PSYC227_fall09_week8

PSYC227_fall09_week8 - Hypothesis Testing

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Unformatted text preview: Hypothesis Testing Was
it
due
to
chance,
or
something
 else? 

Hypothesis
Tes8ng Standardized
method
for
evalua8ng
the
results
of
 a
research
study. Was it du e to chance, o r something else? A hypothesis test is a statistical method that uses sample data to evaluate a hypothesis about a population. Step
1:
state
the
hypothesis Step
2:
set
the
criteria
for
a
decision Step
3:
Collect
data
and
compute
sample
sta8s8cs Step
4:
make
a
decision
 Step
1:
state
the
hypothesis No change The
Null
hypothesis
(H0):
observed
difference
reflects
 chance
varia8on
(zero
effect) The
Alterna1ve
hypothesis
(H1):
 observed
difference
is
real 






≠,







two‐tailed 
 <,
>



one‐tailed
(direc8onal) H0 and H1 are mutually exclusive! Step
1:
state
the
hypothesis Select
an
alterna1ve
hypothesis
as
that
which
the
sampling
 experiment
is
intended
to
establish.

There
are
3
possibili8es 
 
 One‐tailed,
upper‐tailed







> 
 One‐tailed,
lower‐tailed







< 
 Two‐tailed




























≠ Select
the
null
hypothesis
as
the
status
quo.

 No change 
 
 One‐tailed:

parameter
value
closest
to
the
alterna8ve 
 Two‐tailed:
complementary
(only
unspecified)
value Example:
set
up
the
null
and
alterna8ve
hypothesis
for
 tes8ng
the
claim: Science
(1999)
reported
that
the
mean
listening
8me
of
7‐ month‐old
infants
exposed
to
a
three‐syllable
sentence
is
9
 seconds.

 Example:
set
up
the
null
and
alterna8ve
hypothesis
for
 tes8ng
the
claim: Science
(1999)
reported
that
the
mean
listening
8me
of
7‐ month‐old
infants
exposed
to
a
three‐syllable
sentence
is
9
 seconds.

 H0:
µ
=
9 H1:
µ
≠
9 Example:
set
up
the
null
and
alterna8ve
hypothesis
for
 tes8ng
the
claim: Infants
exposed
to
cocaine
in
their
mother’s
womb
are
 thought
to
be
at
a
high
risk
for
major
birth
defects.

 However,
according
to
a
University
of
Florida
study,
 more
than
75%
of
all
cocaine‐exposed
babies
suffer
 no
major
problems. Example:
set
up
the
null
and
alterna8ve
hypothesis
for
 tes8ng
the
claim: Infants
exposed
to
cocaine
in
their
mother’s
womb
are
 thought
to
be
at
a
high
risk
for
major
birth
defects.

 However,
according
to
a
University
of
Florida
study,
 more
than
75%
of
all
cocaine‐exposed
babies
suffer
 no
major
problems. H0: µ = 0.75 H1: µ > 0.75 Example:
set
up
the
null
and
alterna8ve
hypothesis
for
 tes8ng
the
claim: According
to
the
Journal
of
Psychology
and
Aging,
older
 workers
(45
years
old
or
older)
have
a
mean
job
 sa8sfac8on
ra8ng
of
4.3
on
a
5
point
scale. Example:
set
up
the
null
and
alterna8ve
hypothesis
for
 tes8ng
the
claim: According
to
the
Journal
of
Psychology
and
Aging,
older
 workers
(45
years
old
or
older)
have
a
mean
job
 sa8sfac8on
ra8ng
of
4.3
on
a
5
point
scale. H0: µ = 4.3 H1: µ ≠ 4.3 Example:
diet
effect A
researcher
is
using
a
sample
of
16
rats
to
examine
the
effect
of
a
new
diet
drug.

 Under
regular
circumstances,
rats
eat
an
average
of
10gm
of
food
each
day.

The
 distribu8on
of
food
consump8on
is
normal
with
SD=4.

The
expected
effect
of
a
 drug
is
to
reduce
food
consump8on.

The
purpose
of
the
experiment
is
to
 determine
if
the
drug
works. Step 1: Example:
diet
effect A
researcher
is
using
a
sample
of
16
rats
to
examine
the
effect
of
a
new
diet
drug.

 Under
regular
circumstances,
rats
eat
an
average
of
10gm
of
food
each
day.

The
 distribu8on
of
food
consump8on
is
normal
with
SD=4.

The
expected
effect
of
a
 drug
is
to
reduce
food
consump8on.

