Transcript Sampling Theory and Applications

QM 2113 -- Fall 2003
Statistics for Decision Making
Sampling Theory & Applications
Instructor: John Seydel, Ph.D.
Quiz # 2
Put your name in the upper right corner of
the quiz
Answer Problems 1 on the back side of the
quiz (show your work)
You may refer
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To your homework
But not to your text, notes, neighbor, . . .
Normal probability table: ask if you need it
Now, A Normal Distribution
Application: Sampling Theory
Descriptive numerical measures calculated
from the entire population are called
parameters.
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Numeric data: m and s
Categorical data: p (proportion)
Corresponding measures for a sample are
called statistics.
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Numeric data: x-bar and s
Categorical data: p
A Demonstration
Draw a sample of 50 observations
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x ~ N(100,20)
Calculate the average
Note that x-bar doesn’t equal m
Repeat multiple times
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Average the averages
Look at the distribution of the averages
Take a look also at the variances and
standard deviations
Now consider x ~ Exponential(100)
Sampling Distributions
Quantitative data
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Expected value for x-bar is the population or
process average (i.e., m)
Expected variation in x-bar from one sample
average to another is
 Known as the standard error of the mean
 Equal to s/√n
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Distribution of x-bar is approximately normal
(CLT)
Qualitative data: we’ll get to this later
An Example
Do WNB salaries equal industry on average (m
= 41,000?)
But sample results were
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x-bar = $40,080
s = $11,226
If truly m = 41,000
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Assume for now that s = s = 11226
What is P(x-bar < 40080)?
What is P(x-bar < 40080 or x-bar > 42226) ?
Some Answers
Given assumptions about m and s
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Standard error:
s/√n = 11226 /√221 = 755
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An x-bar value of 40080 is -1.22 standard errors
from the supposed population average
 Table probability = 0.3888
 Thus P(x-bar < 40080) = 0.5000 – 0.3888 = 11.1%
 And P(x-bar < 40080 or x-bar > 42226) = 22.2%
Now, consider how this might be put to use in
addressing the question
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Bring action against WNB (low pay?)
What’s the probability of doing so in error?
Maybe a confidence interval estimate could
be helpful . . .
Putting Sampling Theory to Work
We need to make decisions based on
characteristics of a process or population
But it’s not feasible to measure the entire
population or process; instead we do
sampling
Therefore, we need to make conclusions
about those characteristics based upon
limited sets of observations (samples)
These conclusions are inferences applying
knowledge of sampling theory
Schematic View
Statistics
Numeric Data
Informal
Summary Measures
Inferential Analyses
Categorical Data
Informal
Summary Measures
Inferential Analyses
Probability is what allows the linkage between descriptive and inferential
analyses
Appendix
Random Variables
Population or Process
Random Variable
(x)
Parameters
(m,s )
Sampling
Population
Sample
Statistic
Parameter