Homework 03 - Answers and Points

Homework 03 - Answers and Points - 
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Unformatted text preview: 
 Homework
Assignment
3
 
 
 Psychology
60
 Spring
2010
 1. (2
points)
What
does
it
mean
for
a
sample
to
have
a
standard
deviation
of
zero?

Describe
 the
scores
in
such
a
sample.

A
sample
with
a
standard
deviation
of
zero
can
only
occur
 when
the
SS
are
0,
which
will
only
occur
when
all
the
values
in
the
sample
are
the
 same.

 
 2. (2
points)
Explain
why
the
sum
of
squared
deviations,
SS,
can
not
ever
be
negative.
The
SS
 can
never
be
negative
because
it
is
calculated
by
summing
the
squaring
distances
to
 the
mean.
Because
the
square
of
a
negative
number
is
positive,
there
is
no
way
for
the
 sum
of
squared
values
to
be
negative.
 
 
 
 3. Calculate
the
SS,
variance,
and
standard
deviation
for
the
following
population
of
6
scores:
 














































































11,
0,
8,
2,
4,
5
 a. (2
points)
SS
=



 μ
=
5,


SS=
(11­5)2+(0­5)2+(8­5)2+(2­5)2+(4­5)2+(5­5)2
=
80
 
 b. (1
point)
Variance
=

 SS/N,


80/6
=
13.33


























 
 












(must
use
N
not
N­1
for
any
points)
 
 c. (1
point)
Standard
deviation
=


 √ 13.33
=
3.65
 4. Consider
a
sample
consisting
of
4
scores.
The
SS
for
this
sample
=
20.
 a. (2
points)
What
is
the
best
estimate
of
the
standard
deviation
of
the
population
from
 which
this
sample
is
drawn?

 √ (SS/n­1)
=
√ (20/3)
=
√ (6.667)
=
2.58
















(must
use
n­1
for
any
points)
 b. (2
points)
What
is
the
standard
deviation
of
the
scores
in
this
particular
sample?

 √ (SS/N)
=
√ (20/4)
=
√ (5)
=
2.236


























(must
use
n
for
any
points)
 
 5. The
following
sample
was
collected
with
the
explicit
purpose
of
determining
characteristics
 of
the
population
from
which
it
was
drawn.
Calculate
the
SS,
variance,
and
standard
 deviation
using
the
following
sample
of
n
=
4
scores:

 


















































































3,
1,
1,
1.
 a. (2
points)
SS
=
 Xbar
=
1.5,



(3­1.5)2+(1­1.5)2+(1­1.5)2+(1­1.5)2
=
3
 
 b. (1
point)
Variance
=
 SS/(n­1),


3/(4­1)
=
1








































(must
use
n­1
not
N
for
any
points)
 
 c. (1
point)
Standard
deviation
=

 √ 1
=
1
 6. 7. 8. 9. 
 
 
 A
distribution
has
a
standard
deviation
of
σ
=
8.
Find
the
z‐score
for
each
of
the
following
 locations
in
the
distribution.

 a. (0.5
points)
Above
the
mean
by
4
points
 
z=(x­μ)/σ,


4/8
=
0.5,


z
=
0.5
 b. (0.5
points)Above
the
mean
by
16
points
 z=(x­μ)/σ,


16/8
=
2,


z
=
2
 c. (0.5
points)Below
the
mean
by
8
points
 z=(x­μ)/σ,


­8/8
=
­1,


z
=
­1
 d. (0.5
points)Below
the
mean
by
12
points
 z=(x­)/σ,


­12/8
=
­1.5,


z
=
­1.5
 
 For
a
population
with
μ
=
50
and
σ
=
4
 a. Find
the
z‐score
for
each
of
the
following
X
values.

 i. (0.5
points)
X
=
55:
z=(x­μ)/σ,

(55­50)/4
=
5/4,
z
=
1.25
 ii. (0.5
points)
X
=
60:
z=(x­μ)/σ,

(60­50)/4
=
10/4,
z
=
2.5
 iii. (0.5
points)
X
=
35:
z=(x­μ)/σ,

(35­50)/4
=
­15/4,
z
=
­3.75
 
 b. Find
the
score
(X‐value)
that
corresponds
to
each
of
the
following
z‐scores
 i. (0.5
points)
z
=
.80:
x
=
μ+zσ,
50
+
.8(4)
=
53.2

 ii. (0.5
points)
z
=
‐0.50:
x
=
μ+zσ,
50
­
.50(4)
=
48
 iii. (0.5
points)
z
=
0.30:
x
=
μ+zσ,
50
+
.3(4)
=
51.2
 
 
 Find
the
z‐score
corresponding
to
a
score
of
X
=
60
for
each
of
the
following
distributions:
 a. (0.5
points)
μ
=
50,
σ
=
10:
z=(x­μ)/σ,