The
purpose
of
the
experiment
is
to
 determine
if
the
drug
works. Step 1: H0: food consumption is not reduced H1: food consumption is reduced H0


µ
≥
10
(with
the
drug,
food
consump8on
is
at
least
10
gm
per
day) H1:

µ
<
10
(with
the
drug,
food
consump8on
is
less
than
10gm
per
day) Example: one-sided test The distribution of sample means for n = 16 if H0 is true. The null hypothesis states that the drug does not reduce food consumption. A sample mean much smaller than µ = 10 would provide evidence that the drug works and that H0 should be rejected. Step
2:
set
the
criteria
for
a
decision Step
2:
set
the
criteria
for
a
decision Level
of
significance

(the
alpha
level) Alpha
(α)
‐
Probability
level
that
is
used
to
define
the
 very
unlikely
sample
outcomes
if
the
null
hypothesis
 is
true. α = 0.05 α = 0.01 α = 0.001 (5%) (1%) (0.1%) The
cri8cal
region
for
α
=
0.05. Cri8cal
region
depends
on
α.
 The
loca8ons
of
the
cri8cal
region
boundaries
for
three
 different
levels
of
significance
in
a
two‐tailed
test. Step
3:
Collect
data
and
compute
sample
 sta8s8cs Test
sta8s8c
is
used
to
measure
the
difference
between
the
 observed
data
and
what
is
expected
under
the
Null
hypothesis. The
z‐sta8s8c
converts
the
observed
value
to
standard
units,
 on
the
basis
of
the
Null
hypothesis. For a sample mean: Sample data Under the null hypothesis Step
4:
make
a
decision sample data falls into critical region sample data is not in a critical region sample data falls into critical region Example A
vocabulary
test
for
six‐year‐old
children
is
standardized
on
a
large
 na8onwise
sample
to
have
an
average
score
of
50
out
of
100.

School
 authori8es
in
Maine
choose
a
statewise
simple
random
sample
of
400
six‐ year‐olds.

These
children
average
51.3
on
the
test,
with
a
SD
of
12.

Is
it
 safe
to
infer
that
if
the
test
had
been
administered
to
all
six‐year‐olds
in
 Maine,
they
would
have
averaged
above
50?

Or
can
the
1.3
point
 difference
be
explained
as
a
chance
varia8on? Example A
vocabulary
test
for
six‐year‐old
children
is
standardized
on
a
large
 na8onwise
sample
to
have
an
average
score
of
50
out
of
100.

School
 authori8es
in
Maine
choose
a
statewise
simple
random
sample
of
400
six‐ year‐olds.

These
children
average
51.3
on
the
test,
with
a
SD
of
12.

Is
it
 safe
to
infer
that
if
the
test
had
been
administered
to
all
six‐year‐olds
in
 Maine,
they
would
have
averaged
above
50?

Or
can
the
1.3
point
 difference
be
explained
as
a
chance
varia8on? 1. H0
:
µ


50



























2.
α = 0.05 (5%)
 ≤ 






 H1
:
µ
>
50 3.













































P
=
1% 4.
0.01
<
0.05,
reject
H0.
Good
evidence
that
average
in
Maine
is
 bigger
than
50.
 
 Example The population of 2 year old children has an average weight of 26 lb. The distribution of weights is normal with standard deviation 4. Researcher selects a sample of 4 infants and instructs the parents to provide each child with extra handing. The researcher predicts that extra handling will stimulate growth and produce an increase in their weight at age 2. The researcher records the weight of each child at age 2 and obtains a sample mean of 29.5 lb. Step 1: hypothesis H0: µ ≤ 26 H1: µ > 26 (there is no increase in weight) (there is an increase in weight) Step 2: critical region α=0.05 Step 3: compute sample statistic Step 4: make a statistical decision Sample mean Hypothesized population mean test statistic Standard error p-value = 0.0401 1.75 Sample mean is in the critical region (1.75 > 1.65) or p-value = 0.0401 < 0.05 = α Reject the null hypothesis. Conclude that extra handling does result in increased body weight for infants. Compare z-values or Compare probabilities Uncertainty and Errors in Hypothesis Testing Decisions in hypothesis testing: 1. The sample data provide sufficient evidence to reject the null hypothesis. Conclude that the treatment has an effect. 2. The sample data do not provide enough evidence to reject the null hypothesis. Conclude that the treatment does not appear to have an effect. Uncertainty and Errors in Hypothesis Testing α Fail to reject β Level of significance: the alpha level: A researcher is conducting a hypothesis test to determine whether a treatment produces a significant increase in scores. The sample data produces z = 2.3. What would be the decision for a 1. 2. 3. 4. two-tailed test with α = 0.05 one-tailed test with α = 0.05 two-tailed test with α = 0.01 one-tailed test with α = 0.01 Critical region boundary 1.96 1.64 2.58 2.33 α = 0.05 α = 0.01 0 1.64 2.33 z=1.89, p-value=0.0294 Factors that Influence a Hypothesis Test 1. Size of the mean difference 2. The variability of the scores 3. The number of scores in the sample Hypothesis
tes8ng Analogy
to
a
Jury
trial… Analogy
to
a
Jury
trial… A

