stat_exam2A_solutions_spring14
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School
Northwestern University *
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Course
210
Subject
Statistics
Date
Feb 20, 2024
Type
Pages
8
Uploaded by CommodoreUniverseEagle26
Circle your
final
answers
to
each
Problem.
Problem
1 (4
points)
For
#\a
and
#\b:
An SRS of 400
adults asks
if
they agree with
the
statement
"I
support candidate
Smith." Suppose that
80% of all US
adults would agree
if
asked this question,
so the
population
parameter
P =
0.8.
a)
What
are the
mean
and
standard deviation
of the
sample proportion
p
?
Round your answers
to fo
decimal
places
if
necessary.
A
b)
What
is the
probability that
the
survey result
differs
from
the
truth about
the
population
by
more
th
in
4
percentage points?
Use
your
final
answers
from
part
a.
Round your
final
answer
to
four
decimal pla(
es.
0,01.
-2,00
0.0
L
For
#lc and
#ld:
In the
population (all high
school
students),
scores
of
students
on
Test
fFhad
mean
80
and
standard deviation
a =
50. An SRS of 100
students
who
took
the
exam
is
taken.
c)
What
are the
mean
and
standard deviation
of the
sample mean score
x
?
Round your answers
to fo
decimal
places
if
necessary.
i t *
<L^O
^
s
•H
d)
What
is the
probability that
the
mean score
X
of
these
students
is 78 or
higher?
Use
your
final ans
from
part
«.
Round your
final
answer
to
four
decimal
places.
r
,vers
Problem
2 (4
points)
Round
all of
your
final
answers
to
The
joint probability distribution
o
y =
0
y=
1
Four
decimal places
if
necessary.
~Xand
F
appears below:
jc
= 0
0.3
0.1
•
X=l
0.2
0.4
a)
What
is
^7
-
0(^0,4)
+
=
r
O,6
c)
What
i
d)
What
is the
covariance between
X
and
F?
-V-
OS
0,
Problem
3 (4
points)
For
#3a
and
#3b:
Jen and
Elise
are
playing
in the
club bowling tournament. Their scores vary
as the
play
repeatedly. Jen's
score
Jf
has the
./V(180,
6)
distribution,
and
Elise's score
Fhas
the
^(179,
8)
distribution.
Their
scores
are
independent.
a)
What
are the
mean
and
standard deviation
of
(X-Y)1
Round your answers
to
four
decimal places
if
necessary.
~
u
b)
What
is the
probability that
Jen
will
score
at
least
three,
points
/mvgrthan
Elise?
Use
your
final
answers
from
part
a.
Round your
final
answer
to four
decimal places.
/0
For #3c and
#3d:
Tom and
George
are
playing
in the
club
golf tournament. Their scores vary
as
they
play
the
course repeatedly.
Tom's
score
A
has the
N(90,
5)
distribution,
and
George's
score
B
has the
W(89,
12)
distribution. Their scores
are
independent.
fe-**
-^>
A
-
c)
What
are the
mean
and
standard deviation
of
(A-B)7
Round your answers
to
four
decimal places
if
necessary.
1
d)
What
is the
probability that George
will
score
at
least
two
points
higher
than Tom?
Use
your
final
answers
from
part
c.
Round your
final
answer
to four
decimal places.
-0.-L3
^
£
£p—
0.4010
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/
Problem
4 (4
points)
For
#4<3
and
#4b:
The
joint probability distribution
of
Xand
7
appears below:
x = 2
0,5
'=1
0.2
0.3
y
=
i
0.3
0.2
•)
What
is
E(Y\x
=
2)?
Round your
final
answer
to
four decimal places
if
necessary.
,?
)
= f
b)
What
is
E(X\y
=
3)?
Round your
final
answer
to four
decimal
places
if
necessary.
For #4c and
#4c?:
Consider
the
following information about random variables
X
and
Y:
PXY
= 0.2
Consider another random variable:
./?
=
0.4^f+
0.67.
c)
What
is
///;?
Round
your answer
to
four
decimal places
if
necessary.
d)
What
is
cr/??
Round your answer
to
four
decimal places
if
necessary.
-
F
-H
s
+
A
.
Problem
5 (4
points)
Circle
A or B to
indicate whether each statement
is
true
or
false.
1.
Xand
Fare
two
discrete random variables.
If
X
and
Fare
uncorrelated,
then
X
and
I
would
definitely
be
independent.
2.
Xand
Fare
two
discrete random variables.
IfJfand
Fare
not
independent, then
the
correlation between
X
and
F
would
definitely
not
equal zero.
A) true
13))
false
3.
Xand
Tare
two
discrete random variables.
If the
covariance
between
X
and
Fis
negative, then
X
and
F
would
definitely
not be
independent.
true
B)
false
4.
Xand
Fare
two
independent discrete random variables.
The
standard deviation
of
X
equals
the sum of
their standard deviations.
true
false
5.
Decreasing
the
level
of
confidence decreases
the
margin
of
error when constructing
confidence
interval
for the
population mean.
(Ap
true
B)
false
Let
9
be an
estimator
of the
population parameter
9.
lfE\0]=
0,
then
9
is
definitel}
the
most
efficient
estimator
of
9.
,^
true
C.
(&
false
7.
Suppose
the
population
is
equal
to
100,000.
Suppose
the
population proportion
P =
0.96. Here,
one can
safely
use the
normal approximation
for the
sample proportion
when
an SRS of 100
individuals
from
the
population
is
drawn.
true
nf^i-^3
^»-^
?
I'll
SO
/
\
r . i \
. \ /
i
^ *
*—
CIOOJ
(Oflfaj
COiOlH
^
So
'
^-
^
8.
