251x0442 04/26/04
ECO 251 QBA1
FINAL EXAM
MAY 5, 2004
Name
Class ________________
Part I. Do all the Following (14 Points) Make Diagrams! Show your work! Exam is normed on 75 points.
There are actually 127 possible points.
x ~ N 11, 4
1. P0  x  15 
2. P1.00  x  9.68 
3. Px  17 .20 
4. F 23 .43  (Cumulative Probability)
5. P0.20  x  22.20 
6. x.022 (Find
z .022
first)
7. A symmetrical region around the mean with a probability of 35%.
1
251x0442 04/26/04
II. (10 points+-2 point penalty for not trying part a .) Show your work!
x2
x12
x 22
2.3
1.0
-4.2
11.0
2.5
4.4
11.8
2.3
1.2
4.4
36.7
552.25
0.01
166.41
262.44
529.00
501.76
4147.36
104.04
57.76
640.09
6961.12
5.29
1.00
17.64
121.00
6.25
19.36
139.24
5.29
1.44
19.36
335.87
x1
1
2
3
4
5
6
7
8
9
10
23.5
0.1
12.9
16.2
23.0
22.4
64.4
10.2
7.6
25.3
205.6
The data above represents returns of a sample of 10 low-risk mutual funds in 1999( x1 ) and the first quarter
of 2000( x 2 ). Calculate the following.
a. The sample standard deviation s x1 of x1 (2)
and x 2 (1)
b. The sample covariance s x1x2 between x1 and x 2 . (3)
c. The sample correlation rx1 x2 between x1 and x 2 . (2)
d. Given the size and sign of the correlation, what conclusion might you draw on the relation
between x1 and x 2 ? (1)
e. Assume that the return on the funds were .1 lower in the first quarter of 2000. Find
x 2 , s x2 , s x1x2 and rx1 x2 . 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 x1 and x 2 and recompute the
results, you will receive no credit. (4).
f. Do a 95% confidence interval for the mean return in 1999. (4)
g. Was there a significant difference between the return on these funds in 1999 and 2000? A
relatively simplistic way to answer this is to check if the mean return in 2000 was in the confidence interval
for 1999. (2)[33]
2
251x0442 04/26/04
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!) This is not an opinion
questionnaire. Answers without reasons or supporting calculations or table references will not be
accepted!!!!
1. Suppose that you pick up 100 packages sent by a mailer. You believe that this mailer produces packages
with a population mean of 6 oz and a population standard deviation of 2.6 oz. Let x represent the sample
mean weight that you calculate on this particular day.
a. According to the central limit theorem, what distribution, with what mean and standard
deviation should apply to the sample mean weight of the packages?(2)
b. If you assume that the package weights are normally distributed, what is the probability that a
randomly picked package will weigh more than 6.8 oz? (2)
c. What is the probability that the sample mean of the weight of the packages is more than 6.8
oz? (3)
d. What is the probability that the combined weight of all 100 packages is above 40 lbs? (640 oz.)
(3)
e. If you assume that the package weights are Normally distributed, what is the chance that three
randomly picked packages are all above the median weight? (2) [45]
f. How heavy does a package have to be to be heavier than 99% of the packages sent out by this
mailer? (2) [47]
3
251x0442 04/26/04
2. I send out a survey to 200 people. The probability that each person returns the survey is 10%.
This is a Binomial problem.
a. What is the chance of between 20 and 30 returns. Answer the question by showing that you
can use the Normal distribution to solve this problem and solving it.(3)
b. Answer the same question by showing that you can use the Poisson distribution to solve it and
solving it. (3)
c. Let p represent the fraction of the surveys that are returned. Assume that we send out 113
surveys, what is the distribution (including the mean and standard deviation) of p . (2)
d. What is the probability that p is above 11% if (i) n  200 , (ii) n  900 . (4)
e. What is the probability that the first survey returned is between the 10th and the 20th sent out?
(2) [61]
4
251x0442 04/26/04
3. A manufacturer is producing bolts with a nominal length of 5cm. A random sample of 10 bolts is taken
from a box containing a large number of bolts. From the sample we get a sample mean of 5.512 and a
standard deviation of .22754.
a. On the basis of long experience, we know that the standard deviation for the bolts is .2236.
