2009 Midterm

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Unformatted text preview: Statistics
101,
Instructors:
A.
Wyner,
A.

Rakhlin
 

 
 October
29,
2009
 Student
Name:
 
 
 
 
 
 
 
 Section:
 
 Midterm
Exam
 Instructions
 1. Fill
in
your
answers
in
the
blue
book.

 2. You
must
show
work
and
give
explanations.
 3. Rules
of
uniformity:
One‐sided
letter
crib
sheet,
calculator
ok
(even
with
statistical
 functions
and
graphing),
otherwise
closed
notes,
closed
book.
 4. No
cell
phones
visible!
 5. Rules
of
Fairness:
 A. Complete
this
exam
entirely
on
your
own.

Do
not
acquire
information
from
 other
students
or
the
outside
in
any
form.
 B. Do
not
take
copies
of
this
exam
out
of
this
room.
 C. You
have
two
hours
for
this
exam.
Any
student
observed
continuing
to
write
 after
time
has
been
called
will
be
documented
and
points
will
be
deducted.
 6. Rules
of
Consideration:

 A. Be
considerate
to
others
by
not
causing
commotion
and
distraction.
 B. If
you
finish
early,
leave
quietly;
avoid
slamming
doors.
 
 Honor
code
applies
to
the
above
Rules.
 Your
signature:
 __________________________________
 
 
 About
this
test:
When
more
than
one
choice
seems
reasonable,
pick
the
one
that
is
best.
 STOP 
 WAIT UNTIL YOU ARE INSTRUCTED TO PROCEED. 
 
 Problems
1­11:
 A
vendor
would
like
to
anticipate
consumption
of
hot
dogs
at
a
baseball
game.
If
the
vendor
 prepares
too
few
hot
dogs,
it
loses
the
opportunity
to
sell
more.
If
on
the
other
hand
it
 prepares
too
many
hot
dogs,
the
excess
not
sold
is
lost.
The
following
data
shows
the
 number
of
hot
dogs
(NumSold,
in
thousands)
at
50
recent
games,
plotted
against
the
official
 attendance
for
those
games
(Crowd
Size,
in
thousands).
For
each
of
these
50
games,
the
 vendor
had
prepared
excess
hot
dogs
so
that
there
were
no
constraints
of
customer
 demand.
 
 Num Sold 6 7 Crowd Size 8 9 10 11 Quantiles 100.0% maximum 99.5% 97.5% 90.0% 75.0% quartile 50.0% median 25.0% quartile 10.0% 2.5% 0.5% 0.0% minimum 12 14 16 18 20 22 24 26 Quantiles 11.3601 11.3601 11.2235 10.3542 8.99473 8.32376 7.61567 6.89331 6.3296 6.26538 6.26538 Moments Mean Std Dev Std Err Mean Upper 95% Mean Lower 95% Mean N 12 100.0% maximum 99.5% 97.5% 90.0% 75.0% quartile 50.0% median 25.0% quartile 10.0% 2.5% 0.5% 0.0% minimum 24.3517 24.3517 24.0969 21.9885 20.1133 18.5343 17.5796 15.9835 14.3411 13.9986 13.9986 Moments 8.4487513 1.2077504 0.1708017 8.7919901 8.1055124 50 Mean Std Dev Std Err Mean Upper 95% Mean Lower 95% Mean N 18.868184 2.2302545 0.3154056 19.502016 18.234353 50 1. The
average
crowd
size
in
the
50
games
is

 a) Impossible
to
determine
from
the
given
information
 b) A
lot
larger
than
the
median
crowd
size
 c) Smaller
than
the
median
crowd
size
 d) Larger
than
the
75th
quantile
 e) About
equal
to
the
median
crowd
size.
 
