Introduction to Probability and Statistics, 14th Edition by William Mendenhall - Test Bank

Introduction to Probability and Statistics, 14th Edition by William Mendenhall – Test Bank

 

Instant Download – Complete Test Bank With Answers

 

 

Sample Questions Are Posted Below

 

MBB.IntroProb13.ch05sec1-2

 

TRUE/FALSE

 

1. A binomial random variable is an example of a discrete random variable.

 

ANS:  T                    PTS:   1

 

2. In a binomial experiment, the probability of success is the same on every trial.

 

ANS:  T                    PTS:   1

 

3. A coin toss experiment represents a binomial experiment only if the coin is balanced, i.e., p = 0.5.

 

ANS:  F                    PTS:   1

 

4. A jug contains 5 black marbles and 5 white marbles well mixed. A marble is removed and its color noted. A second marble is removed, without replacing the first marble, and its color is also noted. If x is the total number of black marbles in the two draws, then x has a binomial distribution.

 

ANS:  F                    PTS:   1

 

5. As a rule of thumb, if the sample size n is large relative to the population size N in particular, if n / N  0.05 then the resulting experiment will not be binomial.

 

ANS:  T                    PTS:   1

 

6. The binomial random variable is the number of successes that occur in a certain period of time or space.

 

ANS:  T                    PTS:   1

 

7. If x is a binomial random variable with n = 20, and p = 0.5, then P(x = 20) = 1.0.

 

ANS:  F                    PTS:   1

 

8. The binomial probability distribution is an example of discrete probability distributions.

 

ANS:  T                    PTS:   1

 

9. A binomial probability distribution shows the probabilities associated with possible values of a discrete random variable that are generated by a binomial experiment.

 

ANS:  T                    PTS:   1

 

10. A binomial experiment is a sequence of n identical trials such that each trial (1) produces one of two outcomes that are conventionally called success and failure and (2) is independent of any other trial so that the probability of success or failure is constant from trial to trial.

 

ANS:  T                    PTS:   1

 

11. The number of successes observed during the n trials of a binomial experiment is called the binomial random variable.

 

ANS:  T                    PTS:   1

 

12. The binomial experiment requires that the successes and failure probabilities be constant from one trial to the next and also that these two probabilities be equal to each other.

 

ANS:  F                    PTS:   1

 

13. The binomial distribution could be used to describe the spread of tennis balls when the players are serving.

 

ANS:  F                    PTS:   1

 

14. A life insurance salesperson makes 15 sales calls daily. The chance of making a sale on each call is 0.40. The probability that he will make at most 2 sales is less than 0.10.

 

ANS:  F                    PTS:   1

 

15. The distribution of the number of phone calls to a doctor’s office in a one-hour time period is likely to be described by a binomial distribution.

 

ANS:  F                    PTS:   1

 

16. The number of defects in a random sample of 200 parts produced by a machine is binomially distributed with p = .03. Based on this information, the expected number of defects in the sample is 6.

 

ANS:  T                    PTS:   1

 

17. The number of defects in a random sample of 200 parts produced by a machine is binomially distributed with p = .03. Based on this information, the standard deviation of the number of defects in the sample is 5.82.

 

ANS:  T                    PTS:   1

 

18. The binomial distribution is used to describe continuous random variables.

 

ANS:  F                    PTS:   1

 

MULTIPLE CHOICE

 

1. Which of the following is a characteristic of a binomial problem?

 

a.	There are n identical trials, and all trials are independent
b.	Each trial has two possible outcomes which are traditionally labeled “failure” and “success” and the probability of success p is the same on each trial.
c.	We are interested in x, the number of successes observed during the n trials.
d.	All of these are characteristics of a binomial experiment.
e.	None of these

 

 

ANS:  D                    PTS:   1

 

2. Which of the following statements regarding a binomial experiment is false, where n is the number of trials, and p is the probability of success in each trial?

 

a.	The n trials are independent.
b.	The standard deviation is np(1 – p).
c.	The mean is np.
d.	There are only two possible outcomes.
e.	The n trials are independent and the standard deviation is np(1 – p).

