05/11/03 251probl (Open this document in 'Page Layout' view!)
ECONOMICS 251 PROBLEMS
PROBLEM F1 (Weighted Average)
In 1994 The four Scandinavian countries had per capita incomes and populations as follows:
Country
Income Population(Millions)
Finland
$18850
5.1
Sweden
$23530
8.8
Norway
$26390
4.3
Denmark
$27070
5.2
a. Compute the per capita income of Scandinavians, using population as weights.
b. What percent of the population was in each of the four countries? Using these percents as three-place
decimals, compute the average income of Scandinavians.
PROBLEM F2
Take the numbers 100, 200 and 300, and compute four different kinds of means.
PROBLEM F3
Using the numbers below, compute x .50 and x .45 .
1, 5, 7, 9, 9, 11, 13, 14, 17, 19
PROBLEM F4
Compute x .50 and x .01 from the data below.
Group
0- 9.9
10-19.9
20-29.9
30-39.9
40-49.9
50-59.9
60-69.9
70-79.9
80-89.9
90-99.9
f
10
10
10
10
10
10
10
10
10
10
PROBLEM G1. If the mean return for an industry is 10% with a standard deviation of 6%, out of 100
firms how many do you expect to have returns above 22%?
PROBLEM G2. If the mean is 5 and the standard deviation is 2, find an interval that must contain at least
the central two-thirds of the observations.
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251probl
PROBLEM G3. Consider the following sample:
Class
x
0- 4.9
5- 9.9
10-14.9
15-19.9
20-24.9
25-29.9
30-34.9
35-39.9
40-44.9
45-49.9
50-54.9
55-59.9
f
F
1
0
3
7
15
16
12
11
9
9
6
1
90
xf
x2 f
x3 f
Use computational formulas:
a. Complete the cumulative frequency under F .
b. Calculate the mean.
c. Calculate the median.
d. Calculate the mode.
e. Calculate the variance.
f. Calculate the interquartile range.
g. Calculate the standard deviation.
h. Calculate a statistic showing skewness.
i. Show all the data presented on a histogram with six class intervals.
j. Put a box plot below the histogram.
Now repeat Problem G3 using definitional formulas.
PROBLEM G3A. Do Problem G3 substituting the following data for the data in G3. The histogram will
have only five class intervals.
Class
x
0 - 9.999
10 - 19.999
20 - 29.999
30 - 39.999
40 - 49.999
f
F
50
50
100
150
50
PROBLEM G4. For the sample below, compute the following:
b. the mean.
c. the median (hint: put in order first!).
d. the mode.
e. the variance.
f. the interquartile range.
g. the standard deviation.
h. a statistic showing skewness.
1,2,4,5,6,3,3,7,8,3,1,2
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252probl
xf
x2 f
x3 f
PROBLEM H1. A pair of dice is tossed once.
1 2 3 4 5 6
1      
2      
Diagram for dice problems. 3      
4      
5      
6      
Define Event A as rolling an even number (i.e. the sum of both faces is even)
Define Event B as rolling a 5.
Define Event C as a 1 on at least one of the dice.
Find the following probabilities – say what points are in the event and add their probabilities:
a. P A , PB  , PC  , P A , P B , P C
     

 


b. P A  B  , P A  C  , PB  C  , P A  B , P C  B , P A  B  , P A  C  , PB  C  , P A  B
c. Show that the addition rule works for each of the four unions in b).
d. Which of the following are mutually exclusive? A and B ? A and C ? B and C ? A and B ?
e. Which of the following are collectively exhaustive? A , B , and C ? B , B , and C ?

PROBLEM H2. Using the same definitions as H1
a. Find what points are in the events and add the probabilities. P A B , P B A , P A C , P C A ,
   
   
PB C  , PC B , P B C , P C B
   
 
b. Show that the multiplication rule works for A and B . i.e. that P A  B  P A B PB and that
P A  B  PB AP A


c. Do the same for P A  C  , PB  C  and P C  B .
 
   
   
PROBLEM H3. In H2, Demonstrate Bayes' rule for P A B and P B A , P A C and P C A , P B C
   
