Midterm_II_Revision_S2011

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Unformatted text preview: Make
sure
you
focus
on
the
following
concepts
while
preparing
for
Midterm
II
 
 Study
All
the
Quizzes
related
to
Chapters
9,
11‐17.
 Study
All
the
Homework
problems
related
to
Chapters
9,
11‐17.
 Understand
All
the
definitions
and
how
to
use
them
when
needed
(especially
in
chapters
11‐13)
 Understand
Random
Samples
and
ways
to
get
random
samples.
 Understand
the
purpose
of
Simulations
and
how
to
use
them.


 Know
how
to
use
the
table
of
random
digits.
 Study
what
you
have
learned
from
the
related
Lab
activities
(related
to
Chapters
11‐17).
 Use
Pen
to
answer
questions
on
Midterm
(Please
no
Pencils
this
time).
 Understand
Venn
Diagrams
and
the
related
probability
formulas.
 Understand
Bayes
Theorem
and
how
to
apply
it.
 Understand
how
to
construct
a
probability
tree
diagram
and
use
it
to
solve
related
probability
 problems.
 
 Some
Important
Definitions
(not
Comprehensive
list):
 
 Understand
the
following
Definitions:
 • Population
–
Parameters

 • Sample
‐
Statistics
 • Biased
Samples
 • Randomness
 • The
Sample
Size
Not
the
fraction
of
the
Population
 • Census
Sample
 • Equally
Likely
and
simple
random
samples
 • Sample
Variability
 • Strata
and
Stratified
random
sampling
 • Cluster
sampling
and
Multistage
Sampling
 • Systematic
Sampling
 
 Two
Common
Techniques
for
selecting
Samples
 1)
Simple
Random
Sampling

 2)
Stratified
Random
Sampling,
and
others

 
 A
sample
of
size
n
from
a
population
of
size
N
is
obtained
through
simple
random
sampling
if
every
 possible
sample
of
size
n
has
an
equally
likely
chance
of
occurring.

The
sample
is
then
called
a
simple
 random
sample.
 
 Non‐sampling
errors:
are
errors
that
result
from
the
survey
process.


 
 Sampling
error
is
the
error
that
results
in
using
sampling
to
estimate
information
regarding
a
 population.

This
type
of
error
occurs
because
a
sample
does
not
give
complete
information
about
the
 population.
This
can
be
estimated
in
some
situations
based
on
the
sampling
technique
used
for
 collecting
the
data.
 
 • • • • • • • • • • • Observational
Studies:
 a)
Retrospective
Study
 b)
Prospective
Study
 
 Important
Definitions:
 Experiment
 Random
Assignment
 Factor
 Response
Variable
 Experimental
Unit/Participant/Subject
 Levels/Treatments,
Control/Treatment
,
Placebo
 Statistically
Significant
 Blocking
 Lurking
and
Confounding
Variables

 Completely
Randomized
Design,
Randomized
Block
Design.

 
 Component:
A
component
is
the
most
basic
event
we
are
simulating.
 
 Trial:
A
trial
is
a
sequence
of
events
we
want
to
investigate.
(shooting
until
missing)
 
 Response
Variable:
Count
the
number
of
shots
she
made
before
missing.
 
 Statistic:
The
mean
of
the
number
of
shots
made
 
 The
Law
of
Large
Numbers
 
 
 
 A
few
terminologies
 An
experiment
is
the
process
by
which
an
observation
(or
measurement)
is
obtained.
 
 Example
1:
Tossing
a
fair
coin
and
a
balanced
die
together,
and
observe
the
upper
face.
 The
simplest
outcome
from
the
experiment
is
a
simple
Event,
ei
(
also
call:
elementary
outcomes),
e.g.,
a
 head
or
a
tail
in
the
toss
of
a
coin.
 
 E.g.:
A
possible
simple
event
for
Example
1
is
(H,1).
 SAMPLE
SPACE,
S:
The
set
of
all
simple
events
is
called
the
sample
space,
S.