(60­50)/10
=
10/10,
z
=
1
 b. (0.5
points)
μ
=
50,
σ
=
5:
z=(x­μ)/σ,

(60­50)/5
=
10/5,
z
=
2
 
 
 (1
point)
A
score
that
is
6
points
below
the
mean
corresponds
to
a
z‐score
of
z
=
‐2.00.
What
 is
the
population
standard
deviation?
z
=
(x­μ)/σ,


­2
=
­6/σ,


σ
=
3
 10. (1
point)
For
a
population
with
a
standard
deviation
of
σ
=
10,
a
score
of
x
=
44
corresponds
 to
z
=
‐0.50.
What
is
the
population
mean,
μ?

z
=
(x­μ)/σ,


­0.50
=
(44­μ)/10,


μ
=
49
 
 
 11. For
each
of
the
following
populations,
would
a
score
of
X
=
50
be
considered
a
central
score
 (near
the
middle
of
the
distribution)_
or
an
extreme
score
(far
out
in
the
tail
of
the
 distribution)?
 a. (0.5
points)
μ
=
45,
σ
=
10:
z
=
(x­μ)/σ,
(50–45)/10,
z
=
0.50.
Central
score
 b. (0.5
points)
μ
=
40,
σ
=
2,
z
=
(x­μ)/σ,
(50–40)/2,
z
=
5.00.
Extreme
score
 c. (0.5
points)
μ
=
51,
σ
=
2,
z
=
(x­μ)/σ,
(50–51)/2,
z
=
­0.50.
Central
score
 d. (0.5
points)
μ
=
60,
σ
=
20,
z
=
(x­μ)/σ,
(50–60)/20,
z
=
­0.50.
Central
score
 
 
 12. (2
points)
Which
of
the
following
exam
scores
should
lead
to
the
better
grade?
A
score
of
X
=
 43
on
an
exam
with
μ
=
40
and
σ
=
2,
or
a
score
of
X
=
60
on
an
exam
with
μ
=
50
and
σ
=
20?

 Explain
your
answer.
z
=
(x­μ)/σ,
(43–40)/2,
z
=
1.5


or

(60–50)/20,
z
=
0.5.
The
score
 
 of
43
should
lead
to
a
higher
grade
because
that
individual
scored
higher
than
more
 individuals
than
the
person
who
scored
60
on
the
other
exam
 13. Find
each
of
the
following
probabilities
for
a
normal
distribution:
 a. (0.5
points)
p(
z
>
‐0.80
):





.7881
 b. (0.5
points)
p(
z
<
0.25
)




.5987
 c. (0.5
points)
p(
z
>
‐2.10
)




.9821
 
 
 14. Find
each
of
the
following
probabilities
for
a
normal
distribution:
 a. p(
‐1.50
>
z
>
‐0.80
)

DO
NOT
GRADE,
TYPO
IN
QUESTION
 b. (1
point)
p(
­1.00
<
z
<
0.25
):







.3413
+
.0987
=
0.44
 c. (1
point)
p(
‐2.00
<
z
<
1.25
)








.4772
+
.3944
=
0.8716
 
 
 15. The
distribution
of
the
scores
on
the
SAT
is
approximately
normal
with
a
mean
of
μ
=
500
 and
a
standard
deviation
of
σ
=
100.
For
the
population
of
students
who
have
taken
the
SAT:
 a. (2
points
total,
1
point
for
each
step)
What
proportion
have
SAT
scores
greater
than
 700?
 z
=
(x­μ)/σ,
(700–500)/100,
z
=
2.00,
 p(
z
>
2.00)
=
.0228
 
 b. (2
points
total,
1
point
for
each
step)
What
proportion
have
SAT
scores
greater
than
 550?
 z
=
(x­μ)/σ,
(550–500)/100,
z
=
0.50,
 
 p(
z
>
0.50)
=
.
3085
 
 c. (2
points
total,
1
point
for
each
step)
What
is
the
minimum
SAT
score
needed
to
be
 in
the
highest
10%
of
the
population?
 Restated
as
z
problem:
p(X>?)
=
0.10,

p(z
>
?
)
=
0.10
 p(z
>
1.28
)
=
0.10
 X
=
μ+zσ,

X
=
500
+
1.28(100),



X
=
628
 
 d. (2
points
total,
1
point
for
each
step)
If
the
state
college
only
accepts
students
from
 the
top
60%
of
the
SAT
distribution,
what
is
the
minimum
SAT
score
needed
to
be
 accepted?
 
 Restated
as
z
problem:
p(X>?)
=
0.60,

p(z
>
?
)
=
0.60
 p(z
>
­0.255
)
=
0.60
 X
=
μ+zσ,

X
=
500
+
(­0.255)(100),



X
=
474.5
 
 (­0.255
is
most
accurate
for
z,
but
­0.25
or
­0.26
are
also
okay)
 
 
 
 ...
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This note was uploaded on 10/16/2011 for the course PSYCH 60 taught by Professor Parris during the Spring '11 term at UCSD.

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