Exercise on
Step 1, The Modeling Step
Recall Forum F2, where we "completely identified" the random variable of interest as explained
in Types of Variables. Therein we identified one qualitative and two quantitative random
variables of interest. That was assigned because it is the first part of the first step of statistical
inference. Indeed (IMHO), the first step of scientific research in general is to identify the
variables you wish to observe through sampling or experimentation.
Objectives
The objectives of this exercise is to introduce the following concepts.
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The first step of statistical inference (estimation and significance testing) is to formulate
a model for the proposed (observational or experimental) study.
All studies include observation of random variables.
We use what we know, the sample (data), to make inferences about what is unknown, the
parameters of the population (or experimental phenomena).
Choice of the optimal inferential procedure depends on the parameters of interest and the
assumptions we believe are valid for the study at hand.
The assumptions are assumptions about the underlying distributions of the random
variables of interest.
Instructions
For F4, each of us will perform what I call Step 1, the Modeling Step, of my 6-Step Method of
Statistical Inference,
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as summarized briefly in Statistical Inference 6-Step Overview,
and as explained fully in 6-Step Method of Estimation.
The key to, assumptions of, and formulas for, the four models we will entertain are in
Univariate Formulas Tabulated.
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We will perform Step 1 three times:
1. Step 1 for estimation of the (population) proportion of the qualitative random variables
of interest we identified in F2
Example
a. RV of interest
Let Xi denote the sex (male, female) of the i-th randomly sampled Stage 4 lung
cancer patient in southwest Virginia, i = 1, 2, ..., 100
b. Parameter of interest
Let φ denote the proportion of females among all Stage 4 lung cancer patient in
southwest Virginia.
c. Assumptions (about the underlying distribution)
None
2. Step 1 for estimation of the (population) mean of the continuous quantitative random
variables of interest we identified in F2, assuming that the underlying distribution is
the Normal distribution
Example
a. RV of interest
Let Yi denote the age (yr) of the i-th randomly sampled Stage 4 lung cancer patient
in southwest Virginia, i = 1, 2, ..., 100
b. Parameter of interest
Let 𝜇 denote the mean age (yr) of all Stage 4 lung cancer patient in southwest
Virginia.
c. Assumptions (about the underlying distribution)
We assume that Yi is Normally distributed
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3. Step 1 for estimation of the (population) standard deviation of the continuous
quantitative random variables of interest we identified in F2, assuming that the
underlying distribution is the Normal distribution
Example
d. RV of interest
Let Yi denote the age (yr) of the i-th randomly sampled Stage 4 lung cancer patient
in southwest Virginia, i = 1, 2, ..., 100
e. Parameter of interest
Let 𝜎 denote the standard deviation of age (yr) of all Stage 4 lung cancer patient in
southwest Virginia.
f. Assumptions (about the underlying distribution)
We assume that Yi is Normally distributed.
Note well
The modeling step, being the interface between the biology (psychology, engineering, whatever
subject matter) and statistics (a.k.a. analytics) is the most challenging step for both statisticians
and non-statisticians. The modeling step involves imagination and creativity. Once the model is
formulates, the statistical paradigm is framed, and subsequent steps are nearly automatic and, for
the most part, computerized.
In first-semester introductory courses, the models are the simplest models, involving only one or
two variables. In subsequent courses the models get much more elaborate, employing multiple
variables and equations for relationships among the variables.
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