Chapter 6

Compute the mean and variance of the following discrete probability distribution. $$ \begin{array}{|cc|} \hline x & P(x) \\ \hline 0 & .2 \\ 1 & .4 \\ 2 & .3 \\ 3 & .1 \\ \hline \end{array} $$

Compute the mean and variance of the following discrete probability distribution. $$ \begin{array}{|rr|} \hline {}{\underline{\phantom{xx}}} {\boldsymbol{x}} & \boldsymbol{P}(\boldsymbol{x}) \\ \hline 2 & .5 \\ 8 & .3 \\ 10 & .2 \\ \hline \end{array} $$

Which of these variables are discrete and which are continuous random variables? a. The number of new accounts established by a salesperson in a year. b. The time between customer arrivals to a bank ATM. c. The number of customers in Big Nick's barber shop. d. The amount of fuel in your car's gas tank. e. The number of minorities on a jury. f. The outside temperature today.

The information below is the number of daily emergency service calls made by the volunteer ambulance service of Walterboro, South Carolina, for the last 50 days. To explain, there were 22 days on which there were two emergency calls, and 9 days on which there were three emergency calls. $$ \begin{array}{|cc|} \hline \text { Number of Calls } & \text { Frequency } \\ \hline 0 & 8 \\ 1 & 10 \\ 2 & 22 \\ 3 & 9 \\ 4 & 1 \\ \hline \text { Total } & 50 \\ \hline \end{array} $$ a. Convert this information on the number of calls to a probability distribution. b. Is this an example of a discrete or continuous probability distribution? c. What is the mean number of emergency calls per day? d. What is the standard deviation of the number of calls made daily?

The director of admissions at Kinzua University in Nova Scotia estimated the distribution of student admissions for the fall semester on the basis of past experience. What is the expected number of admissions for the fall semester? Compute the variance and the standard deviation of the number of admissions. $$ \begin{array}{|cc|} \hline \text { Admissions } & \text { Probability } \\ \hline 1,000 & .6 \\ 1,200 & .3 \\ 1,500 & .1 \\ \hline \end{array} $$

Levinson's Department Store is having a special sale this weekend. Customers charging purchases of more than \(\$ 50\) to their store credit card will be given a special Levinson's Lottery card. The customer will scratch off the card, which will indicate the amount to be taken off the total amount of the purchase. Listed below are the amount of the prize and the percent of the time that amount will be deducted from the total amount of the purchase. $$ \begin{array}{|rc|} \hline \text { Prize Amount } & \text { Probability } \\ \hline \$ 10 & .50 \\ 25 & .40 \\ 50 & .08 \\ 100 & .02 \\ \hline \end{array} $$ a. What is the mean amount deducted from the total purchase amount? b. What is the standard deviation of the amount deducted from the total purchase?

In a binomial situation, \(n=4\) and \(\pi=.25 .\) Determine the probabilities of the following events using the binomial formula. a. \(x=2\) b. \(x=3\)

In a binomial situation, \(n=5\) and \(\pi=.40 .\) Determine the probabilities of the following events using the binomial formula. a. \(x=1\) b. \(x=2\)

An American Society of Investors survey found \(30 \%\) of individual investors have used a discount broker. In a random sample of nine individuals, what is the probability: a. Exactly two of the sampled individuals have used a discount broker? b. Exactly four of them have used a discount broker? c. None of them has used a discount broker?

The U.S. Postal Service reports \(95 \%\) of first-class mail within the same city is delivered within 2 days of the time of mailing. Six letters are randomly sent to different locations. a. What is the probability that all six arrive within 2 days? b. What is the probability that exactly five arrive within 2 days? c. Find the mean number of letters that will arrive within 2 days. d. Compute the variance and standard deviation of the number that will arrive within 2 days.

Industry standards suggest that \(10 \%\) of new vehicles require warranty service within the first year. Jones Nissan in Sumter, South Carolina, sold 12 Nissans yesterday. a. What is the probability that none of these vehicles requires warranty service? b. What is the probability exactly one of these vehicles requires warranty service? c. Determine the probability that exactly two of these vehicles require warranty service. d. Compute the mean and standard deviation of this probability distribution.

