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STAT 324
Introduction to Statistics for Engineers
Summer 2016
Ismor Fischer
UW Dept of Statistics
1227 Medical Science Center
[email protected]
1 - Introduction
2 - Exploratory Data Analysis
3 - Probability Theory
4 - Classical Probability Distributions
5 - Sampling Distribs / Central Limit Theorem
6 - Statistical Inference
7 - Correlation and Regression
(8 - Survival Analysis)
3
What is “random variation” in the distribution of a population?
Examples: Toasting time, Temperature settings, etc. of a population of toasters…
POPULATION 1: Little to no variation (e.g., product manufacturing)
In engineering situations such as this, we
try to maintain “quality control”… i.e.,
“tight tolerance levels,” high precision,
low variability.
But what about a population of, say, people?
4
What is “random variation” in the distribution of a population?
Example: Body Temperature (F)
POPULATION 1: Little to no variation (e.g., clones)
Most individual values ≈ population mean value
Density
Very little variation
about the mean!
98.6 F
5
What is “random variation” in the distribution of a population?
Example:
Examples:Body
Gender,
Temperature
Race, Age,
(F)
Height, Annual Income,…
POPULATION 2: Much variation (more common)
Density
Much more
variation about the
mean!
6
What are “statistics,” and how can they be applied to real issues?
•
Example: Suppose a certain company insists that it complies with “gender equality”
regulations among its employee population, i.e., approx. 50% male and 50% female.
To test this claim, let us select a random sample of
n = 100 employees, and count X = the number of
males. (If the claim is true, then we expect X  50.)


etc.
 


X = 64 males
(+ 36 females)
Questions:
If the claim is true, how likely is this
experimental result? (“p-value”)
Could the difference (14 males) be
due to random chance variation, or
is it statistically significant?
GLOBAL OPERATION
DYNAMICS, INC.
7
The experiment in this problem can be modeled by a random sequence of n = 100
......
independent coin tosses (Heads = Male, Tails = Female).
It can be mathematically proved that, if the coin is “fair” (“unbiased”), then in 100 tosses:
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
probability of obtaining at…..from
least 0 Heads
away
from 50 is = 1.0000 “certainty”
0 to 100
Heads…..
probability of obtaining at least 1 Head away from 50 is = 0.9204
probability of obtaining at least 2 Heads away from 50 is = 0.7644
probability of obtaining at least 3 Heads away from 50 is = 0.6173
probability of obtaining at least 4 Heads away from 50 is = 0.4841
The  = .05
probability of obtaining at least 5 Heads away from 50 is = 0.3682
cutoff is
probability of obtaining at least 6 Heads away from 50 is = 0.2713
called the
probability of obtaining at least 7 Heads away from 50 is = 0.1933
significance
probability of obtaining at least 8 Heads away from 50 is = 0.1332
level.
probability of obtaining at least 9 Heads away from 50 is = 0.0886
probability of obtaining at least 10 Heads away from 50 is = 0.0569
probability of obtaining at least 11 Heads away from 50 is = 0.0352
probability of obtaining at least 12 Heads away from 50 is = 0.0210
probability of obtaining at least 13 Heads away from 50 is = 0.0120
0.0066 is called
probability of obtaining at least 14 Heads away from 50 is = 0.0066
the p-value of
etc.  0
the sample.
Because our p-value (.0066) is less than the significance level (.05),
our data suggest that the coin is indeed biased, in favor of Heads.
Likewise, our evidence suggests that employee gender in this
company is biased, in favor of Males.
8
What are “statistics,” and how can they be applied to real issues?
•
Example: Suppose a certain company insists that it complies with “gender equality”
regulations among its employee population, i.e., approx. 50% male and 50% female.
HYPOTHESIS
EXPERIMENT
To test this claim, let us select a random sample of
n = 100 employees, and count X = the number of
males. (If the claim is true, then we expect X  50.)


etc. 