 

 

ANS:  B                    PTS:   1

 

3. For a binomial experiment with n trials,  p is the probability of success, q is the probability of failure, and x is the number of successes in n trials. Which one of the following statements is correct?

 

a.	p + q = 1
b.	p(x) = 1 for x = 0, 1, . . ., n
c.	P(x = 0) =
d.	p + q = 1 and p(x) = 1 for x = 0, 1, . . ., n
e.	all of these

 

 

ANS:  E                    PTS:   1

 

4. A telephone survey of American families is conducted to determine the number of children in the average American family. Past experience has shown that 30% of the families who are telephoned will refuse to respond to the survey. Which of the following statements is not a binomial random variable?

 

a.	The number of families out of 50 who respond to the survey.
b.	The number of families out of 50 who refuse to respond to the survey.
c.	The number of children in a family which responds to the survey.
d.	All of these are binomial random variables.
e.	None of these is a binomial random variable.

 

 

ANS:  C                    PTS:   1

 

5. Which of the following is an example of a binomial experiment?

 

a.	A shopping mall is interested in the income level of its customers and is taking a survey to gather information.
b.	A business firm introducing a new product wants to know how many purchases its clients will make each year.
c.	A sociologist is researching an area in an effort to determine the proportion of households with a male head of household.
d.	A study is concerned with the average number of hours worked by high school students.
e.	All of these.

 

 

ANS:  C                    PTS:   1

 

6. The standard deviation of a binomial distribution for which n = 50 and p = 0.15 is:

 

a.	50.15
b.	7.082
c.	6.375
d.	2.525
e.	0.15

 

 

ANS:  D                    PTS:   1

 

7. The National Rifle Association chapter in a university town claims 60% of the students at the university favor gun control. A social scientist at the university conducts a survey of 20 randomly chosen students and finds 9 who favor gun control. Which of the following is a reasonable conclusion?

 

a.	There is no reason to doubt the claim.
b.	The survey results constitute a rare event.
c.	The NRA’s claim is incorrect.
d.	The true percentage of students who favor gun control must be 45%.
e.	None of these.

 

 

ANS:  A                    PTS:   1

 

8. The expected number of heads in 200 tosses of an unbiased coin is:

 

a.	100
b.	50
c.	75
d.	125
e.	200

 

 

ANS:  A                    PTS:   1

 

9. The expected value, E(X), of a binomial probability distribution with n trials and probability p of success is:

 

a.	n / p
b.	np(1 –  p)
c.	np
d.	np – 1
e.	n(1 – p)

 

 

ANS:  C                    PTS:   1

 

10. Which of the following best describes a binomial probability distribution?

 

a.	It is a probability distribution that shows the probabilities associated with possible values of a discrete random variable that are generated by a binomial experiment.
b.	It is a probability distribution that shows, for each of a random variable’s possible values, the probability of the variable being less than or equal to that value.
c.	It is a probability distribution that shows the probabilities associated with possible values of a discrete random variable when these values are generated by sampling without replacement and the probability of success, therefore, changes from one trial to the next.
d.	It is a probability distribution that shows the probabilities associated with possible values of a discrete random variable when these values equal the number of occurrences of a specified event within a specified time or space.
e.	It is a probability distribution that shows, for each of a random variable’s possible values, the probability of the variable is positive.

 

 

ANS:  A                    PTS:   1

 

11. Which of the following best describes the binomial probability distribution?

 

a.	If is a probability distribution that shows the probabilities associated with possible values of a discrete random variable when these values are generated by sampling without replacement and the probability of success, therefore, changes from one trial to the next.
b.	It is a probability distribution that shows the probabilities associated with possible values of a discrete random variable when these values equal the number of occurrences of a specified event within a specified time or space.
c.	It is a probability distribution that shows the probabilities associated with possible values of a discrete random variable that are generated by a binomial experiment.
d.	It is a probability distribution that shows the probabilities associated with possible values of a continuous random variable that are generated by a binomial experiment.
e.	None of these.