A telemarketer makes six phone calls per hour and is able to make a sale on \(30 \%\) of these contacts. During the next 2 hours, find: a. The probability of making exactly four sales. b. The probability of making no sales. c. The probability of making exactly two sales. d. The mean number of sales in the 2 -hour period.

A recent survey by the American Accounting Association revealed \(23 \%\) of students graduating with a major in accounting select public accounting. Suppose we select a sample of 15 recent graduates. a. What is the probability two select public accounting? b. What is the probability five select public accounting? c. How many graduates would you expect to select public accounting?

It is reported that \(41 \%\) of American households use a cell phone exclusively for their telephone service. In a sample of eight households: a. Find the probability that no household uses a cell phone as their exclusive telephone service. b. Find the probability that exactly 5 households exclusively use a cell phone for telephone service. c. Find the mean number of households exclusively using cell phones.

In a binomial distribution, \(n=8\) and \(\pi=.30 .\) Find the probabilities of the following events. a. \(x=2\) b. \(x \leq 2\) (the probability that \(x\) is equal to or less than 2 ). c. \(x \geq 3\) (the probability that \(x\) is equal to or greater than 3 ).

In a binomial distribution, \(n=12\) and \(\pi=.60 .\) Find the following probabilities. a. \(x=5\). b. \(x \leq 5\) c. \(x \geq 6\)

In a recent study, \(90 \%\) of the homes in the United States were found to have large-screen TVs. In a sample of nine homes, what is the probability that: a. All nine have large-screen TVs? b. Less than five have large-screen TVs? c. More than five have large-screen TVs? d. At least seven homes have large-screen TVs?

A manufacturer of window frames knows from long experience that \(5 \%\) of the production will have some type of minor defect that will require an adjustment. What is the probability that in a sample of 20 window frames: a. None will need adjustment? b. At least one will need adjustment? c. More than two will need adjustment?

The speed with which utility companies can resolve problems is very important. GTC, the Georgetown Telephone Company, reports it can resolve customer problems the same day they are reported in \(70 \%\) of the cases. Suppose the 15 cases reported today are representative of all complaints. a. How many of the problems would you expect to be resolved today? What is the standard deviation? b. What is the probability 10 of the problems can be resolved today? c. What is the probability 10 or 11 of the problems can be resolved today? d. What is the probability more than 10 of the problems can be resolved today?

It is asserted that \(80 \%\) of the cars approaching an individual toll booth in New Jersey are equipped with an E-ZPass transponder. Find the probability that in a sample of six cars: a. All six will have the transponder. b. At least three will have the transponder. c. None will have a transponder.

In a Poisson distribution, \(\mu=0.4\). a. What is the probability that \(x=0 ?\) b. What is the probability that \(x>0 ?\)

In a Poisson distribution, \(\mu=4\) a. What is the probability that \(x=2 ?\) b. What is the probability that \(x \leq 2 ?\) c. What is the probability that \(x>2 ?\)

Ms. Bergen is a loan officer at Coast Bank and Trust. From her years of experience, she estimates that the probability is .025 that an applicant will not be able to repay his or her installment loan. Last month she made 40 loans. a. What is the probability that three loans will be defaulted? b. What is the probability that at least three loans will be defaulted?

Automobiles arrive at the Elkhart exit of the Indiana Toll Road at the rate of two per minute. The distribution of arrivals approximates a Poisson distribution. a. What is the probability that no automobiles arrive in a particular minute? b. What is the probability that at least one automobile arrives during a particular minute?

It is estimated that \(0.5 \%\) of the callers to the Customer Service department of Dell Inc. will receive a busy signal. What is the probability that of today's 1,200 callers, at least 5 received a busy signal?

What is the difference between a random variable and a probability distribution?

For each of the following, indicate whether the random variable is discrete or continuous. a. The length of time to get a haircut. b. The number of cars a jogger passes each morning while running. c. The number of hits for a team in a high school girls' softball game. d. The number of patients treated at the South Strand Medical Center between 6 and 10 p.m. each night. e. The distance your car traveled on the last fill-up. f. The number of customers at the Oak Street Wendy's who used the drive- through facility. g. The distance between Gainesville, Florida, and all Florida cities with a population of at least 50,000 .

