Homework 10 - Answers

Homework 10 - Answers - Homework
Assignment
10
...

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Unformatted text preview: Homework
Assignment
10
 NOT
GRADED
 Solutions
 
 
 
 1. What
information
does
the
sign
of
the
Pearson
correlation
provide?
How
does
this
 information
relate
to
the
sign
of
the
slope
(b)
and
will
they
always
correspond?

 
 The
sign
of
the
Pearson
correlation
coefficient
describes
whether
the
relationship
 between
the
variables
is
positive
or
negative.
A
positive
sign
of
r
indicates
a
positive
 relationship,
which
means
that
increases
in
the
X
variable
are
associated
with
 increase
in
the
Y
variable.
For
example,
a
positive
relationship
exists
between
height
 and
weight;
people
who
are
taller
also
tend
to
be
heavier.
A
negative
sign
for
r
 indicated
a
negative
relationship,
which
means
that
increases
in
the
X
variable
are
 associated
with
decreases
in
the
Y
variable.
For
example,
a
negative
relationship
exists
 between
the
number
of
security
cameras
in
a
store
and
shoplifting
rates;
the
more
 cameras
a
store
has
the
less
shoplifting
they
have.


 
 The
value
of
b,
the
slope
of
the
linear
regression
function,
also
gives
the
same
 information
and
will
always
correspond.
The
value
of
the
slope
tells
us
how
much
of
a
 change
in
the
Y
variable
we
expect
for
each
1­unit
change
in
the
X
variable.
So,
if
you
 have
a
positive
slope,
it
means
that
you
expect
the
Y
variable
to
increase
when
you
 increase
the
value
of
the
X
variable.
For
example,
increasing
height
would
also
be
 expected
to
increase
weight.
If
you
have
a
negative
slope,
it
means
that
you
expect
the
 Y
variable
to
decrease
when
you
increase
the
value
of
the
X
variable.
For
example,
 increasing
number
of
cameras
would
be
expected
to
decrease
shoplifting
 
 2. Calculate
the
sum
of
products
(SP)
for
the
following
set
of
scores:
 X

 2
 6
 1
 3
 
 
 
 Psychology
60
 Spring
2010
 
 Y
 4
 3
 8
 5
 
 
 
 
 
 Xi − X 
 Dev
for
X
 ­1
 
 3
 
 ­2
 
 0
 
 Yi − Y 
 
 Dev
forY
 ­1
 
 ­2
 
 3
 
 0
 
 ( Xi − X )(Yi − Y ) 
 
 SP
 
 
 1
 
 6
 
 6
 
 0
 
 SP
=
­11
 3. Assuming
that
you
are
running
a
two‐tailed
test
with
alpha
=
0.05,
how
large
of
a
correlation
 is
needed
to
be
statistically
significant
or
each
of
the
following
samples?
 a. A
sample
of
n
=
10
 For
df
=
8,

rcrit
=
.632
 b. A
sample
of
n
=
20
 For
df
=
18,

rcrit
=
.444
 c. A
sample
of
n
=
30
 For
df
=
28,

rcrit
=
.361
 
 
 4. The
regression
equation
is
intended
to
be
the
best
fitting
straight
line
for
a
set
of
data.
What
 is
the
criterion
for
“best
fitting”
?
 
 The
criterion
for
best
fitting
is
that
the
sum
of
the
squared
distances
from
the
actually
 ˆ observed
Y
values
(the
Yi)
to
the
values
for
Y
predicted
by
the
regression
line
( Yi )
is
as
 small
as
it
possibly
can
be.
The
best
fitting
regression
line
will
have
a
sum
of
these
 squared
errors
that
is
smaller
than
for
any
other
straight
line

 
 ˆ 5. For
the
following
scores,
find
the
best
fitting
line,
and
find
the
predicted
values
of
Y
( Yi )
at
 each
X
score
(that
is,
what
values
the
fitted
line
predicts
for
each
observed
value
of
X)
 
 X
 Y
 3
 0
 8
 10
 7
 8
 5
 3
 7
 7
 6
 8
 
 
 SP 
 
 ( Xi − X )(Yi − Y ) ( Xi − X )2 (Yi − Y )2 Xi − X Yi − Y 
X Y 
3 0 -3 -6 18 9 36 
8 10 2 4 8 4 16 8 1 2 2 1 4 
7 3 -1 -3 3 1 9 
5 7 7 1 1 1 1 1 
 6 8 0 2 0 0 4 
 
 Y =6 SP = 32 SSx=16 SSY=70 
X =6 
 SP 32 b= , b= , b = 2 


















 a = Y − bX , a = 6 − 2(6 ), a = 6 − 12, a = −6 
 SSX 16 
 ˆ ˆ Y = bX + a, Y = 2 X − 6 
 
 ˆ X Y = 2X − 6 
 
 
 3 8 7 5 7 6 0 10 8 4 8 6 
 6. A
set
of
n
=
20
pairs
of
scores
(X
and
Y
values)
has
SSx
=
25,
SSY
=
16,
and
SP
=
12.50.
If
the
 mean
for
the
X
values
is
 X = 6 
and
the
mean
for
the
Y
values
is
 Y = 4 
 Find
the
best
fitting
line
for
these
data
 
 SP 12.5 b= , b= , b = 0.5 


















 a = Y − bX , a = 4 − 0.5(6 ), a = 4 − 3, a = 1 
 SSX 25 
 ˆ ˆ Y = bX + a, Y = 0.5 X − 1 
 
 a. Calculate
the
Pearson
correlation
for
the
scores
 
 SP 12.50 12.50 r= , r= , r= , r = .625 
 20 SSX SSY 25 × 16 
 
 b. Assuming
everything
else
stays
the
same,
does
the
value
of
the
Pearson
correlation
 coefficient
change
if
the
mean
for
the
X
values
is
now
 X = 600 
and
the
mean
for
the
Y
 values
is
 Y = 400 ?
Why
or
why
not?
 
 SP 12.50 12.50 r= , r= , r= , r = .625 
 20 SSX SSY 25 × 16 
 
 The
value
of
r
is
unchanged
by
the
change
in
the
mean.
The
sample
means
have
 nothing
to
do
with
the
correlation
coefficient
as
the
correlation
measures
 association.

 
 
 c. Assuming
these
data
represent
a
sample
from
some
population,
do
we
have
evidence
 that
there
is
a
linear
relationship
in
the
population
from
which
the
sample
is
drawn?
 
 
 For
df
=
18,

rcrit
=
.444.




 
 robserved
=
0.625.
Because
robserved
>
rcrit,
we
reject
H0.
The
data
provide
evidence
that
 there
is
a
relationship
in
the
population
from
which
the
sample
is
drawn.

 ...
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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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