PSYC227_fall09_t_test_repeated_measures

PSYC227_fall09_t_test_repeated_measures -...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: t‐test
for
2
Related
Samples Learning
Objec-ves Know
the
difference
between
independent‐measures
and
repeated‐measures
 research
designs. Know
the
difference
between
a
repeated‐measures
and
a
matched‐subjects
 research
design. Be
able
to
perform
computa-ons
for
the
repeated‐measures
t
test. Be
able
to
compute
measures
of
effect
size
for
the
repeated‐measures
t. Understand
the
advantages
and
disadvantages
of
the
repeated
measures
 design. Related
samples
designs Repeated
measures
study:
single
sample
of
 individuals
is
measures
more
than
once
on
the
 same
dependent
variable Measu re individu different a them a ls, but treat s same Measu re individu same als twic e Matched‐subjects
study:
each
individual
in
one
 sample
is
matched
with
a
subject
in
the
other
 sample.

(two
individuals
are
equivalent)
 The
t
sta-s-c
for
related
samples What is the sample statistic? Reac-on
-me
measurements
taken
before
and
aLer
taking
an
over‐the‐counter
cold
 medica-on.

Note
that
MD
is
the
mean
for
the
sample
of
D
scores. Sample statistic Stating the Hypothesis we
are
interested
in
the
mean
difference
for
the
general
popula-on.
 The
t
sta-s-c
for
related
samples Example Does
birth
order
affect
introversion? Are
first‐borns
more
introverted
than
second‐borns? Administer
an
introversion
“test”
to
siblings
from
families
that
 have
2
children. Suppose
that
the
introversion
test
yields
interval
scale
data
with
 high
values
reflec-ng
high
introversion.
 Example Pair 1 2 3 4 5 6 7 8 9 10 First
 Born 65 48 63 52 61 53 63 70 65 66 Second
 Born 61 42 66 52 47 58 65 62 64 69 Difference 4 6 ‐3 0 14 ‐5 ‐2 8 1 ‐3 Example 1.

State
the
hypothesis
and
 select
an
alpha
level. 2.

Locate
the
cri-cal
region df = n – 1 = 10 – 1 = 9 Critical value 3.

Compute
the
test
sta-s-c. Pair 1 2 3 4 5 6 7 8 9 10 First
Born 65 48 63 52 61 53 63 70 65 66 Second
Born 61 42 66 52 47 58 65 62 64 69 Difference 4 6 ‐3 0 14 ‐5 ‐2 8 1 ‐3 variance Example 4.

Make
a
decision. Fail to reject the Null hypothesis Test statistic Critical value Example Would
like
to
assess
whether
birth
order
affects
introversion. Are
first‐borns
more
or
less
introverted
than
 second‐borns? How
about
a
two‐sided
test? Wha t chan ges? Does
birth
order
affects
introversion? Are
first‐borns
more
or
less
introverted
than
second‐ borns? How
about
a
two‐sided
test? The
cri-cal
value
for
a
5%
significance
level: Related
samples
designs Repeated
measures
study:
single
sample
of
 individuals
is
measures
more
than
once
on
the
 same
dependent
variable Measu re individu same als twic e H0 : µD = 0 H1 : µD = 0 Example Does stress make asthma symptoms worse? 1.

State
the
hypothesis
and
select
an
alpha
level. 2.

Locate
the
cri-cal
region The critical regions with α = .05 and df = 4 Example 3.

Compute
the
test
sta-s-c. variance standard error test statistic 4.

Make
a
decision. -4.47 < -2.776 Reject the null hypothesis. Relaxation training does effect the amount of medication needed to control the asthma symptoms. Another way to look at the data The
data
show
a
consistent
 decrease
in
scores
and
 suggest
that
 µD
=
0
(no
effect)
is
not
a
 reasonable
hypothesis. Imagine
that
variability
is
large Effect
Size
for
the
Repeated‐Measures
t Effect size Mean difference Example: The percentage of variance accounted for Example: Assump-ons
for
Related‐Samples
t
Tests Independent
observa-ons Popula-on
distribu-on
of
difference
scores
is
normal What
about
homogeneity
of
variance
assump-on? When
to
use
Repeated‐Samples
t
 tests • Number
of
subjects • Study
changes
over
-me • Individual
differences Es-ma-on Learning
Objec-ves • Use
sample
data
to
make
a
point
es-mate
or
an
 interval
es-mate
of
an
unknown
popula-on
mean – Single
sample – Independent
samples – repeated
measures • Factors
influencing
width
of
the
confidence
interval Es-ma-on
vs.
Hypothesis
Tes-ng Hypothesis
tes;ng:
determine
whether
or
not
a
 treatment
has
an
effect 
 H0:
“no,
treatment
has
no
effect” 
 H1:
“yes,
treatment
has
an
effect” Es;ma;on:
es-mate
popula-on
parameters Both
procedures
use
the
same
t‐sta-s-c
equa-ons,
and
 both
use
the
t‐distribu-on. Es-ma-on
equa-on Original
equa-on Es-ma-on
equa-on Point estimate: t=0 Interval estimates: range of t values is obtained from the t distribution table using the df value for the t statistic and the percentage of confidence for the interval Es-ma-on
equa-on popula;on
=
sample
±
some
error High probability value for t Assume that the equation produces a high probability value for the mean or mean difference Es-ma-on 
–
the
inferen-al
process
of
using
sample
sta-s-cs
to
es-mate
popula-on
 parameters ? Point
es;mate
–
single
number Interval
es;mate
–
range
of

