Homework 05 - Answers and Points

Homework 05 - Answers and Points - 
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Unformatted text preview: 
 Homework
Assignment
5
 
 Psychology
60
 Spring
2010
 
 
 
 ‐ For
homework
problems
requiring
calculation,
you
must
show
all
of
your
work,
 including
intermediate
steps,
in
order
to
receive
credit.


 
 
 1) The
value
of
 z X in
a
hypothesis
test
is
influenced
by
a
variety
of
factors.
Assuming
that
all
 other
variables
are
held
constant,
explain
how
the
value
of
 z X 
is
influenced
by
each
of
the
 following:
 a. (1
point)
Increasing
the
difference
between
the
sample
mean
 X 
and
the
untreated
 population
mean
μ
(the
mean
given
H0:
True)
 The
value
of
 z X will
be
more
extreme
(further
from
zero)

 b. (1
point)
Increasing
the
population
standard
deviation,
σ
 The
value
of
 z X will
be
less
extreme
(closer
to
zero)
because

 c. (1
point)
Increasing
the
number
of
scores
in
the
sample,
n
 The
value
of
 z X will
be
more
extreme
(further
from
zero)

 [note:
the
above
is
true
as
long
as
there
is
a
non­zero
mean
difference
between
 the
sample
mean
 X 
and
the
untreated
population
mean
μ]
 
 
 2) A
researcher
is
investigating
the
effectiveness
of
a
new
study‐skills
training
program
for
 elementary
school
children.
A
sample
of
n
=
25
third‐grade
children
is
selected
to
participate
 in
the
program
and
each
child
is
given
a
standardized
achievement
test
at
the
end
of
the
 year.
For
the
regular
population
of
third‐grade
children,
scores
on
the
test
form
a
normal
 distribution
with
mean
of
μ
=
150
and
a
standard
deviation
of
σ
=
25.
The
mean
for
the
 sample
is
 X 
=
158.
Assuming
a
non‐directional
(two
tailed)
hypothesis,
with
 α =0.05:
 a. (1
point)
Identify
the
independent
and
dependent
variables
 IV:
whether
a
child
participates
in
the
study
skills
training
program

 DV:
Score
on
the
standardized
achievement
test
 b. (1
point)
In
a
sentence,
state
the
null
hypothesis
being
tested
 Null:
The
study
skills
program
will
not
affect
the
score
on
the
 standardized
achievement
test.
 
 c. (1
point)
In
symbols,
state
the
null
hypothesis
and
the
alternative
hypothesis
 H 0 : µtreatment = µno treatment H 0 : µtreatment = 150 



or




 
 H 1 : µtreatment ≠ µno treatment H 1 : µtreatment ≠ 150 
 d. (1
point)
Identify
the
critical
values
of
 z X .
That
is,
beyond
what
values
will
you
reject
 the
null
hypothesis?
 zcritical
:
± 
1.96
 
 e. (2
points)
Calculate
 z X 
for
the
sample
mean
 zX = X−µ , σX zX = X−µ σ2 n or z X = X−µ 158 − 150 zX = σ 625 n 25 or z X = 158 − 150 
 25 25 
 zX = 8 , 5 z X = 1.6 
 
 f. (1
point)
What
decision
should
be
made
about
the
null
hypothesis
and
the
 effectiveness
of
this
study‐skills
program?
 Fail
to
reject
the
Null:
There
is
not
enough
evidence
to
conclude
that
the
study
 skills
program
affects
the
score
on
the
standardized
achievement
test
 
 
 3) If
the
alpha
level
of
a
study
is
changed
from
 α =0.05
to
 α =0.01,
 a. (1
point)
What
happens
to
the
boundaries
of
the
critical
region?

 (report
the
actual
critical
values
for
each
 α =0.05
and
 α =0.01)
 
 1.96
 
2.576
 
 b. (1
point)
What
happens
to
the
probability
of
a
Type
I
error?
 (report
the
actual
probability
of
a
Type
I
error
for
each
 α =0.05
and
 α =0.01)
 
 P(Type
I
error)
will
be
reduced
from
0.05
 
0.01
 
 c. (1
point)
What
happens
to
the
probability
of
a
Type
II
error
given
H1
is
true?
 (report
­in
general­
what
happens
to
the
probability
of
a
Type
II
error)
 
 P(Type
II
error)
will
increase
when
alpha
is
reduced.