OBSERVATIONS
X = 64 males
(+ 36 females)
Questions:
If the claim is true, how likely is this
experimental result? (“p-value”)
Could the difference (14 males) be
due to random chance variation, or
is it statistically significant?
GLOBAL OPERATION
DYNAMICS, INC.
9
The experiment in this problem can be modeled by a random sequence of n = 100
......
independent coin tosses (Heads = Male, Tails = Female).
It can be mathematically proved that, if the coin is “fair” (“unbiased”), then in 100 tosses:
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
probability of obtaining at least 0 Heads away from 50 is = 1.0000 “certainty”
probability of obtaining at least 1 Head away from 50 is = 0.9204
probability of obtaining at least 2 Heads away from 50 is = 0.7644
probability of obtaining at least 3 Heads away from 50 is = 0.6173
probability of obtaining at least 4 Heads away from 50 is = 0.4841
The  = .05
probability of obtaining at least 5 Heads away from 50 is = 0.3682
cutoff is
probability of obtaining at least 6 Heads away from 50 is = 0.2713
called the
probability of obtaining at least 7 Heads away from 50 is = 0.1933
significance
probability of obtaining at least 8 Heads away from 50 is = 0.1332
level.
probability of obtaining at least 9 Heads away from 50 is = 0.0886
probability of obtaining at least 10 Heads away from 50 is = 0.0569
probability of obtaining at least 11 Heads away from 50 is = 0.0352
ANALYSIS
probability of obtaining at least 12 Heads away from 50 is = 0.0210
probability of obtaining at least 13 Heads away from 50 is = 0.0120
0.0066 is called
probability of obtaining at least 14 Heads away from 50 is = 0.0066
the p-value of
etc.  0
the sample.
Because our p-value (.0066) is less than the significance level (.05),
our data suggest that the coin is indeed biased, in favor of Heads.
Likewise, our evidence suggests that employee gender in this
company is biased, in favor of Males.
CONCLUSION
10
“Classical Scientific Method”

Hypothesis – Define the study population...
What’s the question?

Experiment – Designed to test hypothesis

Observations – Collect sample measurements

Analysis – Do the data formally tend to
support or refute the hypothesis, and with
what strength? (Lots of juicy formulas...)

Conclusion – Reject or retain hypothesis; is
the result statistically significant?