Draw independent observations
at
random
from
any
population with
finite
mean
//.
Decide
how
accurately
you
would like
to
estimate
//.
As the
number
of
observations
drawn
increases,
the
mean
x
of the
observed values eventually approaches
the
mean
of the
population
as
closely
as you
specified
and
then stays that close. This result
is 1
central limit theorem.
true
£LT
—*
X^H
frlk*
h
false
ie
2.
3.
B.
C.
A
Each of the 10 multiple choice questions is worth 1 point (10 points total). Circle your
answers (circle
the
letter).
1.
Test
Z is a
standardized test that
all
high school seniors take.
The
mean
test
score
fo
random sample
of 30
students
who
take
the
test
is
65.
The
sample standard
deviatioi
YL.
Using this
data,
a 90%
confidence interval
for the
population mean
is
constructe
What
is the
margin
of
error (rounded
to
four
decimal places) associated with this
confidence
interval?
3.6040 points
3.6369
points
3.6456
points
3.7179
points
3.7223 points
3.7267
points
In
a
random sample
of 80
registered
female
voters,
48
said they supported Obama.
I
random sample
of 80
registered male
voters,
24
said they supported Obama.
Let's
denote
the
respective population proportions
as
P
x
(female)
and
P
y
(male). Using
thi
data,
a 95%
confidence interval
for the
population
difference
(P
x
-P
y
)
is
constructed
What
is the
margin
of
error (rounded
to
four
decimal places) associated with this
confidence
interval?
A)
B)
C)
SP
E)
F)
0.1209
0.1234
0.1440
0.1470
0.1893
0.1932
ffo
-
_
"50
O,(o
-
0,3
Let
the
computer randomly generate
a
number that ranges
from
0 to 10 and
call this
random variable
X.
Consider
the
following
function:
*-'-'
x
-—
y
^
r
function
of
x = 0
if
x <
0
function
of
jc
=
(0.1
);c
if
0
<x
<
10
function
of
jc
= 1
if
x >
10
The
function
described above
is .
the
cumulative distribution
function
of
X
the
probability
density
function
of
X
neither
the
cumulative distribution
function
of
X
nor the
probability density
function
X
Suppose
that
X
and
Fare
two
discrete random variables that have correlation
p,
and
p < 0.
I
.
v
_ _
this
case,
cr
x
+<Ty
!_>
4-
is
i
a
rf
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5.
6.
A
The
mean monthly rent
for a
random sample
of 30
apartments advertised
in the
local
newspaper
is
$1000.
The
population standard deviation
is
$300. Using this data,
a 9
confidence
interval
for the
population mean rent
is
constructed. What
is the
margin
error (rounded
to two
decimal places) associated with this confidence interval?
$107.35
$108.67
$111.84
$112.01
$112.17
$112.39
The
insurance company sees that
in the
entire population
of
homeowners,
the
mean
1
from
fire is
ju
=
$300
and the
standard deviation
of the
loss
is
a
=
$200. Losses
on
separate policies
are
independent.
The
random variable
W
is the
total loss
for 8
randomly
selected policies. What
is the
standard deviation
of
W
(rounded~to
two
decimal
places)?
$70.71
$106.07
$565.69
$848.53
8
A
number other than
any of the
other choices listed.
7.
Suppose
fiy—
100
and
ay
=
50. An SRS
of
n = 6
is
drawn
from
a
population
of
10,00
and
>>,
is a
weighted average
in
which
the
observations
are
alternatively weighted
by
1/4
and
7/4:
>V
Suppose
fiy=
100
and
ay
=
50. An SRS
of
n
= 6 is
drawn
from
a
population
of
10,00
and
y
2
is a
weighted
average
in
which
the
observations
are
alternatively weighted
fr
1/2
and
3/2:
h
In
this example,
y
}
this example,
y
2
_>
is, is
is, is not
is
not,
is
is
not,
is not
_ an
unbiased estimator
of the
population mean
of
Y
(u
y).
I
an
unbiased estimator
of the
population mean
of
Y
(jiy).
5%
)SS
0.
9.
10.
/-
8.
Test
A
is a
standardized test that
all
high school juniors take. Suppose
we
wish
to
calculate
a 95%
confidence interval
for the
population mean test score.
The
desired
margin
of
error
is 5
points. Suppose that
the
population standard deviation
is 14. W
is the
minimum required sample size
to
estimate
/u
with
a
margin
of
error less than
or
equal
to 5
points?
Students
are
randomly assigned into
a
treatment group
and a
control group.
Membeis
of
the
treatment group
use
method
A
to
learn
how to
read. Members
of the
control
groi
p
follow
the
traditional curriculum.
After
eight weeks,
all
students
are
given
a
standardized
test.
Consider
the
information
in the
table below:
Group
Treatment
(x)
Control
(y)
n
30
25
Sample mean
60
55
ftrj
10
8
Using this
data,
a 95%
confidence interval
for
(p.
x
-
fj.
y
)
is
constructed. What
is the
margin
of
error (rounded
to
four
decimal places) associated with this confidence
interval?
AtF
c:
3.
4.7581 points
f
4.8164
points
4.9572 points
4.9645 points
5.0009 points
5.0106
points
Gallup polled
US
adults
and
asked,
"Do you
support President
Obama?"
Of the 80
individuals
who
were polled,
44
responded,
"Yes."
Using this
data,
a 95%
confident
interval
for
P,
the
proportion
of all US
adults
who
support President Obama,
is
constructed. What
is the
margin
of
error (rounded
to
four
decimal places) associated
with
this confidence interval?
0.0818
0.0913
0.0915
0.0975
0.1088
0.1090
/
w i
?0
iat
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