 
 For
the
Example
1:

Sample
Space
is
(complete
the
rest):
 S
=
{(H,1),(H,2),…

 
 
 
 
 
 (T,6)}

 
 An
event
is
a
collection
of
one
or
more
simple
events
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Bayes’s
Rule:

 
 
 
 
 
 
 • Discrete
Random
Variable:
A
variable
can
be
discrete,
such
as
#
of
accidents
occurred
at
an
 intersection
per
month,

or

 Continuous
Random
Variable:
The
possible
values
are
in
a
real
interval,

such
as
SAT
score
 (between
0
to
1600)
,
gas
price
(between
$0
to
$10
per
gallon).

 In
these
two
chapters,
we
will
focus
on
discrete
variable
and
introduce
a
very
useful
special
discrete
 distribution:
Binomial
distribution.
 
 A
variable
X
is
a
random
variable
if
the
value
that
it
assumes
corresponding
to
the
outcome
of
an
 experiment,
is
a
chance
or
random
event.

 NOTE:
In
Chapter
15,
the
outcomes
can
be
anything
such
as
Head,
Tail,
Win/Loss,
and
so
on.
In
this
 chapter,
similar
process
is
applied
to
develop
the
probability
distribution,
except
that
the
possible
 outcomes
will
be
numerical
values.


 
 The
probability
distribution
for
a
discrete
random
variable
is
a
formula,
table,
or
graph
that
provides
 p(x),
the
probability
associated
with
each
of
the
values
of
x.
 Requirements
for
a
Discrete
Probability
Distribution

 
 The
Center:
The
mean
or
expected
value
of
the
variable
X
:
µ
=
E(x)
=
Sum(
xP(x))
 NOTE:
The
Expected
Value
m,
the
population
mean.
For
a
given
population,
there
is
ONLY
one
Expected
 value,
m.


 DO
NOT
mix
up
with
the
Sample
Mean
.
We
can
take
many
different
samples
from
the
same
population.
 Sample
means
are
different
from
sample
to
sample.
BUT,
there
is
only
one
population
mean.
You
can
 not
compute
population
mean
from
a
random
sample
that
is
not
the
entire
population.

 The
Variation:
The
variance
and
standard
deviation
of
X:

 Let
x
be
a
discrete
random
variable
with
probability
distribution
p(x)
and
mean
m.
The
variance
of
x
is
 
 
 
 sigma
2
=
E
[(x
‐
m
)
2
]
=
Sum(x
‐
m
)
2
p(x)
=
 
 
 
 The
standard
deviation
s

of
a
random
variable
X
:

 NOTE:
s
2
and
s

are
population
parameters.
DO
NOT
mix
up
with
Sample
variance,
sigma2
,
and
sample
 s.d.,
sigma.


 
 The
Binomial
Probability
Distribution
 Definition:
A
binomial
experiment
is
one
that
has
these
five
characteristics:
 
 1.
The
experiment
consists
of
n
identical
results.
 
 2.
Each
trial
results
in
one
of
two
outcomes:
one
outcome
is
called
a
success,
S,
and
the
other
a
 failure,
F.
 
 3.
The
probability
of
success
on
a
single
trial
is
equal
to
p
and
remains
the
same
from
trial
to
 trial.
The
probability
of
failure
is
equal
to
(1
‐
p)
=
q.
 
 4.
The
trials
are
independent.
 
 5.
We
are
interested
in
x,
the
number
of
successes
observed
during
the
n
trials,
for
x
=
0,
1,
2,
…,
 n.
 Mean
and
Standard
Deviation
for
the
Binomial
Random
Variable:
 For
binomial
random
variable
X,
there
is
a
very
easy
way
to
obtain
the
expected
value,
E(X),
Variance,
 and
standard
deviation
[
must
know
formulas
for
Binomial
distribution]:


 
 
 Mean:
m
=
np,


Variance:
sigma
2
=
n
p(1‐p),



 For
a
fixed
probability
of
success,
p,
as
the
number
of
trials
n
in
a
binomial
experiment
increase,
the
 probability
distribution
of
the
random
variable
X
becomes
bell‐shaped.

As
a
general
rule
of
thumb,
if
np
 ≥10
and

 n(1
–
p)
>
10,
then
the
probability
distribution
will
be
approximately
bell‐shaped.
 The
Success/Failure
Condition
states
that
we
must
expect
at
least
10
“successes”
and
10
“failures.”
 • ...
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This note was uploaded on 04/09/2012 for the course STATS 10 taught by Professor Ioudina during the Winter '08 term at UCLA.

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