Interpretation – Translate findings in context!
Statistics is implemented in each step of the
classical scientific method!
11
Example
• Click on image
for full .pdf article
• Links in article
to access datasets
Study Question:
How can we estimate
“mean age at first birth”
of women in the U.S.?
POPULATION
Women in U.S. who
have given birth
“Random Variable”
X = Age at first birth
Suppose we know that X follows a “normal
distribution” (a.k.a. “bell curve”) in the population.
That is, the Population Distribution of X ~ N(, ).
 and  are
“population
characteristics”
i.e., “parameters”
(fixed, unknown)
Without knowing every
value in the population, it
is
not
possible
to
determine the exact value
of  with 100% “certainty.”
standard
deviation
σ
mean μ = ???
{x1, x2, x3, x4, … , x400}
FORMUL
A
mean x = 25.6
Study Question:
How can we estimate
“mean age at first birth”
of women in the U.S.?
POPULATION
Women in U.S. who
have given birth
“Random Variable”
X = Age at first birth
x = 25.6 is an example of a “sample
characteristic” = “statistic.”
(numerical info culled from a sample)
Suppose we know that X follows a “normal
This is called a “point estimate“ of 
distribution” (a.k.a. “bell curve”) in the population.
from the one sample.
That is, the Population Distribution of X ~ N(, ). Can it be improved, and if so, how?
• Choose a bigger sample, which
standard
should reduce “variability.”
 and  are
deviation
• Average the sample means of
“population
σ
many samples, not just one.
characteristics”
(introduces “sampling variability”)
i.e., “parameters”
“Sampling Distribution” ~ ???
(fixed, unknown)
mean μ = ???
{x1, x2, x3, x4, … , x400}
FORMUL
A
mean x = 25.6
Study Question:
How can we estimate
“mean age at first birth”
of women in the U.S.?
POPULATION
Women in U.S. who
have given birth
Statistical Inference
and
Hypothesis Testing
“Random Variable”
X = Age at first birth
“Null Hypothesis”
Year 2010: Suppose we know that
X follows a “normal distribution”
(a.k.a. “bell curve”) in the population.
That is, X ~ N(25.4, 1.5).
• public education,
awareness programs
• socioeconomic
conditions, etc.
standard
standard
deviation
deviation
σ
μ < 25.4
σ = 1.5
Present: Is H0: μ = 25.4 still true?
Or,
is
the
“alternative
hypothesis” HA: μ ≠ 25.4 true?
i.e., either μ < 25.4 or μ > 25.4 ?
(2-sided)
μ > 25.4
Does
the
sample
statistic x = 25.6 tend to
support H0, or refute H0
in favor of HA?
25.4
mean μ = ???
{x1, x2, x3, x4, … , x400}
FORMUL
A
mean x = 25.6
In order to answer this question, we must account for the amount of variability of
different x values, from one random sample of n = 400 individuals to another.
We will see three things:
95% CONFIDENCE INTERVAL FOR µ
 = 25.4 25.453
x = 25.6
25.747
BASED ON OUR SAMPLE DATA, the true value of μ today is between
25.453 and 25.747, with 95% “confidence” (…akin to “probability”).
95% ACCEPTANCE REGION FOR H0
25.253
 = 25.4
25.547 x = 25.6
IF H0 is true, then we would expect a random sample mean x
to lie between 25.253 and 25.547, with 95% probability.
IF H0 is true, then we would expect a random sample mean x
that is at least 0.2 away from 25.4 (as ours was), to occur with
probability .00383 (= 0.383%)… VERY RARELY! ,which is less t
“P-VALUE” of our sample
25.4 25.6
In order to answer this question, we must account for the amount of variability of
different x values, from one random sample of n = 400 individuals to another.
We will see three things:
95% CONFIDENCE INTERVAL FOR µ
 = 25.4 25.453
x = 25.6
25.747
BASED ON OUR SAMPLE DATA, the true value of μ today is between
25.453 and 25.747, with 95% “confidence” (…akin to “probability”).
95% ACCEPTANCE REGION FOR H0
25.253
 = 25.4
25.547 x = 25.6
IF H0 is true, then we would expect a random sample mean x
to lie between 25.253 and 25.547, with 95% probability.
IF H0 is true, then we would expect a random sample mean x
that is at least 0.2 away from 25.4 (as ours was), to occur with
probability .00383 (= 0.383%)… VERY RARELY! ,which is less t
“P-VALUE” of our sample
25.4 25.6
In order to answer this question, we must account for the amount of variability of
different x values, from one random sample of n = 400 individuals to another.
We will see three things:
95% CONFIDENCE INTERVAL FOR µ
 = 25.4 25.453
x = 25.6
25.747
BASED ON OUR SAMPLE DATA, the true value of μ today is between
25.453 and 25.747, with 95% “confidence” (…akin to “probability”).
95% ACCEPTANCE REGION FOR H0