Hypothesis
Test Analogy
to
a
Jury
trial… A

Hypothesis
Test 





H0:
assume
there
is
no
treatment
effect
un8l
 there
is
enough
evidence
to
show
otherwise Analogy
to
a
Jury
trial… A

Hypothesis
Test 





H0:
assume
there
is
no
treatment
effect
un8l
 there
is
enough
evidence
to
show
otherwise A
Jury
Trial Analogy
to
a
Jury
trial… A

Hypothesis
Test 





H0:
assume
there
is
no
treatment
effect
un8l
 there
is
enough
evidence
to
show
otherwise guilty. A
Jury
Trial 




Assume
individual
is
innocent
un8l
proven
 Analogy
to
a
Jury
trial… A

Hypothesis
Test 





H0:
assume
there
is
no
treatment
effect
un8l
 there
is
enough
evidence
to
show
otherwise The
alpha
level.

We
are
confident
that
 there
is
a
treatment
effect
because
it
is
very
 unlikely
that
the
data
could
occur
simply
by
 chance
(without
any
treatment
effect). guilty. A
Jury
Trial 




Assume
individual
is
innocent
un8l
proven
 Analogy
to
a
Jury
trial… A

Hypothesis
Test 





H0:
assume
there
is
no
treatment
effect
un8l
 there
is
enough
evidence
to
show
otherwise The
alpha
level.

We
are
confident
that
 there
is
a
treatment
effect
because
it
is
very
 unlikely
that
the
data
could
occur
simply
by
 chance
(without
any
treatment
effect). guilty. The
jury
must
be
convinced
beyond
a
 reasonable
doubt
before
they
find
that
a
 person
is
guilty. A
Jury
Trial 




Assume
individual
is
innocent
un8l
proven
 Analogy
to
a
Jury
trial… A

Hypothesis
Test 





H0:
assume
there
is
no
treatment
effect
un8l
 there
is
enough
evidence
to
show
otherwise The
alpha
level.

We
are
confident
that
 there
is
a
treatment
effect
because
it
is
very
 unlikely
that
the
data
could
occur
simply
by
 chance
(without
any
treatment
effect). The
sample
data:

gather
data
to
 demonstrate
that
the
treatment
has
an
 effect guilty. The
jury
must
be
convinced
beyond
a
 reasonable
doubt
before
they
find
that
a
 person
is
guilty. A
Jury
Trial 




Assume
individual
is
innocent
un8l
proven
 Analogy
to
a
Jury
trial… A

Hypothesis
Test 





H0:
assume
there
is
no
treatment
effect
un8l
 there
is
enough
evidence
to
show
otherwise The
alpha
level.

We
are
confident
that
 there
is
a
treatment
effect
because
it
is
very
 unlikely
that
the
data
could
occur
simply
by
 chance
(without
any
treatment
effect). The
sample
data:

gather
data
to
 demonstrate
that
the
treatment
has
an
 effect guilty. The
jury
must
be
convinced
beyond
a
 reasonable
doubt
before
they
find
that
a
 person
is
guilty. The
prosecutor
presents
evidence
to
 demonstrate
that
the
defendant
is
 guilty A
Jury
Trial 




Assume
individual
is
innocent
un8l
proven
 Analogy
to
a
Jury
trial… A

Hypothesis
Test 





H0:
assume
there
is
no
treatment
effect
un8l
 there
is
enough
evidence
to
show
otherwise The
alpha
level.