An investment will be worth \(\$ 1,000, \$ 2,000,\) or \(\$ 5,000\) at the end of the year. The probabilities of these values are \(.25, .60,\) and \(.15,\) respectively. Determine the mean and variance of the investment's dollar value.

Croissant Bakery Inc. offers special decorated cakes for birthdays, weddings, and other occasions. It also has regular cakes available in its bakery. The following table gives the total number of cakes sold per day and the corresponding probability. Compute the mean, variance, and standard deviation of the number of cakes sold per day. $$ \begin{array}{cc} \hline \text { Number of Cakes } & \\ \text { Sold in a Day } & \text { Probability } \\ \hline 12 & .25 \\ 13 & .40 \\ 14 & 25 \\ 15 & .10 \\ \hline \end{array} $$

In a recent study, \(35 \%\) of people surveyed indicated chocolate was their favorite flavor of ice cream. Suppose we select a sample of 10 people and ask them to name their favorite flavor of ice cream. a. How many of those in the sample would you expect to name chocolate? b. What is the probability exactly four of those in the sample name chocolate? c. What is the probability four or more name chocolate? \(?\) ?

Thirty percent of the population in a Southwestern community are Spanishspeaking Americans. A Spanish-speaking person is accused of killing a non-Spanishspeaking American and goes to trial. Of the first 12 potential jurors, only 2 are Spanish-speaking Americans, and 10 are not. The defendant's lawyer challenges the jury selection, claiming bias against her client. The government lawyer disagrees, saying that the probability of this particular jury composition is common. Compute the probability and discuss the assumptions.

An auditor for Health Maintenance Services of Georgia reports \(40 \%\) of policyholders 55 years or older submit a claim during the year. Fifteen policyholders are randomly selected for company records. a. How many of the policyholders would you expect to have filed a claim within the last year? b. What is the probability that 10 of the selected policyholders submitted a claim last year? c. What is the probability that 10 or more of the selected policyholders submitted a claim last year? d. What is the probability that more than 10 of the selected policyholders submitted a claim last year?

Tire and Auto Supply is considering a 2 -for- 1 stock split. Before the transaction is finalized, at least two-thirds of the 1,200 company stockholders must approve the proposal. To evaluate the likelihood the proposal will be approved, the CFO selected a sample of 18 stockholders. He contacted each and found 14 approved of the proposed split. What is the likelihood of this event, assuming two-thirds of the stockholders approve?

A federal study reported that \(7.5 \%\) of the U.S. workforce has a drug problem. A drug enforcement official for the state of Indiana wished to investigate this statement. In her sample of 20 employed workers: a. How many would you expect to have a drug problem? What is the standard deviation? b. What is the likelihood that none of the workers sampled has a drug problem? c. What is the likelihood at least one has a drug problem?

The Bank of Hawaii reports that \(7 \%\) of its credit card holders will default at some time in their life. The Hilo branch just mailed out 12 new cards today. a. How many of these new cardholders would you expect to default? What is the standard deviation? b. What is the likelihood that none of the cardholders will default? c. What is the likelihood at least one will default?

Recent statistics suggest that \(15 \%\) of those who visit a retail site on the Internet make a purchase. A retailer wished to verify this claim. To do so, she selected a sample of 16 "hits" to her site and found that 4 had actually made a purchase. a. What is the likelihood of exactly four purchases? b. How many purchases should she expect? c. What is the likelihood that four or more "hits" result in a purchase?

Acceptance sampling is a statistical method used to monitor the quality of purchased parts and components. To ensure the quality of incoming parts, a purchaser or manufacturer normally samples 20 parts and allows one defect. a. What is the likelihood of accepting a lot that is \(1 \%\) defective? b. If the quality of the incoming lot was actually \(2 \%,\) what is the likelihood of accepting it? c. If the quality of the incoming lot was actually \(5 \%,\) what is the likelihood of accepting it?