25.253
 = 25.4
25.547 x = 25.6
IF H0 is true, then we would expect a random sample mean x
to lie between 25.253 and 25.547, with 95% probability.
IF H0 is true, then we would expect a random sample mean x
that is at least 0.2 away from 25.4 (as ours was), to occur with
Less than
.05t
probability .00383 (= 0.383%)… VERY RARELY! ,which
is less
“P-VALUE” of our sample
< SIGNIFICANCE LEVEL (α)
25.4 25.6
In order to answer this question, we must account for the amount of variability of
x values,
different
from one random sample of n = 400 individuals to another.
FORMAL
CONCLUSIONS:
We will see three things:
95% CONFIDENCE INTERVAL FOR µ
 The 95% confidence interval corresponding to our sample mean does not
contain the “null value” of the population mean, μ = 25.4.
 = 25.4 25.453
x = 25.6
25.747
 The
95% ON
acceptance
regionDATA,
for the
contain the
BASED
OUR SAMPLE
thenull
truehypothesis
value of μ does
todaynot
is between
value25.453
of our sample
mean,
.
x =95%
25.6“confidence”
and 25.747,
with
(…akin to “probability”).
 The p-value of our sample, .00383, is less than the predetermined α = .05
significance level.
95% ACCEPTANCE REGION FOR H0
Based on our sample data, we may reject the null hypothesis H0: μ = 25.4 in
= 25.6
= 25.4
25.547
favor of the two-sided25.253
alternativehypothesis
HA: μ ≠ x25.4,
at the α = .05
significance
IF Hlevel.
0 is true, then we would expect a random sample mean x
to lie between 25.253 and 25.547, with 95% probability.
INTERPRETATION: According to the results of this study, there exists a
statistically
difference between the mean ages at first birth in
IF Hsignificant
0 is true, then we would expect a random sample mean x
2010 (25.4
old) and
today,from
at the
5%(as
significance
Moreover,
the
thatyears
is at least
0.2 away
25.4
ours was),level.
to occur
with
evidenceprobability
from the sample
suggests that
population
age
Lessmean
than
.05
.00383data
(= 0.383%)…
VERYthe
RARELY!
,which
is less
t today
is older than in 2010, rather than younger, by about 0.2 years.
“P-VALUE” of our sample
SIGNIFICANCE LEVEL (α)
25.4
<
25.6
SUMMARY: Why are these methods so important?
 They help to distinguish whether or not differences between
populations are statistically significant, i.e., genuine, beyond
the effects of random chance.
 Computationally intensive techniques that were previously
intractable are now easily obtainable with modern PCs, etc.
 If your particular field of study involves the collection of
quantitative data, then eventually you will either:
1 - need to conduct a statistical analysis of your own, or
2 - read another investigator’s methods, results, and
conclusions in a book or professional research journal.
 Moral:
You can run, but you can’t hide….
Study Question:
How can we estimate
“mean age at first birth”
of women in the U.S.?
• Arithmetic Mean
POPULATION
Women in U.S. who
have given birth
“Random Variable”
X = Age at first birth
“population
characteristics”
i.e., “parameters”
(fixed, unknown)
x1  x2 
n
 xn
• Geometric Mean
xG  n x1 x2
xn
• Harmonic Mean
Suppose we know that X follows a “normal
distribution” (a.k.a. “bell curve”) in the population.
That is, the Population Distribution of X ~ N(, ).
 and  are
xA 
standard
deviation
σ
xH 
1
x1
 x12
n

Each of these gives an
estimate of  for a particular
sample.
Any
general
sample
estimator for  is denoted by
the symbol ˆ .
Likewise for

and
mean μ = ???
{x1, x2, x3, x4, … , xn}
 x1n
FORMUL
A
mean x
ˆ .
Study Question:Other possible parameters:
How can we estimate
• standard POPULATION
deviation
“mean age at first birth”• median
Women in U.S. who
of women in the U.S.?
• minimum
have given birth
•
maximum
“Random Variable”
x = 25.6 is an example of a “sample
characteristic” = “statistic.”
(numerical info culled from a sample)
Suppose we know that X follows a “normal
This is called a “point estimate“ of 
distribution” (a.k.a. “bell curve”) in the population.
from the one sample.
That is, the Population Distribution of X ~ N(, ). Can it be improved, and if so, how?
• Choose a bigger sample, which
standard
should reduce “variability.”
 and  are
???
deviation
• Average the sample means of
“population
σ
many samples, not just one.
characteristics”
(introduces “sampling variability”)
i.e., “parameters”
“Sampling Distribution” ~ ???
(fixed, unknown)
X = Age at first birth
?????????
How big???
mean μ = ???
{x1, x2, x3, x4, … , x400}
FORMUL
A
mean x = 25.6