values Confidence
interval
–
described
in
 terms
of
the
level
(percentage)
of
 confidence
in
the
accuracy
of
the
 es-ma-on. Sampling
 error Uses
of
Es-ma-on ALer
the
Null
hypothesis
is
rejected. Want
to
find
out
how
large
is
the
effect,
once
it
is
know
that
effect
exists. Want
to
know
basic
informa-on
about
a
popula-on. The
goal
is
to
use
the
sample
data
to
 answer
ques-ons
about
the
unknown
 popula-on
mean
aLer
treatment. ffec ere
an
e Is
th t? Hypothesis
Test Test
a
hypothesis
about
a
popula-on
 parameter • • • Hypothesize
a
value
for
unknown
 popula-on
parameter Compute
t
(test
sta-s-c) Test
whether
or
not
a
value
for
 the
t
sta-s-c
is
“reasonable” How ect? of
an
eff 
much
 Es;ma;on Es-mate
the
value
of
an
unknown
 popula-on
parameter • • Es-mate
what
t
ought
to
be
 (select
a
“reasonable”
value) Computa-on
will
produce
a
 “reasonable”
es-mate
of
the
 popula-on
parameter,
since
we
 used
a
“reasonable”
value “reasonable”
–
high
probability
outcome
located
near
the
center
of
the
distribu-on. “reasonable”
–
high
probability
outcome
located
near
 the
center
of
the
distribu-on. Confidence
Intervals Selected
based
on
 confidence
level Popula-on
mean

=
sample
mean
±
t
(es-mated


SE) Sample
data Sample
data Single
sample Repeated
measures Independent
samples Confidence
Intervals Confidence Intervals are wide if • • population variance of X is large sample size is small Wide CI means that we are not very precise in estimating the population parameter (here the population mean) Example A
researcher
begins
with
a
normal
popula-on
with
mean
60.

The
 researcher
is
evalua-ng
a
specific
treatment
that
is
expected
to
 increase
scores.

The
treatment
is
administered
to
a
sample
of
16
 individuals,
and
the
mean
of
the
treated
sample
is
M
=
66
with
SS
=
 1215. What
type
of
study
is
this?

(single
sample,
independent
samples,
 repeated
measures) 1.

Test
the
hypothesis
that
there
is
a
treatment
effect
(two‐ sided,
alpha
=
0.1). 2.

Es-mate
the
size
of
the
treatment
effect. Hypothesis
tes-ng STEP
1 H0:
µ
=
60 H1:
µ

≠
60 STEP
2 df=n‐1=15,
 t=1.753 STEP
3 M
=66 STEP
4 Test
sta-s-c
=
2.67
>
1.735
=
cri-cal
value Reject
the
Null
hypothesis The
treatment
has
an
effect.

How
much
of
an
effect? Step
1

Basic
formula
for
es-ma-on. Es-ma-ng
the
size
of
the
treatment
effect. Step
2

Point
or
an
interval
es-mate?

Specify
a
level
of
confidence
for
an
interval. 
 
 
 90%
confidence
interval
for
the
unknown
popula-on
mean Step
3

Find
the
appropriate
t
values
to
subs-tute
in
the
equa-on. 
 
 
 
 Point
es-mate
 t
=
0

 
 
 
 
 
 
 Interval
es-mate df=n‐1=15,
t=+1.753,

t=‐1.753 Step
4

Compute
M
and
sM

from
the
sample
data. M
=66
(given) Step
5

Subs-tute
the
appropriate
values
in
the
es-ma-on
equa-on. Es-mate
is
66 Es-mate
is
between
62.06
and
69.94 We
are
90%
confident
that
the
unknown
popula-on
 mean
falls
into
this
interval. Example A
developmental
psychologist
would
like
to
determine
how
much
fine
motor
 skill
improves
for
children
from
age
3
to
age
4.

A
random
sample
of
n=15
 3‐year‐old
children
and
a
second
sample
of
n=15
4‐year‐olds
are
obtained.

 Each
child
is
given
a
manual
dexterity
test
that
measures
fine
motor
skills.

