Download Chapter 11 Sampling and Sampling Distributions

Sampling and Sampling Distributions
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Aims of Sampling
Probability Distributions
Sampling Distributions
The Central Limit Theorem
Types of Samples
Aims of sampling
Reduces cost of research (e.g. political
polls)
 Generalize about a larger population (e.g.,
benefits of sampling city r/t neighborhood)
 In some cases (e.g. industrial production)
analysis may be destructive, so sampling
is needed
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Probability
Probability: what is the chance that a given
event will occur?
 Probability is expressed in numbers
between 0 and 1. Probability = 0 means
the event never happens; probability = 1
means it always happens.
 The total probability of all possible event
always sums to 1.
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Probability distributions: Permutations
What is the probability distribution of number
of girls in families with two children?
2 GG
1 BG
1 GB
0 BB
0.6
Probability Distribution of
Number of Girls
0.5
0.4
0.3
0.2
0.1
0
0
1
2
How about family of three?
Num. Girls
0
1
1
1
2
2
2
3
child #1
B
B
B
G
B
G
G
G
child #2
B
B
G
B
G
B
G
G
child #3
B
G
B
B
G
G
B
G
Probability distribution of number of girls
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
How about a family of 10?
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
8
9
10
As family size increases, the binomial
distribution looks more and more normal.
0.0
1.0
2.0
Number of Successes
3.0
-0
1
2
3
4
5
6
Number of Successes
7
8
9
10
Normal distribution
Same shape, if you adjusted the scales
B
A
C
Coin toss
Toss a coin 30 times
 Tabulate results
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Coin toss
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Suppose this were 12 randomly selected
families, and heads were girls
If you did it enough times distribution would
approximate “Normal” distribution
Think of the coin tosses as samples of all
possible coin tosses
Sampling distribution
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.
Central Limit Theorem

No matter what we are measuring, the
distribution of any measure across all
possible samples we could take
approximates a normal distribution, as
long as the number of cases in each
sample is about 30 or larger.
Central Limit Theorem
If we repeatedly drew samples from a
population and calculated the mean of a
variable or a percentage or, those sample
means or percentages would be normally
distributed.
Most empirical distributions are not normal:
U.S. Income distribution 1992
N
u
m
b
e
r
o
f
s
a
m
p
l
e
s
Number of samples
But the sampling distribution of mean income over
many samples is normal
18
19
20
21
22
23
24
25
26
Sampling Distribution of Income, 1992 (thousands)
Standard Deviation
Measures how spread
out a distribution is.
Square root of the sum
of the squared
deviations of each
case from the mean
over the number of
cases, or

2


X


 i
N
Example of Standard Deviation
Amount
600
350
275
430
520
X
435
435
435
435
435
(X  X)
n 1
2
s=
=
Deviation from Mean
(X - X)
600 - 435 = 165
350 - 435 = -85
275 - 435 = -160
430 -435 =
-5
520 - 435 = 85
0
2
(X-X)
27,225
7,225
25,600
25
7,225
67,300
67,300
= 16,825 = 129.71
4
Standard Deviation and Normal Distribution
Distribution of Sample Means with 21
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
43
44
45
46
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
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
43
44
45
46
The standard deviation of the sampling
distribution is called the standard error
The Central Limit Theorem
Standard error can be estimated from a single sample:
Where
s is the sample standard deviation (i.e., the
sample based estimate of the standard deviation of
the population), and
n is the size (number of observations) of the sample.
Confidence intervals
Because we know that the sampling distribution is normal,
we know that 95.45% of samples will fall within two
standard errors.
95% of samples fall within 1.96
standard errors.
99% of samples fall within
2.58 standard errors.
Sampling
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.
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Random Sampling
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
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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 (K is a ration obtained by dividing
the population size by the desired sample
size) 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.
Systematic Random Sampling
Stratified Random Sampling
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Proportionate stratified sample – The size
of the sample selected from each subgroup is
proportional to the size of that subgroup in
the entire population. (Self weighting)
Disproportionate stratified sample – The
size of the sample selected from each
subgroup is disproportional to the size of that
subgroup in the population. (needs weights)
Disproportionate Stratified Sample
Stratified Random Sampling
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Stratified random sample – A method of
sampling obtained by (1) dividing the
population into subgroups based on one or
more variables central to our analysis and
(2) then drawing a simple random sample
from each of the subgroups