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Statistics for Business and Economics: Global Edition

Newbold P., Carlson W.L., Thorne B.M.

Chapter 4

Discrete Probability Distributions - all with Video Answers

Educators

Chapter Questions

00:31

Problem 1

A store sells from 0 to 12 computers per day. Is the amount of daily computer sales a discrete or continuous random variable?

Christopher Stanley
Christopher Stanley
Numerade Educator
00:37

Problem 2

A factory production process produces a small number of defective parts in its daily production. Is the number of defective parts a discrete or continuous random variable?

Christopher Stanley
Christopher Stanley
Numerade Educator
01:25

Problem 3

For each of the following, indicate if a discrete or a continuous random variable provides the best definition:
a. The number of cars that arrive each day for repair in a two-person repair shop
b. The number of cars produced annually by General Motors
c. Total daily e-commerce sales in dollars
d. The number of passengers that are bumped from a specific airline flight 3 days before Christmas

Christopher Stanley
Christopher Stanley
Numerade Educator
00:27

Problem 4

An equity actor auditions 100 times a year and obtains a contract for a play $8 \%$ of the time. Is her work schedule (number of plays) a discrete or random variable?

Christopher Stanley
Christopher Stanley
Numerade Educator
01:46

Problem 5

List four examples of discrete random variables that could be observed in a new consulting business.

Christopher Stanley
Christopher Stanley
Numerade Educator
01:25

Problem 6

Define three continuous random variables that a marketing vice president should regularly examine.

Christopher Stanley
Christopher Stanley
Numerade Educator
00:39

Problem 7

A presidential election poll contacts 2,000 randomly selected people. Should the number of people that support candidate $\mathrm{A}$ be analyzed using discrete or continuous probability models?

Christopher Stanley
Christopher Stanley
Numerade Educator
00:41

Problem 8

A salesperson contacts 20 people each day and requests that they purchase a specific product. Should the number of daily purchases be analyzed using discrete or continuous probability models?

Christopher Stanley
Christopher Stanley
Numerade Educator
00:31

Problem 9

What is the probability distribution function of the number of heads when a fair coin is tossed once?

Christopher Stanley
Christopher Stanley
Numerade Educator
00:36

Problem 10

Show the probability distribution function of the face values of a single die when a fair die is rolled.

Christopher Stanley
Christopher Stanley
Numerade Educator
01:27

Problem 11

Show the probability distribution function of the number of heads when three fair coins are tossed independently.

Christopher Stanley
Christopher Stanley
Numerade Educator
02:11

Problem 12

Let the random variable represent the number of times that you will miss class this semester. Prepare a table that shows the probability distribution and the cumulative probability distribution.

Christopher Stanley
Christopher Stanley
Numerade Educator
02:48

Problem 13

The number of computers sold per day at Dan's Computer Works is defined by the following probability distribution:
$$
\begin{array}{lccccccc}
\hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline P(x) & 0.05 & 0.10 & 0.20 & 0.20 & 0.20 & 0.15 & 0.10 \\
\hline
\end{array}
$$
a. $P(3 \leq x<6)=$ ?
b. $P(x>3)=$ ?
c. $P(x \leq 4)=$ ?
d. $P(2<x \leq 5)=$ ?

Christopher Stanley
Christopher Stanley
Numerade Educator
02:52

Problem 14

In a geography assignment the grade obtained is the random variable $X$. It has been found that students have these probabilities of getting a specific grade:
A: 0.18
B: 0.32
C: 0.25
D. 0.07
E: 0.03
F: 0.15
Based on this, calculate the following.
a. The cumulative probability distribution of $X$.
b. The probability of getting a higher grade than B.
c. The probability of getting a lower grade than C.

Christopher Stanley
Christopher Stanley
Numerade Educator
03:18

Problem 15

Consider the probability distribution function.
$$
\begin{array}{lcc}
\hline x & 0 & 1 \\
\hline \text { Probability } & 0.40 & 0.60 \\
\hline
\end{array}
$$
a. Graph the probability distribution function.
b. Calculate and graph the cumulative probability distribution.
c. Find the mean of the random variable $X$.
d. Find the variance of $X$.

Christopher Stanley
Christopher Stanley
Numerade Educator
08:34

Problem 16

Given the probability distribution function:
$$
\begin{array}{lccc}
\hline x & 0 & 1 & 2 \\
\hline \text { Probability } & 0.25 & 0.50 & 0.25 \\
\hline
\end{array}
$$
a. Graph the probability distribution function.
b. Calculate and graph the cumulative probability distribution.
c. Find the mean of the random variable $X$.
d. Find the variance of $X$.

Barsha Rana
Barsha Rana
Numerade Educator
02:44

Problem 17

Consider the probability distribution function
$$
\begin{array}{lcc}
\hline x & 0 & 1 \\
\hline \text { Probability } & 0.50 & 0.50 \\
\hline
\end{array}
$$
a. Graph the probability distribution function.
b. Calculate and graph the cumulative probability distribution.
c. Find the mean of the random variable $X$.
d. Find the variance of $X$.

Christopher Stanley
Christopher Stanley
Numerade Educator
08:24

Problem 18

An automobile dealer calculates the proportion of new cars sold that have been returned a various numbers of times for the correction of defects during the warranty period. The results are shown in the following table.
$$
\begin{array}{lccccc}
\hline \text { Number of returns } & 0 & 1 & 2 & 3 & 4 \\
\hline \text { Proportion } & 0.28 & 0.36 & 0.23 & 0.09 & 0.04 \\
\hline
\end{array}
$$
a. Graph the probability distribution function.
b. Calculate and graph the cumulative probability distribution.
c. Find the mean of the number of returns of an automobile for corrections for defects during the warranty period.
d. Find the variance of the number of returns of an automobile for corrections for defects during the warranty period.

Barsha Rana
Barsha Rana
Numerade Educator
09:12

Problem 19

A company specializes in installing and servicing central-heating furnaces. In the prewinter period, service calls may result in an order for a new furnace. The following table shows estimated probabilities for the numbers of new furnace orders generated in this way in the last two weeks of September.
$$
\begin{array}{lcccccc}
\hline \text { Number of orders } & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline \text { Probability } & 0.10 & 0.14 & 0.26 & 0.28 & 0.15 & 0.07 \\
\hline
\end{array}
$$
a. Graph the probability distribution function.
b. Calculate and graph the cumulative probability distribution.
c. Find the probability that at least 3 orders will be generated in this period.
d. Find the mean of the number of orders for new furnaces in this 2 -week period.
e. Find the standard deviation of the number of orders for new furnaces in this 2-week period.