We
are
confident
that
 there
is
a
treatment
effect
because
it
is
very
 unlikely
that
the
data
could
occur
simply
by
 chance
(without
any
treatment
effect). The
sample
data:

gather
data
to
 demonstrate
that
the
treatment
has
an
 effect The
cri1cal
region:
either
the
sample
data
fall
in
 the
cri8cal
region
(enough
evidence
to
reject
H0)
 or
the
data
do
not
fall
in
the
cri8cal
region
(not
 enough
evidence
to
reject
H0)
 guilty. The
jury
must
be
convinced
beyond
a
 reasonable
doubt
before
they
find
that
a
 person
is
guilty. The
prosecutor
presents
evidence
to
 demonstrate
that
the
defendant
is
 guilty A
Jury
Trial 




Assume
individual
is
innocent
un8l
proven
 Analogy
to
a
Jury
trial… A

Hypothesis
Test 





H0:
assume
there
is
no
treatment
effect
un8l
 there
is
enough
evidence
to
show
otherwise The
alpha
level.

We
are
confident
that
 there
is
a
treatment
effect
because
it
is
very
 unlikely
that
the
data
could
occur
simply
by
 chance
(without
any
treatment
effect). The
sample
data:

gather
data
to
 demonstrate
that
the
treatment
has
an
 effect The
cri1cal
region:
either
the
sample
data
fall
in
 the
cri8cal
region
(enough
evidence
to
reject
H0)
 or
the
data
do
not
fall
in
the
cri8cal
region
(not
 enough
evidence
to
reject
H0)
 guilty. The
jury
must
be
convinced
beyond
a
 reasonable
doubt
before
they
find
that
a
 person
is
guilty. The
prosecutor
presents
evidence
to
 demonstrate
that
the
defendant
is
 guilty Either
there
is
enough
evidence
to
 convince
the
jury
that
the
 defendant
is
guilty,
or
there
is
not. A
Jury
Trial 




Assume
individual
is
innocent
un8l
proven
 Analogy
to
a
Jury
trial… A

Hypothesis
Test 





H0:
assume
there
is
no
treatment
effect
un8l
 there
is
enough
evidence
to
show
otherwise The
alpha
level.

We
are
confident
that
 there
is
a
treatment
effect
because
it
is
very
 unlikely
that
the
data
could
occur
simply
by
 chance
(without
any
treatment
effect). The
sample
data:

gather
data
to
 demonstrate
that
the
treatment
has
an
 effect The
cri1cal
region:
either
the
sample
data
fall
in
 the
cri8cal
region
(enough
evidence
to
reject
H0)
 or
the
data
do
not
fall
in
the
cri8cal
region
(not
 enough
evidence
to
reject
H0)
 The
conclusion:
if
the
data
are
not
in
 the
cri8cal
region
–
“fail
to
reject
H0“.

 We
have
not
proved
that
H0
is
true,
we
 simply
have
failed
to
reject
it. guilty. The
jury
must
be
convinced
beyond
a
 reasonable
doubt
before
they
find
that
a
 person
is
guilty. The
prosecutor
presents
evidence
to
 demonstrate
that
the
defendant
is
 guilty Either
there
is
enough
evidence
to
 convince
the
jury
that
the
 defendant
is
guilty,
or
there
is
not. A
Jury
Trial 




Assume
individual
is
innocent
un8l
proven
 Analogy
to
a
Jury
trial… A

Hypothesis
Test 





H0:
assume
there
is
no
treatment
effect
un8l
 there
is
enough
evidence
to
show
otherwise The
alpha
level.

We
are
confident
that
 there
is
a
treatment
effect
because
it
is
very
 unlikely
that
the
data
could
occur
simply
by
 chance
(without
any
treatment
effect). The
sample
data:

gather
data
to
 demonstrate
that
the
treatment
has
an
 effect The
cri1cal
region:
either
the
sample
data
fall
in
 the
cri8cal
region
(enough
evidence
to
reject
H0)
 or
the
data
do
not
fall
in
the
cri8cal
region
(not
 enough
evidence
to
reject
H0)
 The
conclusion:
if
the
data
are
not
in
 the
cri8cal
region
–
“fail
to
reject
H0“.

 We
have
not
proved
that
H0
is
true,
we
 simply
have
failed
to
reject
it. guilty. The
jury
must
be
convinced
beyond
a
 reasonable
doubt
before
they
find
that
a
 person
is
guilty. The
prosecutor
presents
evidence
to
 demonstrate
that
the
defendant
is
 guilty Either
there
is
enough
evidence
to
 convince
the
jury
that
the
 defendant
is
guilty,
or
there
is
not. A
Jury
Trial 




Assume
individual
is
innocent
un8l
proven
 If
there
is
not
enough
evidence,
the
 decision
is
“not
guilty ”.

The
trial
 has
not
proved
that
the
defendant
 is
innocent. ...
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This note was uploaded on 02/24/2011 for the course PSYC 227 taught by Professor Shinkareva during the Fall '09 term at South Carolina.

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