Intro
to Probability and Statistics
Sample
Midterm #1 Questions Only
Professor Brian Shydlo
Instructions:
1) Please
write your name: _____________________________________
2) There
are 7 questions totaling 100 points. Please be careful to answer all questions.
Partial credit will be given (so show your work).
Question 1)
15 Points
Question 2)
22 Points
Question 3)
10 Points
Question 4)
15 Points
Question 5)
15 Points
Question 6)
5 Points
Question 7)
12 Points
Question 8)
6 Points
Total
100 Points
Question
1) (15 points in total) Basic Statistics
You have
the following set of numbers.
10, 15, 16,
8, 17, 18, 20, 31, 4, 5, 10
Question
1a) (3 points) What
is the Mean?
Answer: __________________
Question
1b) (3 points) What
is the Median?
Answer: __________________
Question
1c) (3 points) What
is the Mode?
Answer: __________________
|
A |
B |
|
|
|
Question
1d) (3 points)
Observe the Boxplots above (they are each based on a different set of numbers,
not shown). The interquartile range of
BoxPlot A is 9 to 35. What is the
significance of the interquartile Range?
Answer:
_____________ญญญญญญญ__________________________________________
Question
1e) (3 points)
Observe the Boxplots above. Which plot
appears to have a lower variance.
Answer: __________________
Question
2) (22 points in total) Basic Probability
The
following is a chart of a probability distribution. It shows a Random Variable called x and the
probability function for x (known as f(x)).
For example, there is a 5% chance that x would equal 0.
|
x |
f(x) |
|
0 |
5% |
|
2 |
10% |
|
3 |
20% |
|
5 |
Purposely left blank |
|
9 |
35% |
Question
2a) (3 points) What
is the probability that x = 5?
Answer: __________________
Question
2b) (3 points) Fill
in the chart to produce the Cumulative Probability Function (known as
F(x)).
|
x |
F(x) |
|
0 |
|
|
2 |
|
|
3 |
|
|
5 |
|
|
9 |
|
Question
2c) (5 points) An article in this week's Business Week says that
if there is a War in Iraq, there are exactly three possible ways a war in Iraq
can affect Oil prices. They are listed
below along with the estimated probability of each.
|
Scenario # |
Scenario Description |
Probability Estimates (from Brian,
not from BW) |
|
1 |
Quick
conflict, oil fields undamaged, leads to a "Moderate Decline" in
Oil prices as they go to $20/ barrel range |
70% |
|
2 |
Stiff
resistance, oil fields somewhat damaged, leads to a "stubborn high"
prices as they stay in the $30s for a year. |
25% |
|
3 |
Iraq
destroys various oil facilities in and about Iraq, leading to "Dangerous
Spike" of $80 to $100 |
5% |
Furthermore,
I read that if there is no war, it is expected (100% chance) that Oil prices will
exhibit a Moderate Decline to the $20/barrel range (an identical scenario to
Scenario 1 above)
Brian
estimates the following probabilities for whether or not there will be a war:
|
Scenario # |
Scenario Description |
Probability
Estimates (from Brian, not from BW) |
|
1 |
War |
89.5% |
|
2 |
No War |
10.5% |
Define Two
Events, A and B such that:
A = Prices
go Down the $20/barrel range. (Meaning
that the Moderate Decline described in scenario A Happens)
B = There
is a War.
Question: What are the odds that there is a War and
that Prices go down to the $20/barrel range.
(In other words, both A and B happen)
Hint:
Think Intersection
Answer: __________________
Question
2d) (5 points) Based on the same data and assumptions as in the
previous question, what is the probability of A occurring (remember A = Prices
go Down the $20/barrel range and can occur if there is or is not a war).
Answer: __________________
Question
2e) (3 points) Are
A and B statistically Independent based on the data in the above problems, and
if so, why (in other words, show a proof, don't just put a yes or a no for full
credit.)?
Recall
that:
A = Prices
go Down the $20/barrel range.
B = There
is a War.
Answer: __________________
Question
2f) (3 points)
Assume A
and B are Mutually Exclusive. (These are
new variables, unrelated to any in previous problems)
Give one
example where A' and B' are mutually exclusive and one example where A' and B'
are not mutually exclusive.
Answer:
_________________________________________________________
Question
3) (10 points) You
have the following distribution for stock market price returns:
|
Market Return |
Probability |
|
20% |
60% |
|
2% |
30% |
|
-5% |
10% |
Question
3a) (5 points) What
is the Expected Value of your return?
Answer: __________________
Question
3b) (5 points) What
is the Standard Deviation of Your Return?
Answer: __________________
Question
4) (15 points) My friend,
David, is an excellent student. The odds
of him getting an A in any class he takes are 95%. There is a 5% chance of him getting a B. There are no other grades he can get. He is in the MBA program at Stern. There are 20 classes that he'll take to
graduate.