Barsha Rana
Barsha Rana
Numerade Educator
21:24

Problem 20

Forest Green Brown, Inc., produces bags of cypress mulch. The weight in pounds per bag varies, as indicated in the accompanying table.
$$
\begin{array}{lccccccc}
\hline \text { Weight in pounds } & 44 & 45 & 46 & 47 & 48 & 49 & 50 \\
\hline \text { Proportion of bags } & 0.04 & 0.13 & 0.21 & 0.29 & 0.20 & 0.10 & 0.03 \\
\hline
\end{array}
$$
a. Graph the probability distribution.
b. Calculate and graph the cumulative probability distribution.
c. What is the probability that a randomly chosen bag will contain more than 45 and less than 49 pounds of mulch (inclusive)?
d. Two packages are chosen at random. What is the probability that at least one of them contains at least 47 pounds?
e. Compute-using a computer-the mean and standard deviation of the weight per bag.
f. The cost (in cents) of producing a bag of mulch is $75+2 X$, where $X$ is the number of pounds per bag. The revenue from selling the bag, regardless of weight, is $$\$ 2.50$$. If profit is defined as the difference between revenue and cost, find the mean and standard deviation of profit per bag.

Barsha Rana
Barsha Rana
Numerade Educator
16:48

Problem 21

A municipal bus company has started operations in a new subdivision. Records were kept on the numbers of riders on one bus route during the early-morning weekday service. The accompanying table shows proportions over all weekdays.
$$
\begin{array}{lcccccccc}
\hline \text { Number of riders } & 20 & 21 & 22 & 23 & 24 & 25 & 26 & 27 \\
\hline \text { Proportion } & 0.02 & 0.12 & 0.23 & 0.31 & 0.19 & 0.08 & 0.03 & 0.02 \\
\hline
\end{array}
$$
a. Graph the probability distribution.
b. Calculate and graph the cumulative probability distribution.
c. What is the probability that on a randomly chosen weekday there will be at least 24 riders from the subdivision on this service?
d. Two weekdays are chosen at random. What is the probability that on both of these days there will be fewer than 23 riders from the subdivision on this service?
e. Find the mean and standard deviation of the number of riders from this subdivision on this service on a weekday.
f. If the cost of a ride is $$\$ 1.50$$, find the mean and standard deviation of the total payments of riders from this subdivision on this service on a weekday.

Barsha Rana
Barsha Rana
Numerade Educator
00:24

Problem 22

a. A very large shipment of parts contains $10 \%$ defectives. Two parts are chosen at random from the shipment and checked. Let the random variable $X$ denote the number of defectives found. Find the probability distribution of this random variable.
b. A shipment of 20 parts contains 2 defectives. Two parts are chosen at random from the shipment and checked. Let the random variable $Y$ denote the number of defectives found. Find the probability distribution of this random variable. Explain why your answer is different from that for part (a).
c. Find the mean and variance of the random variable $X$ in part (a).
d. Find the mean and variance of the random variable $Y$ in part (b).

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:28

Problem 23

A student needs to know details of a class assignment that is due the next day and decides to call fellow class members for this information. She believes that for any particular call, the probability of obtaining the necessary information is 0.40 . She decides to continue calling class members until the information is obtained. But her cell phone battery will not allow more than 8 calls. Let the random variable $\mathrm{X}$ denote the number of calls needed to obtain the information.
a. Find the probability distribution of $X$.
b. Find the cumulative probability distribution of $X$.
c. Find the probability that at least three calls are required.

Christopher Stanley
Christopher Stanley
Numerade Educator
01:40

Problem 24

Your school Ping-Pong team is not performing very well this season. After some rough calculations, you found out that your team's probability of winning a game is about 0.45 . A fellow team member wants to know more and asked you also to determine the following.
a. The probability of the team winning 2 games out of 5 .
b. The probability of winning 10 times out of 25 .

Christopher Stanley
Christopher Stanley
Numerade Educator
08:36

Problem 25

A professor teaches a large class and has scheduled an examination for 7:00 p.m. in a different classroom. She estimates the probabilities in the table for the number of students who will call her at home in the hour before the examination asking where the exam will be held.
$$
\begin{array}{lcccccc}
\hline \text { Number of calls } & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline \text { Probability } & 0.10 & 0.15 & 0.19 & 0.26 & 0.19 & 0.11 \\
\hline
\end{array}
$$
Find the mean and standard deviation of the number of calls.

Barsha Rana
Barsha Rana
Numerade Educator
05:17

Problem 26

Students in a large accounting class were asked to rate the course by assigning a score of $1,2,3,4$, or 5 to the course. A higher score indicates that the students received greater value from the course. The accompanying table shows proportions of students rating the course in each category.
$$
\begin{array}{lccccc}
\hline \text { Rating } & 1 & 2 & 3 & 4 & 5 \\
\hline \text { Proportion } & 0.07 & 0.19 & 0.28 & 0.30 & 0.16 \\
\hline
\end{array}
$$
Find the mean and standard deviation of the ratings.

Barsha Rana
Barsha Rana
Numerade Educator
08:00

Problem 27

A store owner stocks an out-of-town newspaper that is sometimes requested by a small number of customers. Each copy of this newspaper costs her 70 cents, and she sells them for 90 cents each. Any copies left over at the end of the day have no value and are destroyed. Any requests for copies that cannot be met because stocks have been exhausted are considered by the store owner as a loss of 5 cents in goodwill. The probability distribution of the number of requests for the newspaper in a day is shown in the accompanying table. If the store owner defines total daily profit as total revenue from newspaper sales, less total cost of newspapers ordered, less goodwill loss from unsatisfied demand, what is the expected profit if four newspapers are order?
$$
\begin{array}{lcccccc}
\hline \text { Number of requests } & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline \text { Probability } & 0.12 & 0.16 & 0.18 & 0.32 & 0.14 & 0.08 \\
\hline
\end{array}
$$

Barsha Rana
Barsha Rana
Numerade Educator
02:31

Problem 28

A factory manager is considering whether to replace a temperamental machine. A review of past records indicates the following probability distribution for the number of breakdowns of this machine in a week.
$$
\begin{array}{lccccc}
\hline \text { Number of breakdowns } & 0 & 1 & 2 & 3 & 4 \\
\hline \text { Probability } & 0.10 & 0.26 & 0.42 & 0.16 & 0.06 \\
\hline
\end{array}
$$
a. Find the mean and standard deviation of the number of weekly breakdowns.
b. It is estimated that each breakdown costs the company $$\$ 1,500$$ in lost output. Find the mean and standard deviation of the weekly cost to the company from breakdowns of this machine.