 

 

ANS:  C                    PTS:   1

 

12. The binomial distribution formula:

 

a.	computes the number of permutations of x successes (and, therefore, n – x failures) that can be achieved in n trials of a random experiment that satisfies the conditions of the binomial experiment
b.	computes the probability of x successes in n trials of a random experiment that satisfies the conditions of a binomial experiment
c.	produces one of two possible outcomes that are conventionally called success and failure
d.	computes the probability of x successes when a random sample of size n is drawn without replacement from a population of size N within which M units have the characteristic that denotes success
e.	does all of these

 

 

ANS:  B                    PTS:   1

 

13. Given that n is the number of trials and p is the probability of success in any one trial of a random experiment, the expected value of a binomial random variable equals:

 

a.	n
b.	p
c.	np
d.	n/p
e.	n/p(1 – p)

 

 

ANS:  C                    PTS:   1

 

14. A manufacturer of tennis balls uses a production process that produces 5 percent defective balls. A quality inspector takes samples of a week’s output with replacement. Using the cumulative binomial probability table available in your text, the inspector can determine which of the following probabilities?

 

a.	If 5 units are inspected, the probability of zero units being defective is .774.
b.	If 10 units are inspected, the probability of 2 units being defective is .374.
c.	If 15 units are inspected, the probability of 4 units being defective is .999.
d.	If 20 units are inspected, the probability of 6 units being defective is .0569.
e.	If 20 units are inspected, the probability of 6 units being defective is .5.

 

 

ANS:  A                    PTS:   1

 

15. A manufacturer of golf balls uses a production process that produces 10 percent defective balls. A quality inspector takes samples of a week’s output with replacement. Using the cumulative binomial probability table available in your text, the inspector can determine which of the following probabilities?

 

a.	If 5 units are inspected, the probability of at most 3 of these units being defective is .984.
b.	If 10 units are inspected, the probability of 5 or 6 of these units being defective is .002.
c.	If 15 units are inspected, the probability of at least 10 of these units being defective is .547.
d.	If 20 units are inspected, the probability of at least 19 of these units being defective is .0009.

 

 

ANS:  B                    PTS:   1

 

16. It has been alleged that 40 percent of all college students favor Dell computers. If this were true, and we took a random sample of 50 students, the binomial probability table for cumulative values of x available in your text, would reveal which of the following probabilities?

 

a.	The probability of 10 or fewer students in favor is .586.
b.	The probability of fewer than 20 students in favor is 1.000.
c.	The probability of more than 15 students in favor is .013.
d.	All of these.
e.	None of these.

 

 

ANS:  D                    PTS:   1

 

17. Which of the following is not a characteristic of a binomial experiment?

 

a.	The experiment consists of n identical and independent trials.
b.	The probability of success on a single trial remains constant from trial to trial.
c.	The standard deviation of the binomial random variable is the square root of the mean.
d.	Each trial results in one of two outcomes.
e.	All of these.

 

 

ANS:  C                    PTS:   1

 

18. If the random variable x is binomially distributed with n = 10 and p = .05, then P(x = 2) is:

 

a.	0.599
b.	0.914
c.	0.074
d.	0.55
e.	1.00

 

 

ANS:  C                    PTS:   1

 

19. Which of the following statements is true with respect to the binomial distribution?

 

a.	The binomial distribution tends to be more symmetric as the probability of success p approaches 0.5.
b.	As the number of trials increase, the expected value of the random variable decreases.
c.	As the number of trials increase for a given probability of success, the binomial distribution becomes more skewed.
d.	The binomial distribution tends to be more symmetric as the probability of success p approaches 1.0.
e.	None of these.

 

 

ANS:  A                    PTS:   1

 

PROBLEM

 

1. A fellow student tells you he is working with a discrete random variable, x, which takes on integer values from 0 to 50 and has a mean of 40 and a variance of 10.

 

Is x a binomial random variable?

 

 

______________

 

Explain.

 

 

________________________________________________________

 

ANS:

No; If x is binomial, it has n = 50 trials, since x = 0, 1, 2, . . ., 50. The binomial mean is np, so if x is binomial, np = 50p = 40, so p = 0.8. If x is binomial with n = 50 and p = 0.8, the variance of x is npq. However, npq = (50) .(0.8) .(0.2) = 8, not 10 as stated in the problem. Thus the random variable x is not binomially distributed.