 
and P C B , P B C and P C B .
PROBLEM H4.
Find a formula for P A  B  C  D .
PROBLEM H5. A firm's employees are 15% African-American, 10% Hispanic, and 3% Asian. If 3% are
both African-American and Hispanic, what percent of the employees are minority?
PROBLEM H6. Your firm produces 3 different VCRs, The Deluxe, Super Deluxe, and Incredible
models. You offer a 1 year warranty. 50% of your sales are Deluxe, 30% are Super Deluxe, and 20% are
incredible.
During the warranty period 80% of the Deluxe models fail, 50% of the Super Deluxe models fail, and 30%
of the Incredible models fail. Since the Deluxe and Super Deluxe models are not worth repairing, they are
replaced when they are returned after failure. To decide on how many repair people to hire, find out what
per cent of the returns are incredible.
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251probl
PROBLEM H7.
A machine has two components with the following probabilities of failure:
Day
Component 1
Component 2
Notice that, since neither component has a
life of more than four days, the machine
1
.25
0
cannot last beyond day 4 if it requires both
2
.25
.50
or either component to function. Thus the
3
.25
.50
probabilities of it failing on the four days
4
.25
0
must add to 1.
a. If the machine requires both components to operate, it cannot fail on day 4, since
component 2 cannot last beyond day 3. What is the probability of the machine failing on
day 1? day 2? day 3? Hint!! Before you do either part, make a joint probability table,
with events for component 1 on one axis and events for component 2 on the other.
b. Instead of assuming that the machine needs both components to work, assume that it
will operate as long as either component 1 or component 2 is working. Notice that this
means that it cannot fail on day 1. What about the probability of failure on day 2? Day
3? Day 4?
PROBLEM H8. Explain why mutually exclusive events are also dependent events.
PROBLEM H9 (McClave, Benson and Sincich - edited). Flip a coin three times. Define the following
events: A  At least one head, B  Exactly two heads , C  Exactly two tails and
D  At most one head . You have seen a tree diagram for this in class that showed that
PHHH   PHHT   PHTH             PTTT   18 . We can do this in a more sophisticated way now.
Let H 1 be a head on the first try, H 2 be a head on the second try and H 3 be a head on the third try. These
three events are independent so, by the extended Multiplication Rule, PHHH   PH 3  H 2  H 1 
 PH 3 H 2  H 1 PH 2 H 1 PH 1   PH 3 PH 2 PH 1  
12 12 12   18  . This applies to all the possible
events, which are HHH , HHT , HTH , HTT , THH , THT , TTH and TTT . All are
a) Find P A , PB  , PC  , PD  , P A  B  , P A  D  , PB  C  and PB  D  .
   
1
8
.
 
b) Use your answers in a) to calculate P A B , P A D and P C B .
c) Which pairs of events are independent?
PROBLEM I1.
What is the probability of two jacks in a hand of ten cards?
PROBLEM I2.
What is the probability of five jacks in a hand of ten cards?
PROBLEM I3.
What is the probability of two jacks and three tens in a hand of ten cards?
PROBLEM I4.
In a poker hand (5 cards), what is the probability of getting 2 hearts or two face cards
(or both)?
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251probl
PROBLEM J1. Flip a coin 4 times. Let x be the number of heads. Find the following:
a. The distribution of x .
b. Find the probability of (i) at least one head, (ii) An even number of heads, (iii) at least 2 heads.
c. Find E x 
PROBLEM J2.
In problem J1 find the standard deviation of the number of heads
PROBLEM J3.
Assume that values of x as shown below represent sales of your product.
Sales
Probability
Do the following. Except for part a do these two ways - first by
x
figuring out each value of the function of x and its probability,
P x 
and, second, by using a formula for a function of x .
0
.50
1
.10
2
.10
3
.10
4
.20
a. What is E x , E x 2 ?
b. If costs are C1  10 x  10 , what is E C1  ?
 
