251x0831 4/16/08
ECO251 QBA1
THIRD EXAM
Apr 18, 2007
Name: _____________________
Student Number: _____________________
Class time: _____________________
Part I. (16 points) Do all the following (2 points each unless noted otherwise). Make Diagrams! Show
your work! In particular you must briefly explain how you got the answer to the value of z at the bottom of
this page.
z has the standardized Normal distribution z ~ N 0, 1 for the first four problems.
1. Pz  1.23 
.
2. P3.25  z  0
3. P3.07  z  3.07 
4. z .135
251x0831 4/16/08
x ~ N 4, 7 for problems 5 through 8. Note that all values of z are rounded to the nearest hundredth.
5. Px  1.23 
6. P3.25  x  0
7. P3.07  x  3.07 
8. x.135
251x0831 4/16/08
Part II: (9+ points) Do all the following: All questions are 2 points each except as marked. Exam is normed
on 50 points including take-home. (Showing your work can give partial credit on some problems! In
open-ended questions it is expected. Please indicate clearly what sections of the problem you are
answering and what formulas you are using. Neatness counts!) Remember that you may not be able to
finish this section, so ration your time on each problem. [Numbers in brackets are a cumulative
total].Justify the substitution of one distribution for another.
1. A small life insurance company receives an average of five death claims a day. Assume that the Poisson
distribution is correct. What is the probability that the company will receive more than 10 claims in a given
day (rounded to thousandths)? (2)
a) .986
b) .005
c) .014
d) .032
e) None of the above (Fill in an answer!)
2. A local family planning group serves 20000 teen-age girls. It costs $50 to council each pregnant girl.
There is a 5% chance that each of the 20000 girls will become pregnant during the year. Each pregnancy
can be assumed an independent event. Based on what you know about the expected values of discrete
distributions, how much should the agency budget for counseling this year? (2)
a) $1000
b) $20000
c) $50000
d) $100000
e) None of the above.
3. In the agency in problem two, 20 girls are waiting to see a counselor this morning. Half of them are
pregnant. If Samantha is assigned to counsel 8 of them, what is the chance that all eight are pregnant? (I
want to see your formulas and calculations – this only took me a few minutes.) (3)
4. How many girls would have to be waiting before we could use the Binomial distribution to solve
problem 3? (1)
[8]
5. OK. Assume that there are 4000 girls waiting to see a counselor and Samantha is assigned to counsel 8
of them, what is the chance that all eight are pregnant? The only answer that I will accept here is an answer
gotten from numbers in the tables. If you just write down a solution, you will get half credit. (2)
251x0831 4/16/08
6. A variable has the Binomial distribution. Find the following and explain how you did it.
a) Px  5 p  .02 n  200 (2)
b) Px  60  p  .40 n  200 (2 or 2.5)
[14]
7. Find P11  x  17  for the following distribution: Continuous Uniform with c  12 , d  15 (1)
8. If the amount of time it takes students to find a parking place has a Normal distribution with a mean of
3.5 minutes and a standard deviation of 1 minute, 99% of students’ search time will fall below what
number? (For example it may be true that 99% take 3.6 or fewer minutes to find a parking place, but I
doubt it.)
(2)
9. (Dummeldinger) You decide to invest in four independently moving risky stocks. You guess that each
stock has an independent 40% chance of becoming a total loss. What is the chance that at least one of your
stocks will tank? (2)
[19]
a) 0
a) .0256
b) .4000
c) .8704
d) .9744
e) 1.0000
251x0831 4/16/08
10. (Extra Credit) The amount of rainfall in a 24 – hour period has an exponential distribution with a mean
of 0.2 inches. What is the chance that a randomly picked 24-hour period will have rainfall that exceeds 0.8
inches?
a) .018316
b) .778801
c) .221199
d) .981684
e) None of the above. Produce an alternate answer.
251x0831 4/16/08
ECO251 QBA1
THIRD EXAM
Apr 24, 2008
TAKE HOME SECTION
Name: _________________________
Student Number: _________________________
Throughout this exam show your work! Please indicate clearly what sections of the problem you are
answering and what formulas you are using. Write on one side of the page!
Part III. Do all the Following (25+ Points) Show your work! Neatness counts! Answers of ‘zero’ or
‘one’ especially are unacceptable without an explanation. Do not use one distribution to approximate
another without justifying the replacement!
1. Identify the distribution that you are using in each problem. If you have a number like n  20  g , make
it very clear what value of n you are using. Look at the solved problems for Section L, the solution to
Grass3 and ‘Great Distributions’ (especially the hints on the 3 rd page) before you start.