Manisha Sarker
Manisha Sarker
Numerade Educator
06:30

Problem 29

An investor is considering three strategies for a $$\$ 1,000$$ investment. The probable returns are estimated as follows:
$\bullet$ Strategy 1: A profit of $$\$ 10,000$$ with probability 0.15 and a loss of $$\$ 1,000$$ with probability 0.85
$\bullet$ Strategy 2: A profit of $$\$ 1,000$$ with probability 0.50, a profit of $$\$ 500$$ with probability 0.30 , and a loss of $$\$ 500$$ with probability 0.20
$\bullet$ Strategy 3: A certain profit of $$\$ 400$$
Which strategy has the highest expected profit? Explain why you would or would not advise the investor to adopt this strategy.

Bryan Meares
Bryan Meares
Numerade Educator
00:45

Problem 30

For a Bernoulli random variable with probability of success $P=0.5$, compute the mean and variance.

Christopher Stanley
Christopher Stanley
Numerade Educator
02:01

Problem 31

For a binomial probability distribution with $P=0.5$ and $n=12$, find the probability that the number of successes is equal to 7 and the probability that the number of successes is fewer than 6 .

Christopher Stanley
Christopher Stanley
Numerade Educator
02:01

Problem 32

For a binomial probability distribution with $P=0.3$ and $n=14$, find the probability that the number of successes is equal to 7 and the probability that the number of successes is fewer than 6 .

Christopher Stanley
Christopher Stanley
Numerade Educator
01:50

Problem 33

For a binomial probability distribution with $P=0.4$ and $n=20$, find the probability that the number of successes is equal to 9 and the probability that the number of successes is fewer than 7.

Christopher Stanley
Christopher Stanley
Numerade Educator
02:03

Problem 34

For a binomial probability distribution with $P=0.7$ and $n=18$, find the probability that the number of successes is equal to 12 and the probability that the number of successes is fewer than 6 .

Christopher Stanley
Christopher Stanley
Numerade Educator
02:32

Problem 35

A production manager knows that $5 \%$ of components produced by a particular manufacturing process have some defect. Six of these components, whose characteristics can be assumed to be independent of each other, are examined.
a. What is the probability that none of these components has a defect?
b. What is the probability that one of these components has a defect?
c. What is the probability that at least two of these components have a defect?

Christopher Stanley
Christopher Stanley
Numerade Educator
02:35

Problem 36

A state senator believes that $25 \%$ of all senators on the Finance Committee will strongly support the tax proposal she wishes to advance. Suppose that this belief is correct and that 5 senators are approached at random.
a. What is the probability that at least 1 of the 5 will strongly support the proposal?
b. What is the probability that a majority of the 5 will strongly support the proposal?

Christopher Stanley
Christopher Stanley
Numerade Educator
06:16

Problem 37

A public interest group hires students to solicit donations by telephone. After a brief training period students make calls to potential donors and are paid on a commission basis. Experience indicates that early on, these students tend to have only modest success and that $70 \%$ of them give up their jobs in their first two weeks of employment. The group hires 6 students, which can be viewed as a random sample.
a. What is the probability that at least 2 of the 6 will give up in the first two weeks?
b. What is the probability that at least 2 of the 6 will not give up in the first two weeks?

Nick Johnson
Nick Johnson
Numerade Educator
01:27

Problem 38

In a Godiva shop, $40 \%$ of the cookies are plain truffles, $20 \%$ are black truffles, $10 \%$ are cherry cookies, and $30 \%$ are a mix of all the others. Suppose you pick one at random from a prepacked bag that reflects this composition.
a. What is the probability of picking a plain truffle?
b. What is the probability of picking truffle of any kind?
c. If you instead pick three cookies in a row, what is the probability that all three are black truffles?

Christopher Stanley
Christopher Stanley
Numerade Educator
02:19

Problem 39

A company installs new central-heating furnaces and has found that for $15 \%$ of all installations, a return visit is needed to make some modifications. Six installations were made in a particular week. Assume independence of outcomes for these installations.
a. What is the probability that a return visit will be needed in all these cases?
b. What is the probability that a return visit will be needed in none of these cases?
c. What is the probability that a return visit will be needed in more than 1 of these cases?

Christopher Stanley
Christopher Stanley
Numerade Educator
01:33

Problem 40

In a scuba-diving center in Sipadan (Malaysia), the dive master has tried calculating the probability of encountering some very rare fish underwater. The following are the probabilities of encountering several fish.
Leopard shark: 0.05
Barracuda: 0.41
Lemon shark: 0.04
Scorpion fish: 0.27
Mandarin fish: 0.07
Using these statistics, calculate each likelihood.
a. Of not encountering a shark
b. Of encountering a shark
c. Of not encountering a scorpion fish

Christopher Stanley
Christopher Stanley
Numerade Educator
06:15

Problem 41

A small commuter airline flies planes that can seat up to 8 passengers. The airline has determined that the probability that a ticketed passenger will not show up for a flight is 0.2. For each flight the airline sells tickets to the first 10 people placing orders. The probability distribution for the number of tickets sold per flight is shown in the accompanying table. For what proportion of the airline's flights does the number of ticketed passengers showing up exceed the number of available seats? (Assume independence between the number of tickets sold and the probability that a ticketed passenger will show up.)
$$
\begin{array}{lccccc}
\hline \text { Number of tickets } & 6 & 7 & 8 & 9 & 10 \\
\hline \text { Probability } & 0.25 & 0.35 & 0.25 & 0.10 & 0.05 \\
\hline
\end{array}
$$

Amany Waheeb
Amany Waheeb
Numerade Educator
02:22

Problem 42

You are investigating the punctuality of the airlines in Asia. Your survey tells you that, out of 15 airlines, $80 \%$ of them are likely to be late at least once a month. Assume the punctuality random variable follows a binomial distribution. Determine the following.
a. Which assumptions do you need to make in order to be correct in considering a binomial distribution for your variable?
b. How many airlines will be late in one month?
c. What is the standard deviation of this random variable (i.e., the risk of being late)?
d. What is the probability that they all will be late?

