Introduction
Statistics
The 1.1 collection, organization, summarization, and
analysis of information
The Two Branches of Statistics- 1) Descriptive Statistics
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2) Inferential Statistics
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Types of Data1) Qualitative _________________________________________
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2) Quantitative ________________________________________
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a) Continuous _______________________________________
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b) Discrete ________________________________________
1.2 Types of Studies and Simple Random Sampling
Explanatory variables -
the cause variables
Response variables the effect variables
Their values are determined by the explanatory
variables
Types of studies:
1)Observational _______________________________________
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2) Experimental _______________________________________
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Confounding - Confounding occurs when the effects of two variables
on a response variable are not distinguished from each other.
Lurking variables A lurking variable is a variable which is not
one of the explanatory or response variables and yet it affects
the response variable (confounding)
Example - A plant biologist is studying the amount of sunlight
received by plants in a mountain meadow and their photosynthetic
and finds that plants in full sunlight have lower rates than
plants in shade. A lurking variable may be the amount of moisture
in the soil such that soil with more sunlight is also drier.
Association does not always mean there is causation.
Example. When a large number of different sized cities are
compared it is found that as the number of fire hydrants increase
the number of crimes also increase. Of course an increase in the
number of fire hydrants does not cause an increase in the number
of crimes. In this case the population of the city is a lurking
variable and as the population increases so does both the number
of both hydrants and crimes.
In science, researchers test hypotheses and theories. To perform
these tests data need to be collected and analyzed to determine
which of the hypothesis are most consistent with the data. The
sampling methods researchers use to collect data will be a main
topic of this chapter. Specifically we will be studying: 1)simple
random sampling 2)stratified sampling, 3)Systematic sampling, and
4)cluster sampling
1.3 Simple Random Sampling
Definition -Assume we are taking a sample of size n from a
population of size N. If every possible sample of size n has the
same chance of being selected this is called a simple random
simple.
Note: members of the population are called units. If the units are
people they are ca11ed subjects)
The problem for researchers is to find a method to take a sample
of size n such that all possible samples have the same chance of
selection. For example, we could sample trees in forest a by
moving about the forest in an unorganized fashion and haphazardly
pick trees for measurements. We could also start by looking at the
forest and then try to pick a. "representative group of trees for
measurements. The problem with both these sampling strategies is
that they rely on our judgment as to what is haphazard or
representative and could result in bias in the selection. To
remove the subjective aspect of the selection we use the following
strategy. Assume the population is of size N = 10000 and the
sample is of size 3O. (Note, the list of all the units in the
population is called a frame)
1) Assign each of the units of the population unique number
2) Use a pseudorandom number generator to select 30 integers from
the 10,000 possible integers. The integers selected determine the
population units in the Sample.
1.4 Systematic, Stratified, and Cluster Sampling
Systematic Sampling
Sometimes a good frame is not available and systematic sampling is
used. For example we may want to sample opinions of people living
in a certain section of a city so we go down a street and sample
houses. In a 1 in k systematic sample a random number, x, is
chosen between l and k. If we decide to sample every tenth house k
= 10. We randomly select a number between l and 10 for the
starting point. Lets assume the random value chosen for x is 6,
thus we Sample the 6th, 16th, 26th, 36th, etc' houses.
Advantages of systematic sampling:
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2) ____________________________________________________________
3) ____________________________________________________________
Stratified Sampling
To perform stratified sampling the population is divided into nonoverlapping subpopulations called strata and then simple random
samples are then taken from each stratum. Stratified sampling
yields better estimates of population parameters than simple
random sampling (SRS) when the variability within subgroups is
relatively small while variability between subgroups is relatively
large
Cluster Sampling
Cluster sampling, like stratified sampling divides the population
into groups. However, in Contrast to Stratified Sampling each of
the clusters should be similar to the entire population. For
example, if we want to sample leaves in an orchard it would not be
practical to get a frame of all the leaves in the orchard.
Instead, we could identify the trees in the orchard, take a simple
random sample of the trees to be sampled and then extensively
sample the selected trees. For equal sample sizes cluster sampling
is usually not as effective as simple random sampling. However,
since the units in a cluster are close together we can take larger
samples with cluster sampling and thus on a per unit cost basis
cluster sampling can be more effective.
In Summary, Cluster sampling is more effective than SRS when:
1) The variation within each cluster is about the same as in the
whole population.
And
2) The cost of sampling increases significantly as the distance
between sampled units increases.
Comparison of how Stratified sampling is conducted compared
to Cluster sampling
STRATIFIED
Strata
1
SRS
2
SRS
3
SRS
SAMPLING
4
SRS
census of strata
SRS within strata
CLUSTER
SAMPLING
SRS of clusters
census within clusters
1
2
3
Census
5
6
Census
7
4
Census
8