Find a 90% confidence interval assuming that this population standard deviation is correct. (2)
b. Find a 90% confidence interval for the mean, assuming that the sample standard deviation is
correct. (3)
c. Find a 90% confidence interval for the mean assuming that the sample standard deviation is
correct and that the sample of 10 bolts was taken from a batch of only 50 bolts. (2)
d. On the basis of the tests in a-c, is a population mean of 5 cm. reasonable? (1)
e. Assume that the median length of the bolts is 5cm., and that all 10 bolts in the sample are
more than 5 cm. A p-value is the probability of obtaining a result as extreme or more extreme
than what actually happened. If the p-value is below 1% we strongly doubt that the median
length is 5cm. An event as extreme or more extreme than what actually occurred is either
getting all bolts longer than 5cm or all bolts shorter than 5cm.What is the probability of this
happening? (3)
f. Assume that a basketball team scores at an average rate of 2.0 points per minute. What is the
probability of scoring more than 22 points in a 10 minute interval? What is the probability of
scoring less than 10 points in a 10 minute interval? (2)
g. By eliminating scores in the bottom 5% or less of the distribution and in the top 5% of the
distribution, find an interval between scores with a probability of about 90%. This will look
something like P2  x  26   .90, where Px  2  .05 and Px  26   .055 and these two
numbers are picked so that the first number is the largest with probability below 5% and the
second number is the smallest with probability over 5%. (2)
h. Try a cleaner version of g. Assume that the team plays for a half hour. Use the Normal
distribution to find a 90% symmetrical interval around the mean number of points scored. (3)
i. Re-do a) with a 95.6% confidence interval after looking at page 1(2). [81]
5
251x0442 04/26/04
4a) Moe, Shemp and Curley work in a fast food restaurant. Moe fills 35% of the orders, Shemp fills 45% of
the orders and Curley fills 20% of the orders. Moe gets the order wrong 20% of the time, Shemp gets it
wrong 12% of the time and Curley screws up 5% of the time. If the order was filled correctly, what is the
probability that it was filled by each of the 3 workers? If the order was filled incorrectly, what are the
probabilities? (6)
The following table gives probabilities for x and y.
y
x
1
2
1
.01
.04
2
.03
.12
3
.06
.24
3
.05
.15
.30
b) Find E x , E  y   x ,  y ,  xy , and  xy .(6)
c) x  y can take the values 2, 3, 4, 5 and 6. Find the probabilities for each of these values and show that
they belong to a valid distribution. (5)
d) Find Ex  y  and Var x  y  two different ways – getting the same values each time. (5)
e) If Event A is x  3 and Event B is y  3 find P A and PB  . (0) Find P A  B  (1)
f) Find P A  B  (1)
 
g) Find P B A (1) [105]
6
251x0442 04/26/04
5. Assume that in any batch of soft drink bottles 35% are defective.
a) If I take a sample of 5 bottles from a case of 20, what is the chance that exactly 3 will be
defective? (2)
b) If I take a sample of 5 bottles from a case of 20, what is the chance that at least one will be
defective? (2)
c) If I take a sample of 5 bottles from a shelf with many, many bottles on it, what is the chance
that exactly 3 will be defective? (2)
d) If I take a sample of 5 bottles from a shelf with many, many bottles on it, what is the chance
that at least one will be defective? (2)
e) If I take a sample of 100 bottles from a shelf with many, many bottles on it, what is the chance
that at least 25 will be defective? (2)
f) (extra credit)If I take a sample of 100 bottles from a shelf with 500 bottles on it, what is the
chance that at least 25 will be defective? (2)
g) In an area of 4 square feet, we normally expect 2 paint blisters.
(i) What is the chance of no blisters in one square foot? (1)
(ii) What is the chance of at least two blisters in one square foot? (2)
(iii) 95% of surfaces of one square foot will have less than ? blisters. (1)
(iv) What is the chance of no blisters in 60 square feet? (2)
(v) What is the chance of at least 150 blisters in 60 square feet? (2)
(vi) 95% of 60 square foot surfaces will have less than ? blisters. (2) [127]
7
251x0442 04/26/04 (Blank)
8