 2. The
distribution
of
Crowd
Size
is
 
 a) roughly
bell‐shaped
 b) does
not
obey
the
area
principle
 c) better
represented
by
a
Bar
Chart
 d) clearly
bi‐modal
 e) Better
represented
by
a
pie
chart
than
a
histogram
 
 
 3. The
Empirical
Rule
would
tell
us
that
for
2/3
of
of
the

50
games
the
crowd
size

is
 between
 a) 16.6
and
21.1
thousand
 b) 14.4
and
23.3
thousand
 c) 14.4
and
21.1

thousand
 d) 12.2
and
25.6
thousand
 e) 16.5
and
20.1
thousand
 
 4. The
Inter
Quartile
Range
for
the
NumSold
is
 a) 1.4
thousand
hot
dogs
 b) 0.67
thousand
hot
dogs
 c) 0.71
thousand
hot
dogs
 d) plus
or
minus
1
thousand
hotdogs
without
ketchup
 
 The
following
is
the
regression
of
the
number
of
hotdogs
sold
on
the
size
of
the
crowd:
 Summary of Fit RSquare RSquare Adj Root Mean Square Error Mean of Response Observations (or Sum Wgts) 0.812722 0.808821 0.528078 8.448751 50 12 Num Sold 11 10 9 8 7 6 12 14 16 18 20 Crowd Size 22 24 26 
 5. Judging
from
the
information
above
 a) The
correlation
between
Crowd
Size
and
Num
Sold
is
around
.8
 b) The
correlation
between
Crowd
Size
and
Num
Sold
is
greater
than
.95
 c) The
correlation
between
Crowd
Size
and
Num
Sold
is
around
.4
 d) The
correlation
between
Crowd
Size
and
Num
Sold
is
around
.9
 e) The
correlation
between
Crowd
Size
and
Num
Sold
cannot
be
determined

 
 6. On
average,
each
extra
person
is
expected
to
purchase
about

 a) 3
hot
dogs
 b) 1
hot
dog
 c) ½
a

hot
dog
 d) 4/5th

of
a
hot
dog
 e) 
¼
of
a
hot
dog
 
 7. Given
the
information
above,
calculate
the
equation
of
the
regression
line.
 
 
 
 8. If
16
thousand
people
attend
the
game,
the
model
predicts
that
the
vendor
will
sell
on
 average
how
many
hot
dogs?

 
 9. 
If
you
only
knew
that
7000
hot
dogs
were
sold
at
a
game,
how
big
do
you
think
the
 crowd
was
on
that
day?
 
 10. In
one
game,
attended
by
20
thousand
people,
10
thousand
hot
dogs
were
sold.

Is
this
 amount
fairly
typical
or
unusual?
Explain.
 
 11. Suppose
you
choose
a
game
at
random
from
one
of
the
50
games
in
the
data
set.
Let
X=
 numsold
(in
thousands)
and
Y=crowd
size

(in
thousands).

Calculate
the
Cov(X,Y).

 
 Problems
12­16:
 
 The
baseball
World
Series
between
NY
Yankees
and
the
Phillies
will
alternate
 its
location
between
NY
and
Philadelphia.
The
first
2
games
will
be
played
in
NY,
followed
 by
3
games
in
Philadelphia,
followed
by
2
games
in
NY.
Although
the
schedule
allows
for
7
 games,
 the
 series
 will
 terminate
 as
 soon
 as
 one
 team
 wins
 four
 games.
 
 Thus,
 the
 actual
 series
are
is
not
likely
to
be
a
full
7
games.


 
 Let
N
be
the
random
variable
equal
to
the
number
of
games
played.

Assume
that
 
P(N=7)=

.07,

P(N=6)=
.28

and
P(N=5)=.25.

 
 Let
X
be
the
number
of
games
played
at
Yankee
stadium
in
the
Bronx
(NY)
and
let
Y
be
the
 number
of
games
played
at
Citizens
Bank
Park,
Philadelphia.
 
 12. What
is
the
value
of
N
that
is
most
likely?
 
 13. What
is
the
expected
value
and
standard
deviation
of
N?

 
 14. Are
X
and
Y
independent?
Explain.
 
 15. Calculate
Var(X+Y).
 
 16. A
student
bought
a
ticket
for
Game
7
from
a
season
ticket
holder
for
$200.
If
the
game
is
 not
played,
he
will
be
able
to
recover
$100
of
his
cost.

If
game
7
is
played,
he
will
be
 able
to
sell
his
ticket
on
StubHub.
To
make
this
a
profitable
venture,
the
student
will
 need
to
be
able
to
sell
his
ticket
on
StubHub
for
at
least
how
much?
 
