Problem 1.A random sample of size
n
= 100 was drawn from a population. A histogram of the data is shown
in the figure.
a. (5 pts) Multiple choice. The histogram
indicates the distribution is:
A. normal
B. chisquared
C. lighttailed
D. heavytailed
b. (5 pts)
Multiple choice. Let
X
be a random
variable for an observation from the population
discussed above. Suppose the population mean
μ
and population variance
σ
2
are known. After
assuming that the population is normal a
probablist calculates

≥
=
≥
σ
μ
5
)
5
(
Z
P
X
P
.
Assuming she makes no calculation errors, she:
A. overestimates the true probability
B. underestimates the true probability
C. gets the true probability exactly right
D. still cannot calculate the true probability because of random variation.
Problem 2. Suppose that among engineering students, 13% of men and 9% of women are lefthanded.
a. (5 pts) Assuming 22% of the engineering students are women, what is the probability that a randomly
selected engineering student is lefthanded?
Suppose an engineering student is selected at random. Let
L
be the event that the student is lefthanded and
W
be the
event that the student is a woman. Then:
%
12
.
12
78
.
0
13
.
0
22
.
0
09
.
0
)
(
)

(
)
(
)

(
)
(
=
⋅
+
⋅
=
′
′
+
=
W
P
W
L
P
W
P
W
L
P
L
P
b. (5 pts) Suppose in the course of her Cornell undergraduate career, a woman has 5 boyfriends who are all
engineering students. If the boyfriends are randomly selected, what is the probability that there would be 3 or
more lefthanders in her sample of 5 boyfriends?
Let
X
be a random variable for the number of lefties in a random selection of 5 engineering boyfriends. Then,
assuming
X
~ Binomial(5, 0.13), we can calculate that probability as:
0179
.
0
87
.
0
13
.
0
5
5
87
.
0
13
.
0
4
5
87
.
0
13
.
0
3
5
)
3
(
0
5
1
4
2
3
=
+
+
=
≥
X
P
Problem 3. (5 pts) The probability density function for a random variable
Y
is
f
(
y
) = 0.1*(
y
+ 3) for 1 <
y
< 3
and
f
(
y
) = 0 otherwise. Find
F
(3).
F
(3) = 1 [Obvious.]
Or:
1
)
3
2
/
1
(
9
2
3
1
.
0
3
2
1
.
0
)
3
(
1
.
0
)
(
)
3
(
2
3
1
2
3
1
3
=
+

+
=
+
=
+
=
=
∫
∫
∞

y
y
dy
y
dy
y
f
F
1
10
5
0
5
10
0.0
0.10
0.20
x
Relative Frequency
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Problem 4. (40 points through problem 4) A newspaper reports that on the night of June 7, the glorious
Perseid meteor shower can be viewed. It is claimed that an observer should be able to view an average of 20
meteors per hour. Assume that meteor sightings are independent amd the rate is constant throughout the
night.
a. (5 pts) Let
X
be a random variable for the amount of time an observer would have to wait until first
observing a meteor. Suppose the newspapers' claim that the rate is 20 sighting /hr is correct. Then,
find the
95th percentile of
X
.
Assume
X
~ Exponential(20). Then, the 95
th
percentile would be the number
η
,
such that
95
.
0
20
0
20
=
∫

η
dx
e
x
.
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 Fall '05
 STAFF
 Standard Deviation, Variance, pts

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