 The
average
score
for
the
older
children
was
M
=
40.6
with
SS
=
430
and
 the
average
score
for
the
younger
children
was
M
=
35.4
with
SS
=
410. 1. What
is
a
point
es-mate
of
the
popula-on
mean
difference? 2. Test
the
hypothesis
of
mean
difference
in
motor
skills
between
3
and
4‐year‐olds
 at
alpha
level
0.05. 3. Make
an
interval
es-mate
so
you
are
95%
confident
that
the
real
mean
difference
 is
in
your
interval. 4. Test
the
hypothesis
of
mean
difference
in
motor
skills
between
3
and
4‐year‐olds
 at
alpha
level
0.01. 5. Make
an
interval
es-mate
so
you
are
99%
confident
that
the
real
mean
difference
 is
in
your
interval. Example The
counseling
center
at
a
college
offers
a
short
course
in
study
skills
for
 students
who
are
having
academic
difficulty.

To
evaluate
the
effec-veness
 of
this
course,
a
sample
of
n=25
students
is
selected,
and
each
student’s
 grade
point
average
is
recorded
for
the
semester
before
the
course
and
for
 the
semester
immediately
following
the
course.

On
average,
these
 students
show
an
increase
of
0.72
with
SS=24.

 1. What
is
a
point
es-mate
of
the
popula-on
mean
difference? 2. Test
the
hypothesis
for
the
effect
the
course
on
grade
point
average
at
alpha
level
 0.05. 3. Make
an
interval
es-mate
so
you
are
95%
confident
that
the
real
mean
difference
 is
in
your
interval. Another
way
to
look
at
the
data The
data
show
a
 consistent
decrease
 in
scores
and
 suggest
that
 µD
=
0
(no
effect)
is
 not
a
reasonable
 hypothesis. Imagine
that
variability
is
large Es-ma-on Learning
Objec-ves • Use
sample
data
to
make
a
point
es-mate
or
an
 interval
es-mate
of
an
unknown
popula-on
mean – Single
sample – Independent
samples – repeated
measures • Factors
influencing
width
of
the
confidence
interval Es-ma-on
vs.
Hypothesis
Tes-ng Hypothesis
tes;ng:
determine
whether
or
not
a
 treatment
has
an
effect 
 H0:
“no,
treatment
has
no
effect” 
 H1:
“yes,
treatment
has
an
effect” Es;ma;on:
es-mate
popula-on
parameters Both
procedures
use
the
same
t‐sta-s-c
equa-ons,
and
 both
use
the
t‐distribu-on. Es-ma-on
equa-on Original
equa-on Es-ma-on
equa-on Point estimate: t=0 Interval estimates: range of t values is obtained from the t distribution table using the df value for the t statistic and the percentage of confidence for the interval Confidence
Intervals Selected
based
on
 confidence
level Popula-on
mean

=
sample
mean
±
t
(es-mated


SE) Sample
data Sample
data Single
sample Repeated
measures Independent
samples Example A
developmental
psychologist
would
like
to
determine
how
much
fine
motor
 skill
improves
for
children
from
age
3
to
age
4.

A
random
sample
of
n=15
 3‐year‐old
children
and
a
second
sample
of
n=15
4‐year‐olds
are
obtained.

 Each
child
is
given
a
manual
dexterity
test
that
measures
fine
motor
skills.

 The
average
score
for
the
older
children
was
M
=
40.6
with
SS
=
430
and
 the
average
score
for
the
younger
children
was
M
=
35.4
with
SS
=
410. 1. What
is
a
point
es-mate
of
the
popula-on
mean
difference? 2. Test
the
hypothesis
of
mean
difference
in
motor
skills
between
3
and
4‐year‐olds
 at
alpha
level
0.05. 3. Make
an
interval
es-mate
so
you
are
95%
confident
that
the
real
mean
difference
 is
in
your
interval. 4. Test
the
hypothesis
of
mean
difference
in
motor
skills
between
3
and
4‐year‐olds
 at
alpha
level
0.01. 5. Make
an
interval
es-mate
so
you
are
99%
confident
that
the
real
mean
difference
 is
in
your
interval. Example The
counseling
center
at
a
college
offers
a
short
course
in
study
skills
for
 students
who
are
having
academic
difficulty.

To
evaluate
the
effec-veness
 of
this
course,
a
sample
of
n=25
students
is
selected,
and
each
student’s
 grade
point
average
is
recorded
for
the
semester
before
the
course
and
for
 the
semester
immediately
following
the
course.

On
average,
these
 students
show
an
increase
of
0.72
with
SS=24.

 1. What
is
a
point
es-mate
of
the
popula-on
mean
difference? 2. Test
the
hypothesis
for
the
effect
the
course
on
grade
point
average
at
alpha
level
 0.05. 3. Make
an
interval
es-mate
so
you
are
95%
confident
that
the
real
mean
difference
 is
in
your
interval. ...
View Full Document

Ask a homework question - tutors are online