Christopher Stanley
Christopher Stanley
Numerade Educator
00:42

Problem 43

A notebook computer dealer mounts a new promotional campaign. Purchasers of new computers may, if dissatisfied for any reason, return them within 2 days of purchase and receive a full refund. The cost to the dealer of such a refund is $$\$ 100$$. The dealer estimates that $15 \%$ of all purchasers will, indeed, return computers and obtain refunds. Suppose that 50 computers are purchased during the campaign period.
a. Find the mean and standard deviation of the number of these computers that will be returned for refunds.
b. Find the mean and standard deviation of the total refund costs that will accrue as a result of these 50 purchases.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
00:01

Problem 44

A family of mutual funds maintains a service that allows clients to switch money among accounts through a telephone call. It was estimated that $3.2 \%$ of callers either get a busy signal or are kept on hold so long that they may hang up. Fund management assesses any failure of this sort as a $$\$ 10$$ goodwill loss. Suppose that 2,000 calls are attempted over a particular period.
a. Find the mean and standard deviation of the number of callers who will either get a busy signal or may hang up after being kept on hold.
b. Find the mean and standard deviation of the total goodwill loss to the mutual fund company from these 2,000 calls.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:16

Problem 45

We have seen that, for a binomial distribution with $n$ trials, each with probability of success $P$, the mean is as follows:
$$
\mu_X=E[X]=n P
$$
Verify this result for the data of Example 4.7 by calculating the mean directly from
$$
\mu_X=\sum x P(x)
$$
showing that for the binomial distribution, the two formulas produce the same answer.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:58

Problem 46

A campus finance officer finds that, for all parking tickets issued, fines are paid for $78 \%$ of the tickets. The fine is $$\$ 2$$. In the most recent week, 620 parking tickets have been issued.
a. Find the mean and standard deviation of the number of these tickets for which the fines will be paid.
b. Find the mean and standard deviation of the amount of money that will be obtained from the payment of these fines.

Regina Hays
Regina Hays
Numerade Educator
02:49

Problem 47

A company receives a very large shipment of components. A random sample of 16 of these components will be checked, and the shipment will be accepted if fewer than 2 of these components are defective. What is the probability of accepting a shipment containing each number of defectives?
a. $5 \%$
b. $15 \%$
c. $25 \%$

Christopher Stanley
Christopher Stanley
Numerade Educator
02:51

Problem 48

The following two acceptance rules are being considered for determining whether to take delivery of a large shipment of components:
$\bullet$ A random sample of 10 components is checked, and the shipment is accepted only if none of them is defective.
$\bullet$ A random sample of 20 components is checked, and the shipment is accepted only if no more than 1 of them is defective.
Which of these acceptance rules has the smaller probability of accepting a shipment containing $20 \%$ defectives?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:28

Problem 49

A company receives large shipments of parts from two sources. Seventy percent of the shipments come from a supplier whose shipments typically contain $10 \%$ defectives, while the remainder are from a supplier whose shipments typically contain $20 \%$ defectives. A manager receives a shipment but does not know the source. A random sample of 20 items from this shipment is tested, and 1 of the parts is found to be defective. What is the probability that this shipment came from the more reliable supplier?

Anand Jangid
Anand Jangid
Numerade Educator
01:28

Problem 50

Determine the probability of exactly four successes for a random variable with a Poisson distribution with parameter $\lambda=2.4$.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
04:03

Problem 51

Determine the probability of more than 7 successes for a random variable with a Poisson distribution with parameter $\lambda=4.4$.

Barsha Rana
Barsha Rana
Numerade Educator
04:03

Problem 52

Determine the probability of fewer than 6 successes for a random variable with a Poisson distribution with parameter $\lambda=3.4$

Barsha Rana
Barsha Rana
Numerade Educator
01:28

Problem 53

Determine the probability of fewer than or equal to 9 successes for a random variable with a Poisson distribution with parameter $\lambda=8.0$.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
01:18

Problem 54

Customers arrive at a busy checkout counter at an average rate of 3 per minute. If the distribution of arrivals is Poisson, find the probability that in any given minute there will be 2 or fewer arrivals.

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
03:10

Problem 55

The number of accidents in a production facility has a Poisson distribution with a mean of 2.6 per month.
a. For a given month what is the probability there will be fewer than 2 accidents?
b. For a given month what is the probability there will be more than 3 accidents?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:51

Problem 56

A customer service center in India receives, on average, 4.2 telephone calls per minute. If the distribution of calls is Poisson, what is the probability of receiving at least 3 calls during a particular minute?

Christopher Stanley
Christopher Stanley
Numerade Educator
02:00

Problem 57

Records indicate that, on average, 3.2 breakdowns per day occur on an urban highway during the morning rush hour. Assume that the distribution is Poisson.
a. Find the probability that on any given day there will be fewer than 2 breakdowns on this highway during the morning rush hour.
b. Find the probability that on any given day there will be more than 4 breakdowns on this highway during the morning rush hour.

Kari Hasz
Kari Hasz
Numerade Educator
05:41

Problem 58

Blue Cross Health Insurance reported that $4.5 \%$ of claims forms submitted for payment after a complex surgical procedure contain errors. If 100 of these forms are chosen at random, what is the probability that fewer than 3 of them contain errors? Use the Poisson approximation to the binomial distribution.

Sonam Khatri
Sonam Khatri
Numerade Educator
01:25

Problem 59

A corporation has 250 personal computers. The probability that any 1 of them will require repair in a given week is 0.01 . Find the probability that fewer than 4 of the personal computers will require repair in a particular week. Use the Poisson approximation to the binomial distribution.

Christopher Stanley
Christopher Stanley
Numerade Educator
07:13

Problem 60

An insurance company holds fraud insurance policies on 6,000 firms. In any given year the probability that any single policy will result in a claim is 0.001 . Find the probability that at least 3 claims are made in a given year. Use the Poisson approximation to the binomial distribution.

Chris Trentman
Chris Trentman
Numerade Educator
06:39

Problem 61

A state has a law requiring motorists to carry insurance. It was estimated that, despite this law, $6.0 \%$ of all motorists in the state are uninsured. A random sample of 100 motorists was taken. Use the Poisson approximation to the binomial distribution to estimate the probability that at least 3 of the motorists in this sample are uninsured. Also indicate what calculations would be needed to find this probability exactly if the Poisson approximation was not used.

Chris Trentman
Chris Trentman
Numerade Educator
01:16

Problem 62

A new warehouse is being designed and a decision concerning the number of loading docks is required. There are two models based on truckarrival assumptions for the use of this warehouse, given that loading a truck requires 1 hour. Using the first model, we assume that the warehouse could be serviced by one of the many thousands of independent truckers who arrive randomly to obtain a load for delivery. It is known that, on average, 1 of these trucks would arrive each hour. For the second model, assume that the company hires a fleet of 10 trucks that are assigned full time to shipments from this warehouse. Under that assumption the trucks would arrive randomly, but the probability of any truck arriving during a given hour is 0.1 . Obtain the appropriate probability distribution for each of these assumptions and compare the results.

Dominador Tan
Dominador Tan
Numerade Educator
02:01

Problem 63

Compute the probability of 7 successes in a random sample of size $n=14$ obtained from a population of size $N=30$ that contains 15 successes.

