# Chapter 10 Sampling and Sampling Distributions

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```					            Chapter 10
Sampling and Sampling Distributions
Sampling
• Population – A group that includes all the cases
(individuals, objects, or groups) in which the
researcher is interested.
– All third grade children in the United States
– All the wheat crops in North America
– All college students
• Sample – A relatively small subset from a
population.
– 3rd grade children in Mrs. Nozingo’s class at Wiles
Elementary
– Southernmost acre of wheat in each farm
Chapter 10 – 1
– Students taking SOCI 4880 at UNT                Chapter 10 – 2

Notation                                                     Sampling
• 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.

Chapter 10 – 3                                                             Chapter 10 – 4
Sampling: Parameter & Statistic                                                      What is the probability that I would role a 6, assuming
that I have a honest die?

P(6) = 1/6 or .1667

Chapter 10 – 5                                                               Chapter 10 – 6

What is the probability or all possible outcomes, again assuming that I
have a honest die?
P(1) = 1/6 or .1667                                        Probability Sampling
P(2) = 1/6 or .1667
P(3) = 1/6 or .1667                              • Probability sampling – A method of sampling that
P(4) = 1/6 or .1667                                enables the researcher to specify for each case in
P(5) = 1/6 or .1667
P(6) = 1/6 or .1667                                the population the probability of its inclusion in the
total 6/6 or 1.0000                                sample.
And, if it was perfectly                                                            • Goal:
dishonest, such that it          P(1) = 0/6 or .0000                                   – Representative Sample
only rolled 6s?                  P(2) = 0/6 or .0000                                      • Reproduces the important characteristics of the population
P(3) = 0/6 or .0000                                      • If population is 40% female, so should our sample
P(4) = 0/6 or .0000
P(5) = 0/6 or .0000
P(6) = 6/6 or 1.000
total 6/6 or 1.0000               Chapter 10 – 7                                                               Chapter 10 – 8
Random Sampling
EPSEM
• Simple Random Sample – A sample
• Equal Probability of Selection Method                              designed in such a way as to ensure that
– Fundamental Principle of probability sampling                    – (1) every member of the population has an
– Every element or case in the population must                       equal chance of being chosen
have an equal probability of being selected for                  – (2) every combination of N members has an
the sample                                                         equal chance of being chosen.
– EPSEM does not guarantee representativeness                      – (3) Need a complete list of population
• This can be done using a computer,
calculator, or a table of random numbers

Chapter 10 – 9                                                    Chapter 10 – 10

...by selecting a representative sample
from the population

Chapter 10 – 11                                                   Chapter 10 – 12
Random Sampling                                       Systematic Random Sampling

• Systematic random sampling – A method of
sampling in which every Kth member (K is a
ratio 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.

Chapter 10 – 13                                              Chapter 10 – 14

Stratified Random Sampling                                    Stratified Random Sampling
• Proportionate stratified sample – The size
• Stratified random sample – A method of                        of the sample selected from each subgroup
sampling obtained by                                          is proportional to the size of that subgroup
• (1) dividing the population into subgroups                    in the entire population.
based on one or more variables central to                   • Disproportionate stratified sample – The
our analysis                                                  size of the sample selected from each
• (2) then drawing a simple random sample                       subgroup is disproportional to the size of
from each of the subgroups                                    that subgroup in the population.

Chapter 10 – 15                                              Chapter 10 – 16
Disproportionate Stratified Sample
Sampling Distributions
• 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.
– Represents every conceivable combination of
cases from the population

Chapter 10 – 17                                                    Chapter 10 – 18

Sampling Distributions                                                           Example:
• Sampling distribution of the mean – A theoretical                      • We want to know age of a community of
probability distribution of sample means that would                      10,000
be obtained by drawing from the population all                         • We draw and EPSEM sample of 100
possible samples of the same size.                                       residents their age
If we repeatedly drew samples from a population and                      – Mean: 27
calculated the sample means, those sample means would be
normally distributed (as the number of samples drawn                   • Now we toss the first 100 back in the mix
increases.) The next several slides demonstrate this.
• Draw another sample of 100
• Standard error of the mean – The standard
– Mean: 30
deviation of the sampling distribution of the mean.
It describes how much dispersion there is in the                       • We continue to do this over and over again
sampling distribution of the mean.                                       – Ultimately the distribution of sample means is
normal
Chapter 10 – 19                                                    Chapter 10 – 20
Distribution of Sample Means                                                                                                  Distribution of Sample Means
with 21 Samples                                                                                                               with 96 Samples
14
10                                                                                                                                                 S.D. = 1.80
12                                      Mean of Means = 41.12
S.D. = 2.02
Mean of means = 41.0                                                                                              Number of Means = 96
8                                                                                                         10
Number of Means = 21
8

Frequency
Frequency

6
6
4
4

2                                                                                                         2

0
0                                                                                                              37   38   39   40   41      42       43       44      45   46
37    38   39   40   41    42     43     44        45        46
Sample Means                                                                                                 Sample Means

Chapter 10 – 21                                                                               Chapter 10 – 22

Distribution of Sample Means
with 170 Samples                                                                                                        The Central Limit Theorem
• If all possible random samples of size N are
30
S.D. = 1.71                                                        drawn from a population with mean µy and
a standard deviation σ y , then as N becomes
Mean of Means= 41.12
Number of Means= 170

20                                                                                                       larger, the sampling distribution of sample
Frequency

means becomes approximately normal, with
10                                                                                                       mean µy and standard deviation σ y / N  .

0
37    38    39   40    41     42     43        44        45        46

Sample Means

Chapter 10 – 23                                                                               Chapter 10 – 24
Why does our sampling error go
What does CLT mean?
down with increase in sample size?
• Basically, with sufficient sample size the sampling
distribution of the mean will be normal regardless
of the shape of the population distribution
SO………….. We know
• 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.

Chapter 10 – 25                                                         Chapter 10 – 26

How does this all relate to the
Why do we care?
normal curve?
• We said that the sampling distribution of the mean                • Vital to doing inferential statistics
is normally distributed
• Allows us to estimate population parameters
• We can use the properties of the normal curve to
from sample statistics
figure out the probability that a sample mean will
fall within a certain distance of the population                    – Let’s us set our confidence that a statistic is
mean                                                                  accurately reflecting a parameter
• For example, we can expect approximately 68% of
all sample means to fall within plus or minus 1                     – Used for everything we do from this point
standard error of the population mean                                 forward

Chapter 10 – 27                                                         Chapter 10 – 28

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