EXERCISES Section 3.4 (46-67)

46

Compute the following binomial probabilities directly from the formula for $b\left(x;n,p\right)$ : a. $b\left(3;8,.35\right)$ b. $b\left(5;8,.6\right)$ c. $P\left(3\le X\le 5\right)$ when $n=7$ and $p=.6$ d. $P\left(1\le X\right)$ when $n=9$ and $p=.1$

47

The article “Should You Report That Fender-Bender?” (Consumer Reports, Sept. 2013: 15) reported that 7 in 10 auto accidents involve a single vehicle (the article recommended always reporting to the insurance company an accident involving multiple vehicles). Suppose 15 accidents are randomly selected. Use Appendix Table A. 1 to answer each of the following questions.

a. What is the probability that at most 4 involve a single vehicle?

b. What is the probability that exactly 4 involve a single vehicle?

c. What is the probability that exactly 6 involve multiple vehicles?

d. What is the probability that between 2 and 4 , inclusive, involve a single vehicle?

e. What is the probability that at least 2 involve a single vehicle?

f. What is the probability that exactly 4 involve a single vehicle and the other 11 involve multiple vehicles?

48

NBC News reported on May 2, 2013, that 1 in 20 children in the United States have a food allergy of some sort. Consider selecting a random sample of 25 children and let $X$ be the number in the sample who have a food allergy. Then $X\sim \mathrm{Bin}\left(25,.05\right)$ .

a. Determine both $P\left(X\le 3\right)$ and $P\left(X<3\right)$ .

b. Determine $P\left(X\ge 4\right)$ .

c. Determine $P\left(1\le X\le 3\right)$ .

d. What are $E\left(X\right)$ and ${\sigma }_X$ ?

e. In a sample of 50 children, what is the probability that none has a food allergy?

49

A company that produces fine crystal knows from experience that $10\%$ of its goblets have cosmetic flaws and must be classified as “seconds.”

a. Among six randomly selected goblets, how likely is it that only one is a second?

b. Among six randomly selected goblets, what is the probability that at least two are seconds?

c. If goblets are examined one by one, what is the probability that at most five must be selected to find four that are not seconds?

50

A particular telephone number is used to receive both voice calls and fax messages. Suppose that $25\%$ of the incoming calls involve fax messages, and consider a sample of 25 incoming calls. What is the probability that

a. At most 6 of the calls involve a fax message?

b. Exactly 6 of the calls involve a fax message?

c. At least 6 of the calls involve a fax message?

d. More than 6 of the calls involve a fax message?

51

Refer to the previous exercise.

a. What is the expected number of calls among the 25 that involve a fax message?

b. What is the standard deviation of the number among the 25 calls that involve a fax message?

c. What is the probability that the number of calls among the 25 that involve a fax transmission exceeds the expected number by more than 2 standard deviations?

52

Suppose that $30\%$ of all students who have to buy a text for a particular course want a new copy (the successes!), whereas the other $70\%$ want a used copy. Consider randomly selecting 25 purchasers.

a. What are the mean value and standard deviation of the number who want a new copy of the book?

b. What is the probability that the number who want new copies is more than two standard deviations away from the mean value?

c. The bookstore has 15 new copies and 15 used copies in stock. If 25 people come in one by one to purchase this text, what is the probability that all 25 will get the type of book they want from current stock? [Hint: Let $X=$ the number who want a new copy. For what values of $X$ will all 25 get what they want?]

d. Suppose that new copies cost \ {100} and used copies cost \70. Assume the bookstore currently has 50 new copies and 50 used copies. What is the expected value of total revenue from the sale of the next 25 copies purchased? Be sure to indicate what rule of expected value you are using. Hint: Let $h\left(X\right)=$ the revenue when $X$ of the 25 purchasers want new copies. Express this as a linear function.

53

Exercise 30 (Section 3.3) gave the pmf of $Y$ , the number of traffic citations for a randomly selected individual insured by a particular company. What is the probability that among 15 randomly chosen such individuals

a. At least 10 have no citations?

b. Fewer than half have at least one citation?

c. The number that have at least one citation is between 5 and 10, inclusive?

54

A particular type of tennis racket comes in a midsize version and an oversize version. Sixty percent of all customers at a certain store want the oversize version.

a. Among ten randomly selected customers who want this type of racket, what is the probability that at least six want the oversize version?

b. Among ten randomly selected customers, what is the probability that the number who want the oversize version is within 1 standard deviation of the mean value?

c. The store currently has seven rackets of each version. What is the probability that all of the next ten customers who want this racket can get the version they want from current stock?

55

Twenty percent of all telephones of a certain type are submitted for service while under warranty. Of these, $60\%$ can be repaired, whereas the other $40\%$ must be replaced with new units. If a company purchases ten of these telephones, what is the probability that exactly two will end up being replaced under warranty?

56

The College Board reports that $2\%$ of the 2 million high school students who take the SAT each year receive special accommodations because of documented disabilities (Los Angeles Times, July 16, 2002). Consider a random sample of 25 students who have recently taken the test.

a. What is the probability that exactly 1 received a special accommodation?

b. What is the probability that at least 1 received a special accommodation?

c. What is the probability that at least 2 received a special accommodation?

d. What is the probability that the number among the 25 who received a special accommodation is within 2 standard deviations of the number you would expect to be accommodated?

e. Suppose that a student who does not receive a special accommodation is allowed 3 hours for the exam, whereas an accommodated student is allowed 4.5 hours. What would you expect the average time allowed the 25 selected students to be?