Christopher Stanley
Christopher Stanley
Numerade Educator
01:50

Problem 64

Compute the probability of 9 successes in a random sample of size $n=20$ obtained from a population of size $N=80$ that contains 42 successes.

Christopher Stanley
Christopher Stanley
Numerade Educator
02:26

Problem 65

Compute the probability of 3 successes in a random sample of size $n=5$ obtained from a population of size $N=40$ that contains 25 successes.

Aishwarya Krishnakumar
Aishwarya Krishnakumar
Numerade Educator
01:35

Problem 66

Compute the probability of 8 successes in a random sample of size $n=15$ obtained from a population of size $N=100$ that contains 50 successes.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
10:36

Problem 67

A company receives a shipment of 16 items. A random sample of 4 items is selected, and the shipment is rejected if any of these items proves to be defective.
a. What is the probability of accepting a shipment containing 4 defective items?
b. What is the probability of accepting a shipment containing 1 defective item?
c. What is the probability of rejecting a shipment containing 1 defective item?

Willis James
Willis James
Numerade Educator
01:25

Problem 68

A committee of 8 members is to be formed from a group of 8 men and 8 women. If the choice of committee members is made randomly, what is the probability that precisely half of these members will be women?

Christopher Stanley
Christopher Stanley
Numerade Educator
02:04

Problem 69

A bond analyst was given a list of 12 corporate bonds. From that list she selected 3 whose ratings she felt were in danger of being downgraded in the next year. In actuality, a total of 4 of the 12 bonds on the list had their ratings downgraded in the next year. Suppose that the analyst had simply chosen 3 bonds randomly from this list. What is the probability that at least 2 of the chosen bonds would be among those whose ratings were to be downgraded in the next year?

Christopher Stanley
Christopher Stanley
Numerade Educator
01:21

Problem 70

A bank executive is presented with loan applications from 10 people. The profiles of the applicants are similar, except that 5 are minorities and 5 are not minorities. In the end the executive approves 6 of the applications. If these 6 approvals are chosen at random from the 10 applications, what is the probability that fewer than half the approvals will be of applications involving minorities?

Lucas Finney
Lucas Finney
Numerade Educator
16:45

Problem 71

A call center in Perth, Australia receives an average of 1.3 calls per minute. By looking at the date, a Poisson discrete distribution is assumed for this variable. Calculate each of the following.
a. The probability of receiving no calls in the first minute of its office hours.
b. The probability of receiving 1 call in the first minute.
c. The probability of receiving 3 calls in the first minute.

Rowan Ahmed
Rowan Ahmed
Numerade Educator
03:24

Problem 72

Consider the joint probability distribution:
$$
\begin{array}{cccc}
\hline & & \underline {X} \\
& & 1 & 2 \\
\hline Y & 0 & 0.25 & 0.25 \\
& 1 & 0.25 & 0.25 \\
\hline
\end{array}
$$
a. Compute the marginal probability distributions for $X$ and $Y$.
b. Compute the covariance and correlation for $X$ and $Y$.
c. Compute the mean and variance for the linear function $W=X+Y$.

Christopher Stanley
Christopher Stanley
Numerade Educator
03:24

Problem 73

Consider the joint probability distribution:
$$
\begin{array}{cccc}
\hline & & \underline {X} \\
& & 1 & 2 \\
\hline Y & 0 & 0.30 & 0.20 \\
& 1 & 0.25 & 0.25 \\
\hline
\end{array}
$$
a. Compute the marginal probability distributions for $X$ and $Y$.
b. Compute the covariance and correlation for $X$ and $Y$.
c. Compute the mean and variance for the linear function $W=2 X+Y$.

Christopher Stanley
Christopher Stanley
Numerade Educator
02:08

Problem 74

Consider the joint probability distribution:
$$
\begin{array}{cccc}
\hline & & \underline {X} \\
& & 1 & 2 \\
\hline Y & 0 & 0.70 & 0.0 \\
& 1 & 0.0 & 0.30 \\
\hline
\end{array}
$$
a. Compute the marginal probability distributions for $X$ and $Y$.
b. Compute the covariance and correlation for $X$ and $Y$.
c. Compute the mean and variance for the linear function $W=3 X+4 Y$.

Christopher Stanley
Christopher Stanley
Numerade Educator
02:08

Problem 75

Consider the joint probability distribution:
$$
\begin{array}{cccc}
\hline & & \underline {X} \\
& & 1 & 2 \\
\hline Y & 0 & 0.0 & 0.60 \\
& 1 & 0.40 & 0.0 \\
\hline
\end{array}
$$
a. Compute the marginal probability distributions for $X$ and $Y$.
b. Compute the covariance and correlation for $X$ and $Y$.
c. Compute the mean and variance for the linear function $W=2 X-4 Y$.

Christopher Stanley
Christopher Stanley
Numerade Educator
01:01

Problem 76

Consider the joint probability distribution:
$$
\begin{array}{cccc}
\hline & & \underline {X} \\
& & 1 & 2 \\
\hline Y & 0 & 0.70 & 0.0 \\
& 1 & 0.0 & 0.30 \\
\hline
\end{array}
$$
a. Compute the marginal probability distributions for $X$ and $Y$.
b. Compute the covariance and correlation for $X$ and $Y$.
c. Compute the mean and variance for the linear function $W=10 X-8 Y$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
04:21

Problem 77

A researcher suspected that the number of betweenmeal snacks eaten by students in a day during final examinations might depend on the number of tests a student had to take on that day. The accompanying table shows joint probabilities, estimated from a survey.
$$
\begin{array}{ccccc}
\hline \text { Number of } & \underline {\text { Number of Tests }(\mathrm{X})} \\
\text { Snacks }(Y) & 0 & 1 & 2 & 3 \\
\hline 0 & 0.07 & 0.09 & 0.06 & 0.01 \\
1 & 0.07 & 0.06 & 0.07 & 0.01 \\
2 & 0.06 & 0.07 & 0.14 & 0.03 \\
3 & 0.02 & 0.04 & 0.16 & 0.04 \\
\hline
\end{array}
$$
a. Find the probability distribution of $X$ and compute the mean number of tests taken by students on that day.
b. Find the probability distribution of $Y$ and, hence, the mean number of snacks eaten by students on that day.
c. Find and interpret the conditional probability distribution of $Y$, given that $X=3$.
d. Find the covariance between $X$ and $Y$.
e. Are number of snacks and number of tests independent of each other?

