Download Sampling distribution of the mean

POLS 7000X
STATISTICS IN POLITICAL SCIENCE
CLASS 5
BROOKLYN COLLEGE-CUNY
SHANG E. HA
Leon-Guerrero and Frankfort-Nachmias,
Essentials of Statistics for a Diverse Society
Chapter 6: Sampling, Sampling
Distributions, and Estimation
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Aims of Sampling
Probability Sampling
The Concept of the Sampling Distribution
The Sampling Distribution of the Mean
The Central Limit Theorem
Estimation
Procedures for Estimating Confidence Intervals
Confidence Intervals for Proportions
Statistics in Practice: Health Care Reform
Statistics in Practice: The Margin of Error
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Sampling
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Population – A group that includes all the
cases (individuals, objects, or groups) in
which the researcher is interested.
Sample – A relatively small subset from a
population.
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Notation
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Sampling
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Parameter – A measure (for example, mean
or standard deviation) used to describe a
population distribution.
Statistic – A measure (for example, mean or
standard deviation) used to describe a
sample distribution.
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Sampling: Parameter & Statistic
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Probability Sampling
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Probability sampling – A method of
sampling that enables the researcher to
specify for each case in the population the
probability of its inclusion in the sample.
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Random Sampling
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Simple Random Sample – A sample
designed in such a way as to ensure that (1)
every member of the population has an
equal chance of being chosen and (2) every
combination of N members has an equal
chance of being chosen.
This can be done using a computer,
calculator, or a table of random numbers
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Population inferences can be made...
...by selecting a representative sample
from the population
Random Sampling
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Systematic random sampling – A method of
sampling in which every Kth member in the
total population is chosen for inclusion in the
sample after the first member of the sample is
selected at random from among the first K
members of the population.
Population size
Where K =
Sample size
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Sampling Distributions
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Sampling error – The discrepancy between
a sample estimate of a population
parameter and the real population
parameter.
Sampling distribution – A theoretical
distribution of all possible sample values for
the statistic in which we are interested.
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Sampling
SamplingDistributions
Distributions
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Sampling distribution of the mean – A
theoretical probability distribution of sample
means that would be obtained by drawing
from the population all possible samples of
the same size.
If we repeatedly drew samples from a population and
calculated the sample means, those sample means would be
normally distributed (as the number of samples drawn
increases.) The next several slides demonstrate this.
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Standard error of the mean – The standard
deviation of the sampling distribution of the
mean. It describes how much dispersion there
is in the sampling distribution of the mean.
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
The Central Limit Theorem
• If all possible random samples of size N are
drawn from a population with mean y and
a standard deviation y , then as N becomes
larger, the sampling distribution of sample
means becomes approximately normal, with
mean and standard deviation y / N .
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Distribution
Distribution of
of Sample
Sample Means
Means with
with 21
21
Samples
Samples
10
S.D. = 2.02
Mean of means = 41.0
Number of Means = 21
Frequency
8
6
4
2
0
37
38
39
40
41
42
Sample Means
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
43
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Distribution of Sample Means with
96 Samples
14
S.D. = 1.80
Mean of Means = 41.12
Number of Means = 96
12
Frequency
10
8
6
4
2
0
37
38
39
40
41
42
Sample Means
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
43
44
45 46
Distribution of Sample Means with
170 Samples
30
S.D. = 1.71
Mean of Means= 41.12
Number of Means= 170
Frequency
20
10
0
37
38
39
40
41
42
Sample Means
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
43
44
45
46
Estimation Defined:
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Estimation – A process whereby we select a
random sample from a population and use
a sample statistic to estimate a population
parameter.
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Point and Interval Estimation
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Point Estimate – A sample statistic used to
estimate the exact value of a population
parameter
Confidence interval (interval estimate) – A
range of values defined by the confidence
level within which the population parameter is
estimated to fall.
Confidence Level – The likelihood, expressed
as a percentage or a probability, that a
specified interval will contain the population
parameter.
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Estimations Lead to Inferences
Take a subset of the
population
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Estimations Lead to Inferences
Try and reach
conclusions about
the population
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Inferential Statistics Involves
Three Distributions:
A population distribution – variation in the
larger group that we want to know about.
A distribution of sample observations –
variation in the sample that we can observe.
A sampling distribution – a normal
distribution whose mean and standard
deviation are unbiased estimates of the
parameters and allows one to infer the
parameters from the statistics.
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
The Central Limit Theorem Revisited
• What does this Theorem tell us:
– Even if a population distribution is skewed, we know that the sampling
distribution of the mean is normally distributed
– As the sample size gets larger, the mean of the sampling distribution
becomes equal to the population mean
– As the sample size gets larger, the standard error of the mean decreases
in size (which means that the variability in the sample estimates from
sample to sample decreases as N increases).
• It is important to remember that researchers do not
typically conduct repeated samples of the same
population. Instead, they use the knowledge of
theoretical sampling distributions to construct
confidence intervals around estimates.
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications