statistic
¯
x
will get closer
and closer to the population parameter
μ
.
•
The variability or spread of the sampling distribution becomes
smaller.
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 Spring '08
 ABBEY
 Probability

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