Lecture13-February+25th-Hypo+testing2 (1)

Lecture13-February+25th-Hypo+testing2 (1) -  ...

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Unformatted text preview:   Next
week
   Last
tutorial
due
   Exercise
4
due
   Z‐test
=
Testing
the
Null
hypothesis
on
single
 sample
   Statistics
estimates
population
parameters
   Statistics
will
always
differ
from
parameters
due
 to
“sampling
error”
   How
can
we
test
whether
our
“treated”
sample
 differs
“significantly”
from
population?
   Population:
PSC41
students

   DV=
test
scores
   μ
=
38,
σ
=
4
   Sample:
25
students
from
PSC41
   IV=
practice
questions;
DV=
test
scores
   M=
41
(or
39)
   Is
this
group
significantly
different
from
population?
Does
 our
manipulation
have
an
effect?
   Rationale:
what
would
be
the
probability
of
 getting
the
same
mean
if
we
selected
all
of
 possible
samples
(of
same
size)
from
the
 population?

   What
are
the
odds
that
our
result
is
due
to
 chance?
   Test
of
null
hypothesis:

 ▪  H0:
μpractice
=
38
   We
need
to
know
the
“sampling
distribution”
   Distribution
of
“mean
score”
(on
our
DV)
of
all
the
 possible
samples
(of
25
students)
that
could
be
 drawn
from
the
population
   Sampling
distribution
has
 ▪  Mean
=
mean
of
population
μ
 ▪  SD
=

σ/√n
(standard
error:
variation
between
samples)
   Sampling
distribution
quickly
approaches
 normality
(with
increasing
sample
size)
   As
sampling
distribution
is
normal,
we
can
 standardize
scores
to
get
the
probability
 associated
with
their
occurrence
(with
the
 occurrence
of
each
sample)
   Using
z‐scores
we
can
find
the
probability
 associated
with
any
specific
sample
   The
location
of
our
sample
mean
z=


 68% 95% 99% (sample
mean‐μ)
   σ/√n When
will
we
 consider
our
 sample
to
be
 significantly
 different
from
a
 random
sample?

   Level
of
significance:
α
   Determines
the
sample
means
that
could
be
 obtained
with
the
lowest
probability
   68% 95% 99% If
H0
is
true,
most
 sample
means
 should
be
close
 to
distribution
 mean
   α:
conventionally
set
at
5%
and
1%
 The distribution of sample means if the null hypothesis is true (all the possible outcomes) Reject H0 Extreme, low-probability values if H0 is true Sample means close to H0: High probability values if H0 is true µ from H0 Reject H0 Extreme, low-probability values if H0 is true   One‐tailed
vs.
Two
tailed
test
   For
a
directional
alternative
hypothesis
we’ll
 conduct
a
one
tailed
test
 Critical Region ▪  H0:
μ
<
38
 ▪  H1:
μ
>
38
   For
a
non‐directional
alternative
hypothesis
we’ll
 conduct
a
two‐tailed
test
 ▪  H0:
μ
=
38
 ▪  H1:
μ
≠
38
 Critical Region Critical Region   Distribution
of
z‐scores
:
+
1.96
correspond
to
 cutoff
scores
for
critical
region
(2‐tailed)
     If
z‐score
for
our
sample
 falls
between
‐1.96
and
 +1.96,
we
fail
to
reject
H0
 If
z‐score
for
our
sample
 Z=-1.96 falls
below
‐1.96
or
+1.96,
 Reject H0 we
reject
H0
 Z= (41-38) 4/√25 = 3.75 Middle 95%: high-prob values if H0 is true µ from H0 Z=1.96 Reject H0 Critical Region: Extreme 5%   Statistical
Significance
test
 1.  Calculate
critical
values
set
by
α
 2.  Calculate
probability
of

 getting
your
sample
mean:
 p‐value
 ▪  If
p‐value
<
α
then

 reject
null
hypothesis
 ▪  If
p‐value
>
α
then

 fail
to
reject
null
hypothesis
 P-value for M = 39 P-value: Likelihood of getting the sample mean that we got if H0 is true P-value for M = 41   P‐value:
probability
of
getting
a
test
statistic
 (i.e.,
mean)
AT
LEAST
as
extreme
as
the
one
 observed,
assuming
H0
correct
   Psc41:
μ=38;
Practice
sample:
M=41
   Z=3.75,