 Problems
17­20:
 A
car
manufacturer
offers
the
following
options
on
their
2009
models:
 Group
1:

Safety
 Options
 Group
2:
Seat
options
 Group
3:
Other
 Anti‐Lock
brakes
 Heated
seats
 Rear
windshield
wipers
 Traction
control
 Lumbar
control
 CD
player
 Rear
remote
video
 sensor
 Leather

 GPS/Navigation
system
 Side‐impact
air
bags
 Automatic
Height
 adjustable
 Rear
mounted
DVD
 player
 
 
A
survey
is
taken
and
each
respondent
is
asked
to
choose
exactly
three
options
from
the
list
 of
 12
 above.
 
 In
 particular,
 the
 manufacturer
 is
 interested
 in
 knowing
 if
 certain
 grouped
 options
 appear
 together.
 These
 should
 then
 be
 bundled
 together
 in
 a
 package.
 
 The
 marketing
research
department
supposes
to
contrast
the
actual
results
of
the
survey
against
 a
 hypothetical
 group
 of
 customers
 that
 pick
 options
 completely
 randomly.
 We
 will
 use
 probability
to
work
out
the
behavior
of
this
group.


 Assume
 that
 individuals
 randomly
 select
 3
 items
 from
 the
 list
 of
 12
 without
 replacement
 (that
is
each
choice
is
made
equally
likely
from
the
available
options
and
if
they
choose
an
 option
on
one
choice
they
cannot
pick
it
again
on
a
later
choice).
 17. Find
the
probability
a
hypothetical
customer
selects
exactly
one
option
from
each
group.
 
 18. Find
the
probability
that
if
5
hypothetical
customers
each
choose
3
items
independently
 of
each
other,
that
at
least
one
of
the
five
will
choose
an
item
from
Group
3.
 
 19. Suppose
10
hypothetical
customers
each
select
three
options.
Let
Y
be
the
number
of
 customers
that
select
no
items
from
Group
3.

What
is
the
probability
distribution
of
Y?
 Find
E(Y)
and
Var(Y).
 
 
 Problems
20­24:
 A
study
of
261
adults
involved
the
collection
of
a
body
mass
index
(BMI
)
for
each
 individual.
A
BMI
index
25
or
over
is
considered
overweight.
The
histogram
of
BMI
is
given
 below.
The
percentage
in
each
bin
is
marked
on
the
graph.
The
histogram
uses
intervals
that
 contain
the
data
value
equal
to
the
left
endpoint.
 
 
 body
mass
index
 22.6 19.2 14.6 14.2 8.8 6.1 5.7 4.2 1.9 0.8 15 20 25 30 35 1.1 40 0.4 0.0 45 0.4 50 
 
 20. 21. 22. 23. 24. Use
the
histogram
to
approximate
the
median
of
the
BMI
for
the
261
individuals.
 Approximately
how
many
people
in
this
group
are
considered
overweight?
 Is
the
mean
greater
than,
equal
to,
or
less
than
the
median?

 What
is
the
IQR
(approximately)?
 The
mean
weight
is
26
and
the
SD
is
5.

Is
the
empirical
rule
appropriate
for
this
dataset?
 Problems
25­26
 Fuel
economy
of
cars
is
reported
to
consumers
in
Miles
per
Gallon
(MPG).
The
engineers
in
 the
auto
industry
use
a
different
measurement:
Gallons
per
1000
miles
(GP1000).

The
 average
GP1000

of
all
2009
automobiles
is
39.6
(on
the
highway)
with
a
Standard
 Deviation
of
9.9.
The
histogram
of
GP1000
is
Bell
shaped.
 
 
 25. If
possible,
find
median
fuel
economy
of
all
2009
automobiles
in
MPG.

 
 
 26. Fill
in
the
blanks
with
numbers
that
make
a
true
statement:
 
 
 
About
2/3
of
the
cars
have
an
MPG
between




______



and



_________.

 
 Problem
27:
 27. Buggy
software
is
an
every‐day
reality.
In
fact,
70%
of
all
software
is
buggy.
A
buggy
 program
will
crash
within
a
day
with
probability
90%.
However,
nonbuggy
programs
 can
also
crash
because
of
faults
in
the
operating
system
or
the
computer
itself.
In
fact,
 about
20%
of
all
non‐buggy
software
will
crash
within
a
day.

 a) What
is
the
probability
that
the
software
is
non‐buggy
and
it
crashes?
 b) What
is
the
probability
that
the
software
is
buggy
given
that
it
crashed?
 ...
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