Assignment
#3

1. Two six-sided dice are rolled and
the random variable x is the sum of the values produced by each die.

a.
List
the possible outcomes for the variable x, determine the probability for each outcome, and
calculate the expected value of x.

b.
Calculate
the variance of x.

c.
Calculate
the standard deviation of x.

2. Using historical
records, a manufacturing firm has developed the following probability distribution for the number of days
required to get components from its suppliers. The distribution is shown here,
where the random variable x is the number of days.

x |
2 |
3 |
4 |
5 |
6 |

P(x) |
0.15 |
0.45 |
0.30 |
0.075 |
0.025 |

a. What is the average lead time for
the component?

b. What is the coefficient of variation for delivery lead time?

3. A large corporation in search of
a CEO and a CFO has narrowed the fields for each position to a short list. The
CEO candidates graduated from Chicago (C) and three Ivy League universities:
Harvard (H), Princeton (P),and Yale (Y). The four CFO candidates graduated from
MIT (M), Northwestern (N), and two Ivy League universities: Dartmouth (D) and
Brown (B). The personnel director wishes to determine the distribution of the
number of Ivy League graduates who could fill these positions.

a. Assume the selections were made
randomly. Construct the probability distribution of the number of Ivy League
graduates who could fill these positions.

b. Would it be surprising if both
positions were filled with Ivy League graduates?

c. Calculate the expected value and
standard deviation of the number of Ivy League graduates who could fill these
positions.

4. Hewlett-Packard receives large
shipments of microprocessors from Intel Corp. It must try to ensure the
proportion of microprocessors that are defective is small. Suppose HP decides
to test five microprocessors out of a shipment of thousands of these
microprocessors. Suppose that if at least one of the microprocessors is
defective, the shipment is returned.

a. If Intel Corp.’s shipment
contains 10% defective microprocessors, calculate the probability the entire
shipment will be returned.

b. If Intel and HP agree that Intel
will not provide more than 5% defective chips, calculate the probability that
the entire shipment will be returned even though only 5% are defective

c. Calculate the probability that
the entire shipment will be kept by HP even though the shipment has 10%
defective microprocessors

5. Assume that for an ad campaign to
be successful, at least 80% of those seeing a television commercial must be
able to recall the name of the company featured in the commercial one hour
after viewing the commercial. Before distributing an ad campaign nationally, an
advertising company plans to show the commercial to a random sample of 20
people. It will also show the same people two additional commercials for
different products or businesses.

a. Assuming that the advertisement
will be successful (80% will be able to recall the name of the company in the
ad), what is the expected number of people in the sample who will recall the
company featured in the commercial one hour after viewing the three
commercials?

b. Suppose that in the sample of 20
people, 11 were able to recall the name of the company in the commercial one
hour after viewing. Based on the premise that the advertising campaign will be
successful, what is the probability of 11 or fewer people being able to recall
the company name?

a. Based on your responses to parts
a and b, what conclusion might the advertising executives
make about this particular advertising campaign?

6.
When
things are operating properly, E-Bank United, an Internet bank, can process a
maximum of 25 electronic transfers every minute during the busiest periods of
the day. If it receives more transfer requests than this, then the bank’s
computer system will become so overburdened that it will slow to the point that
no electronic transfers can be handled. If during the busiest periods of the
day requests for electronic transfers arrive at the rate of 170 per 10-minute
period on average, what is the probability that the system will be overwhelmed
by requests? Assume that the process can be described using a Poisson distribution.

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