Unilever Inc. recently developed a new body wash with a scent of ginger. Their research indicates that \(30 \%\) of men like the new scent. To further investigate, Unilever's marketing research group randomly selected 15 men and asked them if they liked the scent. What is the probability that six or more men like the ginger scent in the body wash?

Dr. Richmond, a psychologist, is studying the daytime television viewing habits of college students. She believes \(45 \%\) of college students watch soap operas during the afternoon. To further investigate, she selects a sample of \(10 .\) a. Develop a probability distribution for the number of students in the sample who watch soap operas. b. Find the mean and the standard deviation of this distribution. c. What is the probability of finding exactly four students who watch soap operas? d. What is the probability less than half of the students selected watch soap operas?

A recent study conducted by Penn, Shone, and Borland, on behalf of LastMinute .com, revealed that \(52 \%\) of business travelers plan their trips less than two weeks before departure. The study is to be replicated in the tri-state area with a sample of 12 frequent business travelers. a. Develop a probability distribution for the number of travelers who plan their trips within two weeks of departure. b. Find the mean and the standard deviation of this distribution. c. What is the probability exactly 5 of the 12 selected business travelers plan their trips within two weeks of departure? d. What is the probability 5 or fewer of the 12 selected business travelers plan their trips within two weeks of departure?

Suppose \(1.5 \%\) of the antennas on new Nokia cell phones are defective. For a random sample of 200 antennas, find the probability that: a. None of the antennas is defective. b. Three or more of the antennas are defective.

An internal study by the Technology Services department at Lahey Electronics revealed company employees receive an average of two non-work-related e-mails per hour. Assume the arrival of these e-mails is approximated by the Poisson distribution. a. What is the probability Linda Lahey, company president, received exactly one non-work-related e-mail between 4 p.m. and 5 p.m. yesterday? b. What is the probability she received five or more non-work-related e-mails during the same period? c. What is the probability she did not receive any non-work-related e-mails during the period?

Recent crime reports indicate that 3.1 motor vehicle thefts occur each minute in the United States. Assume that the distribution of thefts per minute can be approximated by the Poisson probability distribution. a. Calculate the probability exactly four thefts occur in a minute. b. What is the probability there are no thefts in a minute? c. What is the probability there is at least one theft in a minute?

Recent difficult economic times have caused an increase in the foreclosure rate of home mortgages. Statistics from the Penn Bank and Trust Company show their monthly foreclosure rate is now 1 loan out of every 136 loans. Last month the bank approved 300 loans. a. How many foreclosures would you expect the bank to have last month? b. What is the probability of exactly two foreclosures? c. What is the probability of at least one foreclosure?

According to the "January theory," if the stock market is up for the month of January, it will be up for the year. If it is down in January, it will be down for the year. According to an article in The Wall Street Journal, this theory held for 29 out of the last 34 years. Suppose there is no truth to this theory; that is, the probability it is either up or down is .50. What is the probability this could occur by chance? You will probably need a software package such as Excel or Minitab.

According to sales information in the first quarter of \(2016,2.7 \%\) of new vehicles sold in the United States were hybrids. This is down from \(3.3 \%\) for the same period a year earlier. An analyst's review of the data indicates that the reasons for the sales decline include the low price of gasoline and the higher price of a hybrid compared to similar vehicles. Let's assume these statistics remain the same for 2017 . That is, \(2.7 \%\) of new car sales are hybrids in the first quarter of 2017 . For a sample of 40 vehicles sold in the Richmond, Virginia, area: a. How many vehicles would you expect to be hybrid? b. Use the Poisson distribution to find the probability that five of the sales were hybrid vehicles. c. Use the binomial distribution to find the probability that five of the sales were hybrid vehicles.

A recent CBS News survey reported that \(67 \%\) of adults felt the U.S. Treasury should continue making pennies. Suppose we select a sample of 15 adults. a. How many of the 15 would we expect to indicate that the Treasury should continue making pennies? What is the standard deviation? b. What is the likelihood that exactly eight adults would indicate the Treasury should continue making pennies? c. What is the likelihood at least eight adults would indicate the Treasury should continue making pennies?