c. If costs are C2  0.5x 2  10x  10 , What is E C 2  ?
d. If the product sells for $25 per unit, and R represents revenue, what is E R  ?
e. If Costs are described by C1 , and   R  C represents profit, what is E   ?
PROBLEM J4.
If y  200 x  50,  x  3 and  x2  10 , what are  y and  y2 ?
PROBLEM J5. I have measured the temperature at which successful outcomes of a biochemical process
occur. It turns out that the sample mean is 72 degrees Fahrenheit and the sample standard deviation is 10
degrees Fahrenheit. My European affiliate needs this data. What are the mean and standard deviation in
Celsius? Remember that if x f is the temperature in Fahrenheit and x c is the temperature in Celsius,
x f  9 5 xc  32 .
PROBLEM J6. A package of a product sells for $21.00/lb. plus a $1.00 packing cost, so that A 2 lb.
package would sell for $21.00(2)+$1.00 = $43.00. If the mean package size is 37 oz. and the standard
deviation of package size is 3 oz. , what are the mean and standard deviation of package price?
PROBLEM J7.
dollars):
You are investing in a project with the following distribution of profits( x in millions of
x
10
20
30
40
PROBLEM J8.
variance of y .
P x 
.500
.250
.125
.125
a. What is the mean and variance of the profits?
b. Assume that you must pay me 50% of the profits plus a
finder's fee of $1(million). What is the mean and variance of
your profit?
c. Use your answer in part b. to find the mean and variance
of your profit if I raise my finder's fee to $2(million)?
Assume that x is a standardized random variable. If y  5x  3 , find the mean and
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251probl
PROBLEM J9.
Consider the random variable x in the table below. Make a probability table for a
standardized random variable z , created by subtracting
x
the mean (which is 4) and dividing by the standard deviation
P x 
0
1/16
(which is 2). Show that the mean and standard deviation of
2
1/4
z are 0 and 1.
4
3/8
6
1/4
8
1/16
PROBLEM J10. A component's life has a continuous uniform distribution between 15 and 25 days. Find
the following:
a. The probability that it will last 21 days or more.
b. The probability that it will last between 10 and 18 days.
c. The probability that two such components both last between 18 and 20 days.
d. The mean time that a component lasts.
e. The variance of the duration of a component's life.
f. The cumulative probability F 18  .
PROBLEM J11. If x has the continuous uniform distribution between 10 and 25, show that the Chebyshev
inequality is satisfied for 1, 2 and 3 standard deviations. How does this distribution compare with the
empirical rule?
PROBLEM J12. Find P8  x  12  for the following distributions
a. Continuous Uniform with c  3, d  27 (Make a diagram!).
b. Continuous Uniform with c  3, d  10 (Make a diagram!).
Find these areas in two ways. First, make an appropriate diagram and find the area of the box. Second, use
the cumulative distribution to find P8  x  12   F 12   F 8 .
PROBLEM K1. Let us assume that x has a mean of 2 and a standard deviation of 4, and that y has a
mean of 3 and a standard deviation of 6. Also assume that the correlation between x and y is .9. If
w  7 x  3 and v  4 y  5 , find the covariance and correlation between w and v .
PROBLEM K2. The table below shows average Fahrenheit temperature and yield in lbs./acre for an
industrial crop.
a. Find the covariance and correlation between Fahrenheit
F
Y
70
15
temperature and yield.
75
17
b. If the conversion formula for Celsius temperature is
79
16
C  5 9 F  32  , find the covariance and correlation between
80
20
Celsius temperature and yield.
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251probl
PROBLEM K3. a. What is Covx, x  ?
b. What is (i) Cov5x  3,7 x  9 and (ii) Corr 5 x  3,7 x  9 ?
c. So what is  xx ?
d. If m represents the per item markup and FC is fixed cost, and we sell x units of an
item, the profit is w  mx  FC . Assume that we have two goods, good X and Good
Y, and that the quantity of good X sold is x and the quantity of good Y sold is y . Let
w be the profits on good X and u be the profits on good Y. Assume that
Covx, y   20 , and that data concerning
Good X Good Y sales is as in the table at left. Find the
Markup
5
2
variance of profits on each of the two goods.
Fixed Cost
100
50
Then find the covariance and correlation between
Mean
90
90
profits on the two goods.
Mean
90
90
Variance
50
50
x
x
PROBLEM K4. Find  xy for
1
4
.10
5
0
6
0
y 2
0
.40
0
3
0
0
.50
and
4
5
1
0
0
6
.10
y 2
0
.40
0
3 .50
0
0
PROBLEM L1. Assume that x has the binomial distribution with 10 tries and that the probability of
success on any one try is 25%. Find the following:
a. P4
b. E x  , Var x 
c. The probability that x is at least 1, Px  1 .
d. If the probability of success on any one try is .75, and n  10 , what is P6 ?.
e. Using the cumulative binomial table find the following:
(i) Px  5 when n  10 , p  .2
(ii) Px  5 when n  10 , p  .2
(iii) P2  x  8 when n  10 , p  .4
(iv) P3  x  9 when n  10 , p  .7
f. If ten people are selected at random from a large (continuous) population what is the
probability that at least one makes more than the median income?
g. In f., what is the probability that between two and eight make more than the median income?
h. What is the probability that all make more than the median income?
i. For a binomial distribution with n  10 , p  .4 , find the smallest symmetrical region about the
mean that contains (at least) 95% of the probability.
PROBLEM L2. a. If p  .8 , what is the chance of the first success on the 9th try? The 5th? The first?
b. If p  .8 , what is the mean number of tries to a success? The variance?
c. What is the probability that the first success occurs between the 3rd and the 7th try?