Nick Johnson
Nick Johnson
Numerade Educator
02:28

Problem 78

A real estate agent is interested in the relationship between the number of lines in a newspaper advertisement for an apartment and the volume of inquiries from potential renters. Let volume of inquiries be denoted by the random variable $X$, with the value 0 for little interest, 1 for moderate interest, and 2 for strong interest. The real estate agent used historical records to compute the joint probability distribution shown in the accompanying table.
$$
\begin{array}{cccc}
\hline {\begin{array}{c}
\text { Number } \\
\text { of Lines }(n)
\end{array}} & \underline {\text { Number of Inquiries }(X)} \\
& 0 & 1 & 2 \\
\hline 3 & 0.09 & 0.14 & 0.07 \\
4 & 0.07 & 0.23 & 0.16 \\
5 & 0.03 & 0.10 & 0.11 \\
\hline
\end{array}
$$
a. Find the joint cumulative probability at $X=1, Y=4$, and interpret your result.
b. Find and interpret the conditional probability distribution for $Y$, given $X=0$.
c. Find and interpret the conditional probability distribution for $X$, given $Y=4$.
d. Find and interpret the covariance between $X$ and $Y$.
e. Are number of lines in the advertisement and volume of inquiries independent of one another?

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:00

Problem 79

The accompanying table shows, for credit-card holders with one to three cards, the joint probabilities for number of cards owned $(X)$ and number of credit purchases made in a week $(Y)$.
$$
\begin{array}{cccccc}
\hline \begin{array}{c}
\text { Number of } \\
\text { Cards }(X)
\end{array} & \underline{\text { Number of Purchases in Week }(n)} \\
& 0 & 1 & 2 & 3 & 4 \\
\hline 1 & 0.08 & 0.13 & 0.09 & 0.06 & 0.03 \\
2 & 0.03 & 0.08 & 0.08 & 0.09 & 0.07 \\
3 & 0.01 & 0.03 & 0.06 & 0.08 & 0.08 \\
\hline
\end{array}
$$
a. For a randomly chosen person from this group, what is the probability distribution for number of purchases made in a week?
b. For a person in this group who has three cards, what is the probability distribution for number of purchases made in a week?
c. Are number of cards owned and number of purchases made statistically independent?

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 80

A market researcher wants to determine whether a new model of a personal computer that had been advertised on a late-night talk show had achieved more brand-name recognition among people who watched the show regularly than among people who did not. After conducting a survey, it was found that $15 \%$ of all people both watched the show regularly and could correctly identify the product. Also, $16 \%$ of all people regularly watched the show and $45 \%$ of all people could correctly identify the product. Define a pair of random variables as follows:
$$
\begin{array}{llll}
\hline X=1 & \text { if regularly watch the show } & X=0 & \text { otherwise } \\
\hline Y=1 & \text { if product correctly identified } & Y=0 & \text { otherwise } \\
\hline
\end{array}
$$
a. Find the joint probability distribution of $X$ and $Y$.
b. Find the conditional probability distribution of $Y$, given $X=1$.
c. Find and interpret the covariance between $X$ and $Y$.

Victor Salazar
Victor Salazar
Numerade Educator
06:07

Problem 81

A college bookseller makes calls at the offices of professors and forms the impression that professors are more likely to be away from their offices on Friday than any other working day. A review of the records of calls, $1 / 5$ of which are on Fridays, indicates that for $16 \%$ of Friday calls, the professor is away from the office, while this occurs for only $12 \%$ of calls on every other working day. Define the random variables as follows:
$$
\begin{array}{llll}
\hline X=1 & \text { if call is made on a Friday } & X=0 & \text { otherwise } \\
\hline Y=1 & \text { if professor is away from the office } & Y=0 & \text { otherwise } \\
\hline
\end{array}
$$
a. Find the joint probability distribution of $X$ and $Y$.
b. Find the conditional probability distribution of $Y$, given $X=0$.
c. Find the marginal probability distributions of $X$ and $Y$.
d. Find and interpret the covariance between $X$ and $Y$.

Tatiana Graham
Tatiana Graham
Numerade Educator
02:04

Problem 82

A restaurant manager receives occasional complaints about the quality of both the food and the service. The marginal probability distributions for the number of weekly complaints in each category are shown in the accompanying table. If complaints about food and service are independent of each other, find the joint probability distribution.
$$
\begin{array}{cccc}
\hline \begin{array}{c}
\text { Number } \\
\text { of Food } \\
\text { Complaints }
\end{array} & \text { Probability } & \begin{array}{c}
\text { Number } \\
\text { of Service } \\
\text { Complaints }
\end{array} & \text { Probability } \\
\hline 0 & 0.12 & 0 & 0.18 \\
1 & 0.29 & 1 & 0.38 \\
2 & 0.42 & 2 & 0.34 \\
3 & 0.17 & 3 & 0.10 \\
\hline
\end{array}
$$

Hoan Nguyen
Hoan Nguyen
Numerade Educator
06:41

Problem 83

Refer to the information in the previous exercise. Find the mean and standard deviation of the total number of complaints received in a week. Having reached this point, you are concerned that the numbers of food and service complaints may not be independent of each other. However, you have no information about the nature of their dependence. What can you now say about the mean and standard deviation of the total number of complaints received in a week?

Arulmozhi T
Arulmozhi T
Numerade Educator
09:12

Problem 84

A company has 5 representatives covering large territories and 10 representatives covering smaller territories. The probability distributions for the numbers of orders received by each of these types of representatives in a day are shown in the accompanying table. Assuming that the number of orders received by any representative is independent of the number received by any other, find the mean and standard deviation of the total number of orders received by the company in a day.
$$
\begin{array}{cccc}
\hline \begin{array}{l}
\text { Numbers of } \\
\text { Orders (Large } \\
\text { Territories) }
\end{array} & \text { Probability } & \begin{array}{c}
\text { Numbers } \\
\text { of Orders } \\
\text { (Smaller } \\
\text { Territories) }
\end{array} & \text { Probability } \\
\hline 0 & 0.08 & 0 & 0.18 \\
1 & 0.16 & 1 & 0.26 \\
2 & 0.28 & 2 & 0.36 \\
3 & 0.32 & 3 & 0.13 \\
4 & 0.10 & 4 & 0.08 \\
5 & 0.06 & & \\
\hline
\end{array}
$$

Barsha Rana
Barsha Rana
Numerade Educator
01:46

Problem 85

As an investment advisor, you tell a client that an investment in a mutual fund has (over the next year) a higher expected return than an investment in the money market. The client then asks the following questions:
a. Does that imply that the mutual fund will certainly yield a higher return than the money market?
b. Does it follow that I should invest in the mutual fund rather than in the money market? How would you reply?

