EC305 – Economic Statistics
Boston University, Fall 2010
by Arhan Ertan
HW2 – Answers
1.
Suppose you roll a pair of dice. Let
A
be the event that you observe an even number. Let
B
be
the event that you observe a number greater than seven.
a.
What is the intersection of events
A
and
B
?
Answer:
[8, 10, 12]
b.
What is the union of events A and B?
Answer:
[2, 4, 6, 8, 9, 10, 11, 12]
c.
What is the complement of event
A
?
Answer:
[3, 5, 7, 9, 11]
d.
What is
AB
I
?
Answer:
[2, 4, 6]
e.
What is
U
?
Answer:
[2, 3, 4, 5, 6, 7, 8, 10, 12]
2.
The probability that interest rates on housing loans will go up in the next 6 months is
estimated to be 0.20. The probability that house sales will decrease is estimated to be 0.6.
The probability that interest rates will go up and house sales will decrease is estimated to be
0.15.
a.
The probability that interest rates will go up given that house sales go down is:
Answer:
0.15 / 0.6 = 0.25
b.
The probability of an increase in interest rates and not a decrease in house sales is:
Answer:
0.20 – 0.15 = 0.05
c.
The probability of a decrease in house sales and not an increase in interest rates is:
Answer:
0.6 – 0.15 = 0.45
d.
Are the events of interest rates increasing and house sales decreasing statistically
independent or not? Why?
Answer:
The events of interest rates increasing and house sales decreasing are NOT
statistically independent because the joint probability of interest rates increasing and house
sales decreasing do not equal the product of the marginal probabilities.
3.
A multiple choice quiz has 5 questions, each with 5 answers,
A
through
E
. Assume you just
guess on all of the questions.
a.
What is the probability that you guessed on all five questions right?
Answer:
()
(
)
00032
.
0
5
/
4
5
/
1
5
5
0
5
=
C
b.
What is the probability that you get exactly three questions right?
Answer:
(
)
0512
.
0
5
/
4
5
/
1
5
3
2
3
=
C
1
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A recent marketing survey related consumers’ awareness of a new marketing campaign with
their rating of the product. Consumers rated their awareness as low, medium, or high, and
rated the product as poor, fair, or good. The results are presented below:
Awareness
Low
Medium
High
Poor
0.10
0.15
0.07
Rating
Fair
0.06
0.11
0.06
Good
0.07
0.11
0.27
a.
What is the probability that a consumer had low awareness?
Answer:
0.10 + 0.06 + 0.07 = 0.23
b.
What is the probability that a consumer who ranked the product as fair had a high
awareness of the ad campaign?
Answer:
0.06 / (0.06 + 0.11 + 0.06) = 0.26
c.
What is the probability that a consumer who had high awareness of the ad campaign
ranked the product as good?
Answer:
0.27 / (0.07 + 0.11 + 0.27) = 0.675
5.
The purchasing agent for a municipality has contracted with a local car dealer to purchase
four cars. The dealer has 25 cars on his lot; 10 red, 7 blue, 6 white and 2 purple.
If the
purchasing agent has no control over the colors he receives, what is the probability that he
receives at least one of the purple cars?
Answer:
P(# of purple cars >= 1) = 1 – P(no purple cars)
= 1 – (23/25)*(22/24)*(21/23)*(20/22) = 1 – 0.70 = 0.30
6.
The events
A
and
B
both have nonzero probabilities. Can
A
and
B
be both independent and
mutually exclusive?
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 Spring '10
 Arahan
 Probability, Interest Rates, Probability distribution, Binomial distribution

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