PROBLEM L3. a. If a bunch of fair coins are spilled and I pick them up randomly, what is the chance of
the first head on try 1? Try 2? Try 3? If x is the first success, what is the probability that x is at least 3?
b. On the average, how many coins will I have to pick up to get a head?
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251probl
PROBLEM L4. a. If p  .01 , Use the Chebyshev inequality to give a range in which three quarters of first
successes must lie.
b. Try confirming this using the Cumulative probability for this distribution.
PROBLEM L5. I am a real estate agent who gets an average of 1 customer every three days.
a. If I go away for three days, what is the chance that I miss a customer?
b. If I sit and wait for two days, what is the chance that the first customer
doesn't come in until the third day?
PROBLEM L6. If average sales for a six week period are 20 units, and it takes six weeks to get a delivery,
what is my reorder point if I want a 95% certainty of avoiding a stockout? What about a 90% certainty?
PROBLEM L7. In Frunze there are, on the average, 36 earthquakes a year.
a. If I take a one month vacation in Frunze, what is my chance of at least one
earthquake?
b. What about between five and eight earthquakes?
PROBLEM L8. After showing that the Poisson Distribution can be used for this problem, find the solution
to the Binomial problem Px  3 when p  .01 and n  400 using the Poisson Distribution. How does
this compare with the solution from a binomial table?
PROBLEM L9. We have fifteen units of equipment in a bin of which five are defective. Pull three out at
random. What is the probability that exactly one will be defective if we:
a. sample without replacement?
b. sample with replacement?
PROBLEM L10. In a population of 10 students 60% prefer Coke, the remainder Pepsi. If a sample of four
is taken, (a) what is the chance that exactly three prefer Coke? (b) What is the chance that a majority prefer
Coke? (Hint: The answer to (a) is not the same as the answer to (b) .) (c, d) Redo (a) and (b) assuming that
there are many students in the population (but the sample is still 3 individuals).
PROBLEM L11. Find the probability P5  x  13 for
a) A continuous uniform distribution with c  1, d  21
b) A continuous uniform distribution with c  1, d  11
c) A continuous uniform distribution with c  11, d  21
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251probl
PROBLEM M1. Assume that x ~ N (3,6) . Try the following:
a. P3  x  12 
b. P 6  x  6
c. P6  x  12 
d. P0  x  12 
e. P12  x  25 
f. F 12   Px  12 
g. F  6  Px  6
h. Find the 80th percentile.
i. Find the 20th percentile.
j. Find a symmetric region about the mean with 80% probability.
PROBLEM M2: If x ~ N 3,10  , find the probabilities for the following intervals: Below -3; -3 to 0; 0 to 3;
3 to 6; 6 to 9; above 9.
PROBLEM M3. I am a maker of videotapes. One of my clients has been getting returns of 6-hour tapes
with complaints that they are less than six hours. The client proposes that each time a batch of tapes arrives,
the client will take home three tapes and record on them. If any are less than six hours, the batch will be
rejected.
a. If the tapes are made with a mean length of 6.1 hours and a standard deviation of 0.1 hours and
the normal distribution applies to the actual length of the tape, what is the chance that a given batch is
rejected.
b. To get the probability of rejection down to 1% what should the mean length be?
PROBLEM M4. Assume x ~ N 5,7 . Find:
a. A 60% symmetrical interval about the mean.
b. The 60th percentile.
c. The 30th percentile.
d. z .23
e. x.23
f. x.52
PROBLEM M5. Assume that x has the Poisson distribution with a parameter of 25. Find Px  25  using
both the Poisson and the Normal distributions. Compare the answers.
PROBLEM M6. Do the following binomial problems using the Poisson or the normal distribution as
appropriate:
a. If n  100 , and p  .03 , find Px  5 .
b. If n  25 and p  .80 , find Px  17  .
PROBLEM M7. We send a survey out to 200 people. The probability that any one person will return it is
0.1. What is the chance of between 23 and 33 returns? More than 100 returns?
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251probl
PROBLEM M8. Find the probability P5  x  13 using the formula Pa  x  b  F b  F a  and
cumulative probabilities for
a) A continuous uniform distribution with c  1, d  21
b) A continuous uniform distribution with c  1, d  11
c) A continuous uniform distribution with c  11, d  21 ( a)-c) are the same as problem L?)
d) x ~ N 6,10 
e) (Optional) An exponential distribution with a mean time to success of 12.
PROBLEM N1. The average life of a Toyota Caramba automobile is 44 months with a standard deviation
of 18 months.
a. From a sample of 36, what is the probability that we find an average life below 38 months?
b. Actually only 200 Toyota Carambas were ever produced. Redo part a. continuing to assume a
sample of 36.
c. An average package weighs 44 lbs. with a standard deviation of 18 lbs.. A Toyota Caramba will
carry 1368 lbs. If I must deliver 36 packages, what is the chance that my vehicle will not be overloaded?
PROBLEM N2. For my national fleet (all the same vehicle), mean weekly gasoline consumption is 16.9
gallons with a standard deviation of 3.2 gallons. In a local garage I have 875 gallons of gas and 50 vehicles.
What is the probability of running out of gas this week?
PROBLEM O1. If n  64 and x  11.50 , find 95% confidence intervals for the mean under the following
circumstances:
a.   6.30, N  3000
b.   6.30 , N  300
c. s  6.30, N  3000
d. s  6.30 , N  300
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