Majid Borumand
Majid Borumand
Numerade Educator
01:39

Problem 86

A contractor estimates the probabilities for the number of days required to complete a certain type of construction project as follows:
$$
\begin{array}{lccccc}
\hline \text { Time (days) } & 1 & 2 & 3 & 4 & 5 \\
\hline \text { Probability } & 0.05 & 0.20 & 0.35 & 0.30 & 0.10 \\
\hline
\end{array}
$$
a. What is the probability that a randomly chosen project will take less than 3 days to complete?
b. Find the expected time to complete a project.
c. Find the standard deviation of time required to complete a project.
d. The contractor's project cost is made up of two parts-a fixed cost of $$\$ 20,000$$, plus $$\$ 2,000$$ for each day taken to complete the project. Find the mean and standard deviation of total project cost.
e. If three projects are undertaken, what is the probability that at least two of them will take at least 4 days to complete, assuming independence of individual project completion times?

Christopher Stanley
Christopher Stanley
Numerade Educator
02:40

Problem 87

A car salesperson estimates the following probabilities for the number of cars that she will sell in the next week:
$$
\begin{array}{lcccccc}
\hline \text { Number of cars } & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline \text { Probability } & 0.10 & 0.20 & 0.35 & 0.16 & 0.12 & 0.07 \\
\hline
\end{array}
$$
a. Find the expected number of cars that will be sold in the week.
b. Find the standard deviation of the number of cars that will be sold in the week.
c. The salesperson receives a salary of $$\$ 250$$ for the week, plus an additional $$\$ 300$$ for each car sold. Find the mean and standard deviation of her total salary for the week.
d. What is the probability that the salesperson's salary for the week will be more than $$\$ 1,000$$ ?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
08:57

Problem 88

A multiple-choice test has nine questions. For each question there are four possible answers from which to select. One point is awarded for each correct answer, and points are not subtracted for incorrect answers. The instructor awards a bonus point if the students spell their name correctly. A student who has not studied for this test decides to choose an answer for each question at random.
a. Find the expected number of correct answers for the student on these nine questions.
b. Find the standard deviation of the number of correct answers for the student on these nine questions.
c. The student spells his name correctly:
i Find the expected total score on the test for this student.
ii Find the standard deviation of his total score on the test.

Jeremiah Mbaria
Jeremiah Mbaria
Numerade Educator
02:12

Problem 89

Develop realistic examples of pairs of random variables for which you would expect to find the following:
a. Positive covariance
b. Negative covariance
c. Zero covariance

Christopher Stanley
Christopher Stanley
Numerade Educator
02:40

Problem 90

A long-distance taxi service owns four vehicles. These are of different ages and have different repair records. The probabilities that, on any given day, each vehicle will be available for use are $0.95,0.90,0.90$, and $0.80$ .
Whether one vehicle is available is independent of whether any other vehicle is available.
a. Find the probability distribution for the number of vehicles available for use on a given day.
b. Find the expected number of vehicles available for use on a given day.
c. Find the standard deviation of the number of vehicles available for use on a given day.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:26

Problem 91

Students in a college were classified according to years in school $(X)$ and number of visits to a museum in the last year $(Y=0$ for no visits, 1 for one visit, 2 for more than one visit). The joint probabilities in the accompanying table were estimated for these random variables.
$$
\begin{array}{ccccc}
\hline {\begin{array}{c}
\text { Number of } \\
\text { Visits }(Y)
\end{array}} & \underline {\text { Years in School }(X)} \\
& 1 & 2 & 3 & 4 \\
\hline 0 & 0.07 & 0.05 & 0.03 & 0.02 \\
1 & 0.13 & 0.11 & 0.17 & 0.15 \\
2 & 0.04 & 0.04 & 0.09 & 0.10 \\
\hline
\end{array}
$$
a. Find the probability that a randomly chosen student has not visited a museum in the last year.
b. Find the means of the random variables $X$ and $Y$.
c. Find and interpret the covariance between the random variables $X$ and $Y$.

Nick Johnson
Nick Johnson
Numerade Educator
01:12

Problem 92

A basketball team's star 3-point shooter takes six 3-point shots in a game. Historically, she makes $40 \%$ of all 3-point shots taken in a game. State at the outset what assumptions you have made.
a. Find the probability that she will make at least two shots.
b. Find the probability that she will make exactly three shots.
c. Find the mean and standard deviation of the number of shots she made.
d. Find the mean and standard deviation of the total number of points she scored as a result of these shots.

Nick Johnson
Nick Johnson
Numerade Educator
01:47

Problem 93

It is estimated that $55 \%$ of the freshmen entering a particular college will graduate from that college in four years.
a. For a random sample of 5 entering freshmen, what is the probability that exactly 3 will graduate in four years?
b. For a random sample of 5 entering freshmen, what is the probability that a majority will graduate in four years?
c. 80 entering freshmen are chosen at random. Find the mean and standard deviation of the proportion of these 80 that will graduate in four years.

Foster Wisusik
Foster Wisusik
Numerade Educator
02:47

Problem 94

The World Series of baseball is to be played by team A and team B. The first team to win four games wins the series. Suppose that team A is the better team, in the sense that the probability is 0.6 that team A will win any specific game. Assume also that the result of any game is independent of that of any other.
a. What is the probability that team A will win the series?
b. What is the probability that a seventh game will be needed to determine the winner?
c. Suppose that, in fact, each team wins two of the first four games.
i What is the probability that team A will win the series?
ii What is the probability that a seventh game will be needed to determine the winner?

Joe Lesueur
Joe Lesueur
Numerade Educator
02:52

Problem 95

Using detailed cash-flow information, a financial analyst claims to be able to spot companies that are likely candidates for bankruptcy. The analyst is presented with information on the past records of 15 companies and told that, in fact, 5 of these have failed. He selects as candidates for failure 5 companies from the group of 15 . In fact, 3 of the 5 companies selected by the analyst were among those that failed. Evaluate the financial analyst's performance on this test of his ability to detect failed companies.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:04

Problem 96

A team of 5 analysts is about to examine the earnings prospects of 20 corporations. Each of the 5 analysts will study 4 of the corporations. These analysts are not equally competent. In fact, one of them is a star, having an excellent record of anticipating changing trends. Ideally, management would like to allocate the 4 corporations whose earnings will deviate most from past trends to this analyst. However, lacking this information, management allocates corporations to analysts randomly. What is the probability that at least 2 of the 4 corporations whose earnings will deviate most from past trends are allocated to the star analyst?

