Math Problem

Math Problem

50. A supermarket chain operates five stores of varying
sizes in Bloomington, Indiana. Profits (represented as a
percentage of sales volume) earned by these five stores
are 2.75%, 3%, 3.5%, 4.25%, and 5%, respectively. The
means and standard deviations of the daily sales
volumes at these five stores are given in the file
P04_50.xlsx. Assuming that the daily sales volumes are
independent, find the mean and standard deviation of the
total profit that this supermarket chain earns in one day
from the operation of its five stores in Bloomington.

54. A business manager who needs to make many phone
calls has estimated that when she calls a client, the
probability that she will reach the client right away is
60%. If she does not reach the client on the first call,
the probability that she will reach the client with a
subsequent call in the next hour is 20%.
a. Find the probability that the manager reaches her
client in two or fewer calls.
b. Find the probability that the manager reaches her
client on the second call but not on the first call.
c. Find the probability that the manager is
unsuccessful on two consecutive calls.

55.55. Suppose that a marketing research firm sends questionnaires to two different companies. Based on historical evidence, the marketing research firm believes that each company, independently of the other, will return the questionnaire with probability 0.40.
a. What is the probability that both questionnaires are
returned?
b. What is the probability that neither of the
questionnaires is returned?
c. Now, suppose that this marketing research firm sends
questionnaires to ten different companies. Assuming
that each company, independently of the others,
returns its completed questionnaire with probability
0.40, how do your answers to parts a andb change?
56. Based on past sales experience, an appliance store
stocks five window air conditioner units for the
coming week. No orders for additional air conditioners
will be made until next week. The weekly consumer
demand for this type of appliance has the probability
distribution given in the file P04_56.xlsx.
a. LetX be the number of window air conditioner units
left at the end of the week (if any), and let Y be the
number of special stockout orders required (if any),
assuming that a special stockout order is required
each time there is a demand and no unit is available in
stock. Find the probability distributions of X and Y.
b. Find the expected value of X and the expected
value of Y.
c. Assume that this appliance store makes a $60 profit
on each air conditioner sold from the weekly
available stock, but the store loses $20 for each unit
sold on a special stockout order basis. Let Z be the
profit that the store earns in the coming week from
the sale of window air conditioners. Find the
probability distribution of Z.
d. Find the expected value of Z.
60. The probability distribution of the weekly demand
for copier paper (in hundreds of reams) used in the
duplicating center of a corporation is provided in the
fileP04_58.xlsx. Assuming that it costs the duplicating
center $5 to purchase a ream of paper, find the
mean and standard deviation of the weekly copier
paper cost for this corporation.
66. Suppose there are three states of the economy: boom,
moderate growth, and recession. The annual return on
Honda and Toyota stock in each state of the economy
is shown in the file P04_66.xlsx.
a. Calculate the mean and standard deviation of the
annual return on each stock, assuming the
probability of each state is 1/3.
b. Calculate the mean and standard deviation of the
annual return on each stock, assuming the
probabilities of the three states are 1/4, 1/4, and 1/2.
c. Calculate the covariance and correlation between
the annual returns of the two companies’ stocks,
assuming the probability of each state is 1/3.
d. Calculate the covariance and correlation between
the annual returns of the two companies’stocks,
assuming the probabilities of the three states are
1/4, 1/4, and 1/2.
e. You have invested 25% of your money in Honda
and 75% in Toyota. Assuming that each state is
equally likely, find the mean and variance of your
portfolio’s return.
f. Now check your answer to part e by directly
calculating the return on your portfolio for each
state and use the formulas for mean and variance of
a random variable. For example, in the boom state,
your portfolio earns 0.25(0.25) _ 0.75(0.32).
77. Consider again the supermarket chain described in
Problem 50. Now, assume that the daily sales of the
five stores are no longer independent of one another.
In particular, the file P04_77.xlsx contains the
correlations between all pairs of daily sales volumes.
a. Find the mean and standard deviation of the total
profit that this supermarket chain earns in one day
from the operation of its five stores in Bloomington.
Compare these results to those you found in Problem
50. Explain the differences in your answers.
b. Find an interval such that the regional sales manager
of this supermarket chain can be approximately 95%
sure that the total daily profit earned by its stores in
Bloomington will be contained within the interval.
51. Suppose the annual return on XYZ stock follows a
normal distribution with mean 0.12 and standard
deviation 0.30.
a. What is the probability that XYZ’s value will
decrease during a year?
b. What is the probability that the return on
XYZ during a year will be at least 20%?
c. What is the probability that the return on
XYZ during a year will be between –6% and 9%?
d. There is a 5% chance that the return on XYZ
during a year will be greater than what value?
e. There is a 1% chance that the return on XYZ
during a year will be less than what value?
f. There is a 95% chance that the return on
XYZ during a year will be between which two
values (equidistant from the mean)?
54. A family is considering a move from a midwestern
city to a city in California. The distribution of housing
costs where the family currently lives is normal, with
mean $105,000 and standard deviation $18,200. The
distribution of housing costs in the California city is
normal with mean $235,000 and standard deviation
$30,400. The family’s current house is valued at
$110,000.
a. What percentage of houses in the family’s current
city cost less than theirs?
b. If the family buys a $200,000 house in the new
city, what percentage of houses there will cost less
than theirs?
c. What price house will the family need to buy to be
in the same percentile (of housing costs) in the new
city as they are in the current city?
62. Suppose that if a presidential election were held today,
53% of all voters would vote for Obama over McCain.
(You can substitute the names of the current presidential
candidates.) This problem shows that even if there are
100 million voters, a sample of several thousand is
enough to determine the outcome, even in a fairly close
election.
a. If 1500 voters are sampled randomly, what is the
probability that the sample will indicate (correctly)
that Obama is preferred to McCain?
b. If 6000 voters are sampled randomly, what is the
probability that the sample will indicate (correctly)
that Obama is preferred to McCain?
63. A soft-drink factory fills bottles of soda by setting a
timer on a filling machine. It has generally been
observed that the distribution of the number of ounces
the machine puts into a bottle is normal, with standard
deviation 0.05 ounce. The company wants 99.9% of
all its bottles to have at least 16 ounces of soda. To
what value should the mean amount put in each bottle
be set? (Of course, the company does not want to fill
any more than is necessary.)

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