251x0341 04/21/03
ECO 251 QBA1
FINAL EXAM
MAY 7, 2003
Name
Class ________________
Part I. Do all the Following (14 Points) Make Diagrams! Show your work!
x ~ N 3, 4 .
1. P2.30  x  1.85 
2. P1  x  17 
3. Px  1.85 
4. F 4.00  (Cumulative Probability)
5. P4.00  x  4.00 
6. x.015 (Find
z .015
first)
7. A symmetrical region around the mean with a probability of 25%
Exam is normed on 75 points. There are actually 128 possible points.
1
251x0341 04/21/03
II. (10 points+-2 point penalty for not trying part a .) Show your work!
The following numbers apply to 9 developed countries and give deaths per 100 million miles and speed
limits.
Row
deaths
x
1
2
3
4
5
6
7
8
9
3.1
3.4
3.5
3.6
4.2
4.4
4.8
5.0
6.2
SpLim
y
55
55
55
70
55
60
55
60
75
These sums have been calculated for you.
y
2
 x  38.2 ,  x
2
 169 .86 ,
 y  540 and
 32850 . Please calculate the following:
a. The sample standard deviations of x and y (4) Note that y  60.00 and s y  7.50 .
b. The sample covariance between x and y . (3)
c. The sample correlation between x and y . (2)
d. Given the size and sign of the correlation, what conclusion might you draw on the relation
between speed and safety if this were the only evidence available? (1)
e. Assume that the death rate in all 9 countries fell by .1. What would be the new values of
x , s x , s xy and rxy . Use only the values you computed in a-c and rules for functions of x and y to get
your results. If you state the results without explaining why, or change x and recompute the results, you
will receive no credit. (4).
2
251x0341 04/21/03
III. Do at least 4 of the following 6 Problems (at least 12 each) (or do sections adding to at least 48 points Anything extra you do helps, and grades wrap around) . Show your work! Please indicate clearly what
sections of the problem you are answering! If you are following a rule like E ax  aEx  please state it! If
you are using a formula, state it! If you answer a 'yes' or 'no' question, explain why! If you are using the
Poisson or Binomial table, state things like n , p or the mean. Avoid crossing out answers that you think
are inappropriate - you might get partial credit. Choose the problems that you do carefully – most of us are
unlikely to be able to do more than half of the entire possible credit in this section!)
1. Assume that the amount of paid time (in days) lost by a blue-collar worker during a 3-month period is
N 1.4, 1.3 . I take a random sample of 10 workers and record the time they lost in the last 3 months..
a. What is the probability that a randomly picked worker lost paid time exceeding 1.5 days in the
3-month period? (2)
b. What is the probability that all 10 workers in the sample lost paid time exceeding 1.5 days in
the 3-month period? (2)
c. What is the probability that at least one of the workers in the sample lost paid time exceeding
1.5 days in the 3-month period? (2)
d. What is the probability that the average amount of paid time lost time exceeded 1.5 days in the
three month period? (2)
e. What is the probability that the total amount of time lost by the sample of 10 workers exceeded
15 days in the three month period. (2)
f. Looking at the distribution of the sample mean in this problem, give a value of the sample
mean that will be above the mean we actually observe 95% of the time (the 95th percentile) (2)
3
251x0341 04/21/03
2. (Bowerman and O’Connell) A retailer that sells home entertainment systems accumulated 10,451 sales
invoices during the last year.
a. An auditor takes a sample of 16 invoices and computes mean sales of x  $532 . If the
population standard deviation was known to be $168, find a 99% confidence interval for the
mean sales per invoice. (4)
b. I lied. Though the sample mean was $532, $168 was a sample standard deviation. Do the
99% confidence interval again. (4)
c. I lied. Though it is true that the sample mean was $532 and the sample standard deviation was
$168, the actual sample was 650 invoices out of the 10451 invoices that were collected. Do the
99% confidence interval again. (4)
d. (Extra credit) Assume that the confidence interval in c is correct, and that the 10451 invoices
were all that were generated, using these two facts, create a confidence interval for total sales in
the last year. (3)
e. (Extra credit) The firm claims that its total sales were above $5.75 million last year. In view of
your results in d, does that seem likely? Would you change your mind if I insisted on a
confidence level of 99.8%? (3)
f. Using the data in a) create a 97% Confidence interval for the mean sales per invoice. (You
might want to look at page 1.) (2)
4
251x0341 04/21/03
3. a. Assume that the entire amount of a product made by a supplier is a population of 100 units and that
you buy the whole batch. Assume that 15% of the batch is defective. Take a sample of 10 items and give
me the probability that at least one is defective. (3)
b. Assume that the batch you buy is much larger, well over 200 and that you still take a sample of 10. What
is the chance that at least one is defective? (You should not need to use the number 200 or any larger
number in your calculations.) (2)
c. Assume that you have bought at least a million units, and that 15% still represents the proportion of the
product that is defective. This time you take a sample of 80. Find the probability that at least 10 are
defective using the Poisson distribution. First, show that it is legitimate to use the Poisson distribution in
this case. (3)
d. Do part c using the Normal distribution. That is assume that you have a large population that is 15%
defective and that you take a sample of 80. Show that the Normal distribution can be used here and find the
probability of at least 10 defective items in the sample. (3 points without continuity correction, 3.5 with)
e. (Extra credit) In section 7.3 of the text (on the CD) the author tells us we should use a finite population
correction with the variance if it is justified. Assume that in part d, the population is 200 and we take a
sample of 80, what is the probability of at least 10 defective items in the sample now? (2)
f. If we are taking a sample from a large population, find the probability that the first defective item is
between the 6th to the 10th item we test. (2)
5
251x0341 04/21/03
4. As everyone knows, a jorcillator has two components, a Phillinx and a Flubberall. It seems that the
jorcillator only works as long as both components work.