Christopher Stanley
Christopher Stanley
Numerade Educator
View

Problem 97

A new brand of pizza is going to be sold in Park & Shop, and a market-research company in Admiralty (Hong Kong) has forecast that successful new brands normally obtain a $10 \%$ market share for the product in the first year. However, top management wants to achieve $12 \%$. You may assume a normal distribution with a standard deviation of $3 \%$ (risk on the estimates). Determine each of the following.
a. The probability that the new pizza will actually achieve the target.
b. The probability of failure.
c. The probability of being even more successful, with $18 \%$ of market share in the first year.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:00

Problem 98

A recent estimate suggested that, of all individuals and couples reporting income in excess of $$\$ 200,000$$, $6.5 \%$ either paid no federal tax or paid tax at an effective rate of less than $15 \%$. A random sample of 100 of those reporting income in excess of $$\$ 200,000$$ was taken. What is the probability that more than 2 of the sample members either paid no federal tax or paid tax at an effective rate of less than $15 \%$ ?

Christopher Stanley
Christopher Stanley
Numerade Educator
02:14

Problem 99

Your computer is in serious need of repair. You have estimated that the breakdowns occur on average 3.5 times per week. If you are right and the breakdown variable is a Poisson distribution, calculate the following.
a. The probability that for an entire week your computer runs with no problems.
b. The probability of getting only 1 shutdown.
c. The probability of getting 5 shutdowns.

Christopher Stanley
Christopher Stanley
Numerade Educator
04:49

Problem 100

George Allen has asked you to analyze his stock portfolio, which contains 10 shares of stock D and 5 shares of stock C. The joint probability distribution of the stock prices is shown in Table 4.10. Compute the mean and variance for the total value of his stock portfolio.
Table 4.10 Joint Probability Distribution for Stock Prices
$$
\begin{array}{|ccccc|}
\hline {\begin{array}{c}
\text { Stock C } \\
\text { Price }
\end{array}} & {\text { Stock D Price }} \\
\hline & \$ 40 & \$ 50 & \$ 60 & \$ 70 \\
\$ 45 & 0.00 & 0.00 & 0.05 & 0.20 \\
\$ 50 & 0.05 & 0.00 & 0.05 & 0.10 \\
\$ 55 & 0.10 & 0.05 & 0.00 & 0.05 \\
\$ 60 & 0.20 & 0.10 & 0.05 & 0.00 \\
\hline
\end{array}
$$

James Kiss
James Kiss
Numerade Educator
02:53

Problem 101

Consider a country that imports steel and exports automobiles. The value per unit of cars exported is measured in units of thousands of dollars per car by the random variable $X$. The value per unit of steel imported is measured in units of thousands of dollars per ton of steel by the random variable $Y$. Suppose that the country annually exports 10 cars and imports 5 tons of steel. Compute the mean and variance of the trade balance, where the trade balance is the total dollars received for all cars exported minus the total dollars spent for all steel imported. The joint probability distribution for the prices of cars and steel is shown in Table 4.11.
Table 4.11 Joint Distribution of Automobile and Steel Prices

Dominador Tan
Dominador Tan
Numerade Educator
03:11

Problem 102

Delta International delivers approximately one million packages a day between East Asia and the United States. A random sample of the daily number of package delivery failures over the past six months provided the following results: $15,10,8,16,12,11,9,8$, $12,9,10,8,7,16,14,12,10,9,8,11$. There was nothing unusual about the operations during these days and, thus, the results can be considered typical. Using these data and your understanding of the delivery process answer the following:
a. What probability model should be used and why?
b. What is the probability of 10 or more failed deliveries on a typical future day?
c. What is the probability of less than 6 failed deliveries?
d. Find the number of failures such that the probability of exceeding this number is $10 \%$ or less.

Manisha Sarker
Manisha Sarker
Numerade Educator
03:10

Problem 103

Bright Star Financial Advisers receives a mean of 19.5 applications per week for a personal financial review. Each review requires one day of an analyst's time to prepare a review. Assume that requests received during any week are assigned to an analyst for completion during the following week. If the analysis is not completed during the second week the customer will cancel.
a. How many analysts should be hired so that the company can claim that $90 \%$ of the reviews will be completed during the second week?
b. What is the probability that two of the analysts hired for part a would have no clients for an entire week?
c. Suppose that they decided to hire one less analyst than determined in part (a). What is the probability that customers would cancel given this staffing level?
d. Given the number of analysts hired in part c, what is the probability that two analysts would be idle for an entire week?

Amany Waheeb
Amany Waheeb
Numerade Educator
04:18

Problem 104

Federated South Insurance Company has developed a new screening program for selecting new sales agents. Their past experience indicates that $20 \%$ of the new agents hired fail to produce the minimum sales in their first year and are dismissed. Their expectation is that this new screening program will reduce the percentage of failed new agents to $15 \%$ or less. If that occurs, they would save $$\$ 1,000,000$$ in recruiting and training costs each year. At the end of the first year they want to develop an evaluation to determine if the new program is successful. The following questions are an important part of their research design.
A total of 20 new agents were selected.
a. If this group performs at the same level as past groups, what is the probability 17 or more successfully meet their minimum sales goals in the first year?
b. What is the probability 19 or more reach their minimum sales goals given performance at the same level?
c. If the program has actually increased the probability of success to 0.85 for each new agent, what is the probability that 17 or more meet their minimum sales goals?
d. Given the expected improvement, what is the probability that 19 or more reach their minimum sales goals?

John Long
John Long
Numerade Educator
06:54

Problem 105

Yoshida Toimi is a candidate for the mayor of a medium-sized Midwestern city. If he receives more than $50 \%$ of the votes, he will win the election. Prior to the election, his campaign staff is planning to ask 100 randomly selected voters if they support Yoshida.
a. How many positive responses from this sample of 100 is required so that the probability of $50 \%$ or more voters supporting him is 0.95 or more?
b. Carefully state the assumptions required for your answer in part (a).
c. Suppose the campaign is able to ask 400 randomly selected voters. Now what is your answer to the question in part (a)?

Ivan Kochetkov
Ivan Kochetkov
Numerade Educator
02:38

Problem 106

Faschip, Ltd., is a new African manufacturer of notebook computers. Their quality target is that $99.999 \%$ of the computers they produce will perform exactly as promised in the descriptive literature. In order to monitor their quality performance they include with each computer a large piece of paper that includes a direct-toll-free-phone number to the Senior Vice President of Manufacturing that can be used if the computer does not perform as promised. In the first year Faschip sells 1,000,000 computers.
a. If they are achieving their quality target, what is the probability that they will receive fewer than 5 calls? If this occurs what would be a reasonable conclusion about their quality program?
b. If they are achieving their quality target, what is the probability that they will receive more than 15 calls? If this occurs, what would be a reasonable conclusion about their quality program?

Sheryl Ezze
Sheryl Ezze
Numerade Educator