p<.001
 ▪  Probability
of
practice
sample
performing
the
way
they
 did,
assuming
there’s
no
difference
b/w
them
and
the
 population
they
were
drawn
from,
is
less
than
1%
 ▪  The
lower
the
p‐value,
the
more
"significant"
the
result,
in
 the
sense
of
statistical
significance
(based
on
α)
   REMEMBER:
   p
>
α
means
H0
cannot
be
rejected
AT
THAT
 SIGNIFICANCE
LEVEL.
Does
not
mean
H0
is
true
   Small
p‐values
do
not
IMPLY
that
H1
is
correct
   Significance
level
is
NOT
determine
by
p‐value.
It
is
 decided
upon
BEFORE
viewing
data,
and
is
 COMPARED
to
p‐value

   α=
.05
is
a
convention!
   P‐value
is
not
indication
of
importance
or
magnitude
 of
an
effect
   “…First,
a
main
effect
of
expected
memorability
 was
observed,
specifically

higher
false‐alarm
 rates
for
low
compared
to
high
expected‐ memorability
events
(Low:
M
=
.20,
SD
=
.13;
 High:
M
=
.17,
SD
=
.12),
F
(1,
96)
=
10.22,
p
<
.01,
 ηp2
=
.10...”
 Descriptive Stats Inferential Stats p value if H0 is true   Errors
are
possible
in
the
statistical
decision
 process
   Type
I
error
 ▪  Rejecting
null
hypothesis
that
is
actually
true
 ▪  Always
equals
to
the
alpha
level
   Type
II
error
 ▪  Failing
to
reject
a
null
hypothesis
that
is
really
false
 True State of Affairs Ho True Reject Ho Decision Do not reject Ho Ho False Type I error Correct decision (= α ) Correct decision (confidence level=1- α) (power=1- β) Type II error (=β )   REMEMBER:
   p
>
α
means
H0
cannot
be
rejected
AT
THAT
 SIGNIFICANCE
LEVEL.
Does
not
mean
H0
is
true
   Failure
to
find
significant
effect,
simply
means
you
 didn’t
find
the
effect
BASED
ON
YOUR
STUDY
   It
is
possible
you
didn’t
because
your
statistical
 POWER
was
low
   Power
   Probability
that
the
test
will
correctly
reject
a
false
 null
hypothesis
 ▪  Sensitivity
of
stat
procedure
to
find
the
hypothesized
 differences
   Increase
power
by
larger
sample
size
   Increase
power
by
rigorous
research
design
 ▪  Reliability
and
validity!
   Significance
test:
what
are
the
odds
that
the
 detected
difference
is
due
to
chance?
   Effect
size:
is
the
difference
meaningful?
   Measure
the
absolute
magnitude
of
a
treatment
 independent
of
the
sample
size
   How
substantial
is
the
effect?
   “…First,
a
main
effect
of
expected
memorability
 was
observed,
specifically

higher
false‐alarm
 rates
for
low
compared
to
high
expected‐ memorability
events
(Low:
M
=
.20,
SD
=
.13;
 High:
M
=
.17,
SD
=
.12),
F
(1,
96)
=
10.22,
p
<
.01,
 ηp2
=
.10...”
 Effect size Inferential Stats p value if H0 is true   The
t‐test
   For
single
sample
   For
independent
groups
   For
correlated
groups
   Recap
   The
sample
mean
(noted
as
M
or
x)

is
expected
to
 be
equal
to
the
population
mean
(μ)
   The
standard
error
measures
how
well
a
sample
 mean
approximate
the
population
mean
   Compare
M
to
μ
by
computing
a
z‐score
test
 statistic
 Problem: usually unknown (M - µ ) z= σ/√n = obtained difference between data and hypothesis Standard distance between data and hypothesis   An
alternative
to
Z
test
 (M - µ ) z= σ/√n (M - µ ) t= s/√n The t statistic   Testing
hypotheses
about
an
unknown
population
 mean
μ
when
the
value
of
σ
is
unknown
   T
test
   Data
are
interval
or
ratio
   Population
distribution
is
symmetrical
   mean
μ
is
hypothesized
but
value
of
σ
is
unknown
   We
estimate
population
sd
using
the
sample
sd
   Means
are
not
distributed
normally
   Use
Student
T
distribution
   Student’s
t
distribution
:
   Not
one
distribution,
but
different,
depending
on
 sample
size
   Different
probabilities
associated
with
different
 samples

