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Outline
The
distribution of sample means
Law
of large numbers
Sampling
error
Central
limit theorem
Expected
value
Standard
error
Large
samples will be representative of the population from which they are selected.
Sample
size is one of the primary considerations in determining how well a sample mean
represents the population mean.
The
discrepancy between a sample statistic (M) and its corresponding population
parameter (m).
The
goal of inferential statistics is to use the limited information from samples to
draw conclusions about populations.
We
can predict sample characteristics from the distribution of sample means.
This is a collection of sample means for all possible random samples of a
particular size (n) that can be obtained from a population
So
far we have discussed distributions of scores, now we will discuss distributions
of statistics.
Statistics
are properties of samples.
A distribution of statistics (e.g. Ms) is referred to as a sampling
distribution.
Figure
7-2 (p. 203)
Frequency
distribution histogram for a population of 4 scores: 2, 4, 6, 8.
Figure 7-3
(p. 205)
The
distribution of sample means for n
= 2. The distribution shows the 16 sample means from Table 7.1.
A
distribution of statistics obtained by selecting all the possible samples of a
specific size from a population.
The
sampling distribution of the mean has some characteristics:
1.
The sample means tend to pile up around the population mean.
2.
The distribution is approximately normal in shape.
3.
We can use the distribution of sample means to answer probability
questions about sample means.
The
distribution of sample means has certain known characteristics that can be
applied to any situation. These
characteristics can be expressed in the
CENTRAL
LIMIT THEOREM
For
any population with mean m
and standard deviation s,
the distribution of sample means for sample size n will approach a normal
distribution with a mean of m
and a standard deviation of s
= s
/square root of n, as n approaches infinity.
By
the time n = 30, the distribution is almost perfectly normal, regardless of the
shape of the original population distribution.
The
mean of the sampling distribution is also known as the “expected value of M”
The
standard deviation of the sampling distribution is called the “standard
error”, and expresses the standard distance between M and m.
Think
of the standard error as a numerical index of the extent to which means
vary from one sample to another. An
index of the amount of error that results when a single sample mean is used to
estimate the population mean. In
that sense, it is an index of sampling error.
The
location of each M in the sampling distribution can be specified by a z-score.
The
value of the standard error is determined by two characteristics:
(a) the variability of the original population (b) sample size.
The
primary use of the sampling distribution is to find the probability associated
with any specific sample.