 
 
 4) Imagine
San
Diego
State
is
evaluating
a
new
English
composition
course
for
freshmen.
A
 random
sample
of
n
=
25
freshmen
is
obtained
and
the
students
are
placed
in
the
course
 during
their
first
semester.
One
year
later,
a
writing
sample
is
obtained
for
each
student
and
 the
wiring
samples
are
graded
using
a
standardized
evaluation
technique.
The
mean
score
 for
the
sample
is
 X =76.
For
the
general
population
of
college
students,
writing
scores
form
a
 normal
distribution
with
a
mean
of
μ
=
70.
 a. (3
points)
If
the
writing
scores
for
the
population
have
a
standard
deviation
of
σ
=
 20,
does
the
sample
provide
enough
evidence
to
conclude
that
the
new
composition
 course
has
a
statistically
significant
effect?
Assume
a
two‐tailed
test
with
 α =0.05.
 
 
 zX = X−µ , σX zX = X−µ σ2 n or z X = X−µ 76 − 70 zX = σ 400 n 25 or z X = 76 − 70 
 20 25 
 zX = 6 , 4 z X = 1.5 
















 zcritical = ±1.96 , z X < zcritical , Fail to Reject H 0 
 
 b. (3
points)
If
the
writing
scores
for
the
population
have
a
standard
deviation
of
σ
=
 10,
does
the
sample
provide
enough
evidence
to
conclude
that
the
new
composition
 course
has
a
statistically
significant
effect?
Assume
a
two‐tailed
test
with
 α =0.05.
 
 X−µ X−µ X−µ 76 − 70 76 − 70 zX = , zX = or z X = zX = 
 or z X = 2 σ 10 σX 100 σ 25 n 25 n 
 6 z X = , z X = 3.0 
















 zcritical = ±1.96 , z X > zcritical , Reject H 0 
 2 
 
 c. (1
point)
Comparing
your
answers
in
part
a
and
part
b,
explain
how
the
magnitude
of
 the
standard
deviation
influences
the
outcomes
of
a
hypothesis
test.
 All
else
equal,
the
smaller
the
population
standard
deviation
the
larger
 the
value
of
the
test
statistic.

 
 5) A
researcher
is
evaluating
the
influence
of
a
treatment
using
a
random
sample
selected
from
 a
normally
distribution
population
with
a
mean
of
μ
=
80
and
a
standard
deviation
of
σ
=
20.
 The
researcher
expects
a
12‐point
treatment
effect
and
plans
to
use
a
two‐tailed
hypothesis
 test
with
 α =0.05.
 a. (1
point)
Calculate
Cohen’s
d
for
this
effect.
How
large
(in
words)
is
this
effect
using
 the
general
guidelines
given
by
Cohen?
 
 X − µ 12 estimated Cohen's d = = = 0.6 ,




 σ 20 This
is
between
a
medium
and
large
effect
size
 
 
 b. (4
points)
Compute
the
power
of
the
hypothesis
test
if
the
researcher
uses
a
sample
 size
of
n
=
16
individuals
 
 σ Xcritical = µnull + zcriticalσ X , Xcritical = µnull + zcritical 
 n 
 for
12
point
increase,
critical
value
for
the
upper
tail
would
equal:
 20 Xcritical = 80 + 1.96 × , Xcritical = 80 + 1.96 × 5, Xcritical = 89.8 
 16 
 z= Xcritical − µalternative 89.8 − 92 -2.2 , z= , z= , z = -0.44 
 σX 5 5 
 p(z
>
­0.44)
=
0.6700.














Power
=
0.6700
 
 
 c. (4
points)
Compute
the
power
of
the
hypothesis
test
if
the
researcher
uses
a
sample
 size
of
n
=
25
individuals.
 
 
 σ Xcritical = µnull + zcriticalσ X , Xcritical = µnull + zcritical 
 n 
 for
12
point
increase,
critical
value
for
the
upper
tail
would
equal:
 20 Xcritical = 80 + 1.96 × , Xcritical = 80 + 1.96 × 4, Xcritical = 87.84 
 25 
 X − µalternative 87.84 − 92 -4.16 z = critical , z= , z= , z = -1.04 
 σX 4 4 
 p(z
>
­1.04)
=
0.8508.









Power
=
0.8508
 ...
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