   Student’s
t
distribution
:
   Different
distributions
for
different
sample
sizes
   df=
degrees
of
freedom
 ▪  Sample
size
(n)‐1
 ▪  The
#
of
scores

 in
the
sample
that

 are
independent

 and
free
to
vary
   Comparing
a
sample
mean
to
population
mean
 when
is
σ
unknown
   The
significance
test
 1.  Calculate
critical
values
set
by
α:
t
statistic
 ▪  If

‐tcritical
value<tsample
mean<
tcritical
value
then
you
fail
to
reject
H0
 ▪  If

tsample
mean<‐tcritical
value

or
tsample
mean>
tcritical
value
then
reject
H0
 2.  Calculate
probability
of
getting
your
sample
 mean:
p‐value
 ▪  If
p‐value
<
α
then
reject
null
hypothesis
 ▪  If
p‐value
>
α
then
you
fail
to
reject
null
hypothesis
   This
value
is
used
just
like
a
z‐statistic
   
if
the
value
of
t
exceeds
critical
valued,
tα
,
then
an
 effect
is
detected
(i.e.,
the
null
hypothesis
is
rejected)
 31
   For
a
directional
alternative
hypothesis
we’ll
 conduct
a
one
tailed
test
 Critical Region   For
a
non‐directional
alternative
hypothesis
 we’ll
conduct
a
two‐tailed
test
 Critical Region Critical Region   The
t‐test
   For
single
sample
   For
independent
groups
   For
correlated
groups
   Are
the
two
groups
different?
   Are
they
so
similar
they
could
come
from
 same
population?
 M − µ0 t= sM Difference b/w group means M1 − M 2 t= sM 1− M 2 Variability b/w groups €   Standard
error
of
difference
between
means
 s1 s2 s s sM 1 − M 2 = + n1 n 2 2 1 2 2   Are
means
of
two
groups
significantly
 different
from
each
other?
 µ1 = µ2 : µ1 ≠ µ2 H0 : H1 or H1 : µ1 > µ2 or H1 : µ1 < µ2   To
find
level
of
significance
we
also
need
to
 know
the
DF
   Df
for
independent
sample
is
N1
+N2
‐2
   With
t
and
df
we
check
whether
we
can
reject
 the
null
hypothesis
 µ1 = µ2 : µ1 ≠ µ2 H0 : Reject H1 or H1 : µ1 > µ2 or Ha : µ1 < µ2 As the t statistic & df increase, p-value decreases => more likely to reject null hypothesis   IMPORTANT:
   Comparison
it
at
the
GROUP
level
(not
the
 individual)
   The
mean
are
different.
Does
not
imply
every
 member
of
one
group
is
different
from
every
 member
of
the
other
group
   Correlated/Paired
t‐test:
Comparing
 mean
difference
of
correlated
scores
   When
do
we
use
it?
 1.  In
a
Matched
groups

design

 (aka
paired
comparison
study)
 2.  In
a
Within‐participants
design:

 2
repeated
measures
or
pretest‐posttest
   Are
means
of
two
groups
significantly
 different
from
each
other?
 µ1 - µ2 = 0 : µ1 - µ2 > 0 (one‐tailed)
 H0 : H1   Sampling
distribution
would
not
be
of
means,
 but
of
differences
b/w
means
   To
calculate
t
we
first
need
to
compute
 DIFFERENCE
SCORES
(the
difference
 between
score
in
one
condition
vs.
the
other)
 M − µ0 t= sM MD − 0 t= sD Standard error of difference score €   Standard
error
of
difference
scores
 ∑ (D − M sD = N −1 n D ) 2 ...
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