a. The distribution of failure times for the Phillinx is Normal with a mean of 8 months and a standard
deviation of 2 months.
(i) What is the probability that the Phillinx dies in the first six months Px  6 ?
(ii) What is the probability that the Phillinx dies in months 6 – 12 ?
(iii) What is the probability that the Phillinx lasts beyond 12 months? (4 total)
b. The distribution of the failure times for the Flubberall is described by the continuous uniform
distribution between 0 and 11.
(i) What is the probability that the Flubberall dies in the first six months?
(ii) What is the probability that the Flubberall dies in months 6-12?
(iii) What is the probability that the Flubberall lasts beyond 12 months?
(iv) What is the mean and standard deviation of the Fluberall’s life? (4.5 total)
c. Now let’s see if you learned anything about combining these probabilities.
(i) What is the probability that the jorcillator will fail in the first six months?
(ii) What is the probability that the jorcillator will fail in the second six months?
(iii) What is the probability that the jorcillator will last beyond 12 months? (5 total)
d. Rework (c) assuming that the jorcillator works as long as one component works. (6)
6
251x0341 04/21/03
5. I am a sales representative and I have two visits scheduled for this morning. Let A represent the event
that I make a sale on visit 1 and B represent the event that I make a sale on visit 2.
A A
a. The joint probability table for the two events is
B
.4
B
and the distribution table
.7
w Pw
0
for w , the total number of sales is 1
2
.
Fill in these tables on the assumption that A and B are independent and find the expected
value and standard deviation for w . (5)
b. Fill in the two tables again on the assumption that A and B are collectively exhaustive.
w Pw
A A
0
B
.4
and compute the expected value and variance for
1
B
2
.7
w . (4)
c. Let x represent the number of sales I make on visit 1 ( x can only be 0 or 1) and y represent
the number of sales I make on visit 2. What relation must exist between the variances of x and
y and the variance of w in the case where A and B are independent that cannot exist when
they are mutually exclusive? Why? (2)
d. Assume that PB   .4 . Use the addition rule to show that A and B cannot be both
collectively exhaustive and independent if A has a probability below 1. (3)
e. Let us define the following events S1  No Oil , S 2  Some Oil and S 3  Much Oil . Assume
PS1   .7 , PS 2   .2 and PS 3   .1 . We run a seismic experiment that has three possible
readings H (high), M (medium) and L (low). All you have to know for this problem is
P H S1  .04 , P H S 2  .02 and P H S 3  .96








(i) Explain the difference between P H S 3 and PH  S 3  and show how we get the
second of these from the first. (3)
(ii) Find PH  (2)


(iii) Find P S1 H (4)
7
251x0341 04/21/03(blank page)
8
251x0341 04/21/03
6. The Phillies are in the 2003 World Series. I estimate that they have a constant .6 chance of winning a
game. There are seven games in a series and the series stops if one team wins four games.
a. If there are seven games played what is the mean and variance of the number of games the
Phillies win? (2)
b. What is the chance that they will win at least 4 of the seven games (You can assume all seven
games are played)? (2)
c. What is the chance that they will win the series by winning the first four games? (1)
d. What is the chance that the first game that they win is the third game? (2)
e. What is the chance that they win the series on the fifth game (but not the 4th)? (Ask yourself
what has to happen in the first 4 games so that they do not win their 4th game until the 5th try.)
(3)
f. What is the chance that they lose the first three games and win the series? (2)
g. Let x represent the number of the first game they win (so that Px  3 is the probability that
the first game that they win is the third game). What is the mean and standard deviation of x ?
(2)
9