Question
4a) (5 points) What are the odds that David will graduate with a 4.00
average? Assume that the grade he gets
in each class is INDEPENDENT.
Answer: __________________
Question 4b)
(5 points) What is the expected number of As that David will get?
Answer: __________________
Question
4c) (5 points) My friend Rich is also very smart and is also in the MBA
program. He has a 90% chance of getting
an A in any given class. What are the
odds that Rich will get a better grade than David in any single class? Remember, there are only two grades, A and
Not A (otherwise known as getting a B) and everything is INDEPENDENT.
Hint:
Use a Probability Box to get the answer (you don't have to use a box, if you
don't need to).
Answer: __________________
Question
5) (15 points)
You go to
the supermarket to get ingredients for an omelet for breakfast for you and your
friends. You need eggs for your omelet so
you decide to buy a dozen (one box).
This particular supermarket has a reputation for bad eggs. There is a pretty high rate of getting bad
eggs. The probability that a given egg
is bad is 10%. Each egg's badness is
independent of each other egg's badness.
Question
5a) (5 points)
What are
the odds of getting exactly 2 bad eggs in your dozen?
Answer: __________________
Question
5b) (5 points)
You realize
you need exactly 11 eggs for your omelet so it doesn't matter to you if one of
the eggs is bad. If two or more eggs are
bad, then you can't make your omelet.
What are
the odds that you can NOT make an omelet?
(meaning what are the odds that 2 or more eggs are bad.).
Answer: __________________
Question
5c) (5 points)
What is the
Standard Deviation of the distribution for Number of Bad Eggs in your dozen
(recall that the probability of a Bad egg is 10%).
Answer: __________________
Question
6) (5 points)
You have a
certain kind of computer that you use for your factory. Due to various electrical and mechanical
reasons, your computer gets the wrong answer 5% of the time.
A 5% error
rate is too high for your needs, so you decide to set up a system of 5
computers. (Each computer has the same error rate of 5% and acts independently
of other computers.) With your new
system, the decision is based on the majority of the five computers. For example, if 2 computers say
"no" and the other "3" say yes, then you take the answer to
be yes.
What are
the odds that your new system of 5 computers hooked together (with each one
deciding independently from the other) gives you the correct answer?
To help you
out, I am including this excerpt from a Binomial Distribution table.
|
|
50% |
55% |
60% |
65% |
70% |
75% |
80% |
85% |
90% |
95% |
|
0 |
3.13% |
1.85% |
1.02% |
0.53% |
0.24% |
0.10% |
0.03% |
0.01% |
0.00% |
0.00% |
|
1 |
15.63% |
11.28% |
7.68% |
4.88% |
2.84% |
1.46% |
0.64% |
0.22% |
0.05% |
0.00% |
|
2 |
31.25% |
27.57% |
23.04% |
18.11% |
13.23% |
8.79% |
5.12% |
2.44% |
0.81% |
0.11% |
|
3 |
31.25% |
33.69% |
34.56% |
33.64% |
30.87% |
26.37% |
20.48% |
13.82% |
7.29% |
2.14% |
|
4 |
15.63% |
20.59% |
25.92% |
31.24% |
36.02% |
39.55% |
40.96% |
39.15% |
32.81% |
20.36% |
|
5 |
3.13% |
5.03% |
7.78% |
11.60% |
16.81% |
23.73% |
32.77% |
44.37% |
59.05% |
77.38% |
Answer: __________________
Question
7) (12 points)
Assume X
follows a normal distribution.
The Mean of
X is -40 and the Standard Deviation is 100.
Question
7a) (3 points)
What is the
probability that X > 60?
Answer: __________________
Question
7b) (3 points)
What is the
probability that X <= 180?
Answer: __________________
Question
7c) (3 points)
What is the
probability that X <= 0?
Answer: __________________
Question
7d) (3 points)
What is the
probability that X > 0 and less than 180?
or
P(0 < x
< 180)?
Answer: __________________
Question
8) (6 points)
Question
8a) (6 points)
You decide to
start selling a new kind of exotic drink, a reverse osmosis, non-ionized,
cold-filtered, smartified, vitalified H-2-wow.
You decide to sell your water drink in a ฝ liter bottle. You realize that you'll get fined by the
government if you put any less that 0.5 liters of water in your bottle (false
advertising).
Assume that
your bottling machine (that pours the liquid into the bottles) follows a normal
distribution and has a standard deviation of 0.01 liters.
How much liquid
should you pour into each bottle such that there is only a 0.99% chance (0.99%
= 0.0099) that a particular bottle is less than 0.5 liters?
Answer: _________________