57

A certain type of flashlight requires two type-D batteries, and the flashlight will work only if both its batteries have acceptable voltages. Suppose that $90\%$ of all batteries from a certain supplier have acceptable voltages. Among ten randomly selected flashlights, what is the probability that at least nine will work? What assumptions did you make in the course of answering the question posed?

58

A very large batch of components has arrived at a distributor. The batch can be characterized as acceptable only if the proportion of defective components is at most .10. The distributor decides to randomly select 10 components and to accept the batch only if the number of defective components in the sample is at most 2 .

a. What is the probability that the batch will be accepted when the actual proportion of defectives is .01?.05? .10? .20? .25?

b. Let $p$ denote the actual proportion of defectives in the batch. A graph of $P$ (batch is accepted) as a function of $p$ , with $p$ on the horizontal axis and $P$ (batch is accepted) on the vertical axis, is called the operating characteristic curve for the acceptance sampling plan. Use the results of part (a) to sketch this curve for $0\le p\le 1$ .

c. Repeat parts (a) and (b) with ” 1 ” replacing ” 2 ” in the acceptance sampling plan.

d. Repeat parts (a) and (b) with “15” replacing “10” in the acceptance sampling plan.

e. Which of the three sampling plans, that of part (a), (c), or (d), appears most satisfactory, and why?

59

An ordinance requiring that a smoke detector be installed in all previously constructed houses has been in effect in a particular city for 1 year. The fire department is concerned that many houses remain without detectors. Let $p=$ the true proportion of such houses having detectors, and suppose that a random sample of 25 homes is inspected. If the sample strongly indicates that fewer than $80\%$ of all houses have a detector, the fire department will campaign for a mandatory inspection program. Because of the costliness of the program, the department prefers not to call for such inspections unless sample evidence strongly argues for their necessity. Let $X$ denote the number of homes with detectors among the 25 sampled. Consider rejecting the claim that $p\ge .8$ if $x\le 15$ .

a. What is the probability that the claim is rejected when the actual value of $p$ is . 8 ?

b. What is the probability of not rejecting the claim when $p=.7$ ? When $p=.6$ ?

c. How do the “error probabilities” of parts (a) and (b) change if the value 15 in the decision rule is replaced by 14 ?

60

A toll bridge charges \ {1.00}$forpassengercarsand$$ {2.50}$forothervehicles.Supposethatduringdaytimehours,${60}%$ofallvehiclesarepassengercars.If25vehiclescrossthebridgeduringaparticulardaytimeperiod,whatistheresultingexpectedtollrevenue?Hint:Let$X =$thenumberofpassengercars;thenthetollrevenue$h\left( X\right)$isalinearfunctionof$X$ .

61

A student who is trying to write a paper for a course has a choice of two topics, A and B. If topic A is chosen, the student will order two books through interlibrary loan, whereas if topic B is chosen, the student will order four books. The student believes that a good paper necessitates receiving and using at least half the books ordered for either topic chosen. If the probability that a book ordered through interlibrary loan actually arrives in time is .9 and books arrive independently of one another, which topic should the student choose to maximize the probability of writing a good paper? What if the arrival probability is only . 5 instead of .9 ?

62

a. For fixed $n$ , are there values of p\left( {0 \leq p \leq 1}\right)$forwhich$V\left( X\right) = 0$? Explain why this is so.

b. For what value of $p$ is $V\left(X\right)$ maximized? Hint: Either graph $V\left(X\right)$ as a function of $p$ or else take a derivative.

63

a. Show that $b\left(x;n,1-p\right)=b\left(n-x;n,p\right)$ .

b. Show that $B\left(x;n,1-p\right)=1-B\left(n-x-1;n,p\right)$. Hint: At most $xS$ ‘s is equivalent to at least $\left(n-x\right)$ $F$s.

c. What do parts (a) and (b) imply about the necessity of including values of $p$ greater than .5 in Appendix Table A.1?

64

Show that $E\left(X\right)=np$ when $X$ is a binomial random variable. Hint: First express $E\left(X\right)$ as a sum with lower limit $x=1$ . Then factor out $np$ , let $y=x-1$ so that the sum is from $y=0$ to $y=n-1$ , and show that the sum equals 1.

65

Customers at a gas station pay with a credit card $\left(A\right)$ , debit card $\left(B\right)$ , or cash $\left(C\right)$ . Assume that successive customers make independent choices, with $P\left(A\right)=.5$ , $P\left(B\right)=.2$ , and $P\left(C\right)=.3$ .

a. Among the next 100 customers, what are the mean and variance of the number who pay with a debit card? Explain your reasoning.

b. Answer part (a) for the number among the 100 who don’t pay with cash.

66

An airport limousine can accommodate up to four passengers on any one trip. The company will accept a maximum of six reservations for a trip, and a passenger must have a reservation. From previous records, $20\%$ of all those making reservations do not appear for the trip. Answer the following questions, assuming independence wherever appropriate.

a. If six reservations are made, what is the probability that at least one individual with a reservation cannot be accommodated on the trip?

b. If six reservations are made, what is the expected number of available places when the limousine departs?

c. Suppose the probability distribution of the number of reservations made is given in the accompanying table.

Number of reservations	3	4	5	6
Probability	.1	.2	.3	.4

Let $X$ denote the number of passengers on a randomly selected trip. Obtain the probability mass function of $X$ .

67

Refer to Chebyshev’s inequality given in Exercise 44. Calculate $P\left(\left|X-\mu \right|\ge k\sigma \right)$ for $k=2$ and $k=3$ when $X\sim \mathrm{Bin}\left(20,.5\right)$, and compare to the corresponding upper bound. Repeat for $X\sim \mathrm{Bin}\left(20,.75\right)$.