Let g be the last digit of your student number. If g  0 , change it to 2. Hint: It may help in these problems
to realize that if n  100 , 93 or more is 93 to 100 and 10 or less is 10 to zero. This is especially useful if we
count failures.
a. A basketball player makes 50  5g % of her free throws over the basketball season. In one game she
gets 20 free throws and misses 13  g  of them. The coach will investigate if the probability of doing as
badly as she did or worse is below 5%. Will the coach investigate? Do the math to justify your answer. (2)
b. We believe that 80% of all accidents involve alcohol. If we investigate 6  g  accidents, what is the
probability that 2  g  or fewer involve alcohol? (1)
c. A test consists of 6  g questions and a student must get at least 70% of the questions right to pass the
course. Each question is a multiple choice question with 2 possible answers. Note that if n  16 , 70% of 16
is 11.2, so 12 or more must be right. What is the probability of passing the exam if you just guess? What
happens to the probability as the size of the exam increases to 20 questions? (2)
Assume that the professor instead offers an exam with 6  g questions with four choices and then says that
a passing grade will be 50% or more. Does this raise or lower the chance of passing for the guesser? What
happens to this probability as the size of the exam rises to 20? (2)
d. (Dummeldinger) The diameter of ball bearings has a continuous uniform distribution over the interval
3 .1g millimeters to 5  .1g millimeters.
(i) If ball bearings are only acceptable if they are between 4 and 5 millimeters, what proportion of
those produced is defective?
(ii) If I take a sample of three ball bearings from the assembly line.
What is the chance that none of the ball bearings are defective?
(iii) What is the chance that exactly one is defective?
(iv) What is the chance that at least one is defective?
(v) What is the chance that the third ball bearing that I inspect is the first one that is defective?
[12]
e. Supposedly 60% of drivers wear their seatbelts. 1000 drivers are stopped.
(i) What is the approximate probability that less than half were wearing seatbelts? (1)
(ii) What is the approximate probability that less than 200  g  were wearing seatbelts? (1)[14]
f. If each member of the State Highway patrol writes an average of 10 tickets a day,
251x0831 4/16/08
(i) What is the chance that a given policeman will write more than 10  g  tickets in a day? (1)
(ii) Assuming that a policeman averages 50 tickets during a work week, what is the chance that a
given policeman will write more than 510  g  tickets in a week? (I strongly doubt that the answer
is the same as the answer to (i)) (2)
g. A ‘psychic’ is shown 5 cards and then is expected to be able to guess which one of the cards is randomly
drawn. If the ‘psychic’ is a fake and is merely randomly guessing what card is picked and the experiment is
repeated 20 times, what is the chance that the psychic will pick the right card 4  g  or more times? (1)
h. The average number of small business bankruptcies in Fredonia are 13  .1g  a month. What is the
probability of at least one bankruptcy in a month? What is the probability of at least 3 in a month? What is
the probability of more than 180 in a year? (3)
[21]
i. There are exactly 14 firms in the US that produce jorcillators and 1  g  produce computers as well. If
you pick a random sample of 3 firms to examine, what is the chance that at least one will be a computer
producer . What is the chance that all the firms in your sample are computer producers? Assume that the
same proportion of 70 companies produce computers what would the probability that at least one and all
three in a 3-firm sample be computer producers? (4)
[25}
j. (Extra credit) The time between landings at an airport follows an exponential distribution with a mean of
30 seconds. What is the probability that there will be a gap of between 15  g  and 45  g  seconds before
the next plane lands?
2. As everyone knows, a jorcillator has two components, a phillinx and a flubberall. It seems that the
jorcillator only works as long as either component works (so that it fails in the first period only if both
components fail). Note
The probability of the phillinx failing is given by a Normal distribution with  18  g and   3. , For
example, if the life of the phillinx is represented by x1 , the chance of the phillinx failing in the first ten
years is P0  x1  10  and the probability of it failing in the second ten years is P10  x  20  For
example: Ima Badrisk has the number 375292, so her distribution has a mean of   18  .g  20 . The
flubberall has exactly the same distribution.
In order to maintain my sanity, use the following events. Period 1 is the first ten years, period 2 is the
second ten years and period three is happily ever after.
Failure of the phillinx in period 1, 2, 3 are events A1, A2 and A3
Failure of the flubberall in period 1, 2, 3 are events B1, B 2 and B 3
Failure of the jorcillator in period 1, 2, 3 are events C1, C 2 and C 3 .
a) What is the probability that the phillinx will fail in period 1? Period 2? Period 3? (2)
b) What is the probability that the jorcillator will fail in the first period? (1)
c) What is the probability that the jorcillator will fail in the second period 2? (1)
d) What is the probability that the jorcillator will fail in the third period? (1)
If you haven’t figured it out already, one of the easiest ways to do this is to make a joint probability
table. Put the A events across the top. Put the B events down the side. Figure out what the
probability of the joint events must be if they are independent. Now make a similar table. This time,
instead of probabilities, fill in the period in which the jorcillator fails.
e) (Extra Credit) Find the probability that the jorcillator and the Phillinx both fail in the third
period (1)
f) (Extra Credit) Find the probability that the jorcillator fails in the third period, given that the
phillinx fails in the third month i.e. P C3 A3 (1)




g) Demonstrate Bayes’ rule by finding P A3 C3 and showing that Bayes’ rule explains the
relationship between the conditional probabilities that you have found. (2) [34]
251x0831 4/16/08
3. Assume that you have a variable with a Poisson distribution with a mean of 1.5  .1g  . (My value of
g is 2, so I have a mean of 1.5  .12  1.5  .2  1.3 . Write down the values of P0 , P 1 etc for all
values of x that have probabilities of .0001or larger by subtracting values from the Poisson table.
Compute the expected value and standard deviation using the probabilities and values of x and show
that these are the same as stated in our write-ups on the Poisson distribution. (4) [37]