PSYC227_fall09_week9

PSYC227_fall09_week9 - EXAM 2 on
Thursday Hypothesis...

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Unformatted text preview: EXAM 2 on
Thursday Hypothesis Testing Was
it
due
to
chance,
or
something
 else? Learning Objectives: 1. Effect size 2. Power 

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 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
beforethey
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. 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 Effect size Population: normal distribution with µ=80, σ=10. M = 81, α=0.05 =1 n = 25 n =400 0.025 0.025 Cohen’s d = (81 – 80)/10 = .1 0.5 1.96 2 Effect size Magnitude 0 < d < 0.2 0.2 <d < 0.8 d > 0.8 Evaluation of Effect Size small effect (mean difference less than 0.2 SD) medium effect large effect Statistical Power The power of a statistical test is the probability that the test will correctly reject a false null hypothesis. It is the probability that the test will identify a treatment effect if one really exists. µ σ M=83.92 P(z > -2.04) = 0.9793 M=83.92 0 The power of the test is 97.93% z 0.9793 M=83.92 0 P(z > -2.04) = 0.9793 The power of the test is 97.93% Factors that affect power Sample size Alpha level z=2.58 for α=0.01 Previous example, n=25 here, n=4 One-tailed vs. two-tailed tests z=1.65 for onetailed test Example The average grade for a seniors at the local high school is 78, with a standard deviation of 12. The distribution is normal. A researcher would like to evaluate the effectiveness of a new study-skills training program using a sample of 16 students. The researcher plans to examine the results using a two-tailed hypothesis test with alpha 0.05. 1. Compute the power of the hypothesis test if the training program has a 3-point effect. Compute the power of the hypothesis test if the training program has a 6-point effect. 2. Assuming
H0 n

 regio 5 cal
 Cri8 2
=
0.02 / 0.05 6
points Assuming

H1 Power
=
P(z
>‐0.04) 78











84 83.88 



‐1.96







0







1.96 ‐0.04
0 REVIEW 25 Probability propor8on
of
all
the
possible
outcomes Normal
distribu1on Useful
Approxima8ons Normal
approxima8on
of
binomial
distribu8on Note:
upper
and
lower
real
limits Distribu8on
of
sample
mean Standard
Error ...
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