 

PTS:   1

 

2. The taste test for PTC (phenylthiourea) is a common class demonstration in the study of genetics. It is known 70% of the American people are “tasters” and 30% are “non-tasters.” Suppose a genetics class of size 20 does the test to see if they match the U.S. percentage of “tasters” and “non-tasters.” (Assume the assignment of students to classes constitutes a random process.)

 

For the random variable x, the number of “non-tasters” in the class,

 

Find P(3 < x < 9).

 

 

______________

 

Find the mean of x.

 

 

______________

 

Find the variance of x.

 

 

______________

 

ANS:

0.78; 6; 4.2

 

PTS:   1

 

3. An oil firm plans to drill 20 wells, each having probability 0.2 of striking oil. Each well costs $20,000 to drill; a well which strikes oil will bring in $750,000 in revenue. Find the expected gain from the 20 wells.

 

 

______________

 

ANS:

$2,600,000

 

PTS:   1

 

4. From past experience, it is known 90% of one-year-old children can distinguish their mother’s voice from the voice of a similar sounding female. A random sample of 20 one-year-olds are given this voice recognition test.

 

Find the probability at least 3 children do not recognize their mother’s voice.

 

 

______________

 

Find the probability all 20 children recognize their mother’s voice.

 

 

______________

 

Let the random variable x denote the number of children who do not recognize their mother’s voice. Find the mean of x.

 

 

______________

 

Let the random variable x denote the number of children who do not recognize their mother’s voice. Find the variance of x.

 

 

______________

 

Find the probability that at most 4 children do not recognize their mother’s voice.

 

 

______________

 

ANS:

0.323; 0.122; 2; 1.8; 0.957

 

PTS:   1

 

5. A quiz consists of 15 multiple choice questions. Each question has 5 choices, with exactly one correct choice. A student, totally unprepared for the quiz, guesses on each of the 15 questions.

 

How many questions should the student expect to answer correctly?

 

 

______________

 

What is the standard deviation of the number of questions answered correctly?

 

 

______________

 

If at least 9 questions must be answered correctly to pass the quiz, what is the chance the student passes?

 

 

______________

 

ANS:

3; 1.549; 0.01

 

PTS:   1

 

6. Suppose 40% of the TV sets in use in the U.S. on a particular night were tuned in to game 7 of the World Series.

 

If we were to take a sample of six in-use TV sets that night, what is the probability exactly three are tuned to the World Series?

 

 

______________

 

If, instead, the sample consisted of 15 in-use TVs, what is the probability five or more are tuned to the World Series?

 

 

______________

 

ANS:

0.277; 0.783

 

PTS:   1

 

7. In an effort to persuade customers to reduce their electricity consumption, special discounts have been offered for low consumption customers. In Florida, the EPA estimates that 70% of the residences have qualified for these discounts.

 

If we randomly select four residences in Florida, what is the probability exactly two of the four will have qualified for the discounts?

 

 

______________

 

If, instead, a sample of 20 residences is selected, what is the probability 12 or more will have qualified for the discounts?

 

 

______________

 

ANS:

0.264; 0.887

 

PTS:   1

 

8. Hotels, like airlines, often overbook, counting on the fact that some people with reservations will cancel at the last minute. A certain hotel chain finds 20% of the reservations will not be used.

 

If we randomly selected 15 reservations, what is the probability more than 8 but less than 12 reservations will be used?

 

 

______________

 

If four reservations are made, what is the chance fewer than two will cancel?

 

 

______________

 

ANS:

0.334; 0.8192

 

PTS:   1

 

9. To determine the effectiveness of a diet to reduce the amount of serum cholesterol, 20 people were randomly selected to try the diet. After they had been on the diet a sufficient amount of time, their blood was tested to determine the level of serum cholesterol. The clinic running the experiment will approve and endorse the diet only if at least 16 of the 20 people lower their cholesterol significantly.

 

What is the random variable of interest?

 

 

________________________________________________________

 

What is the probability the clinic will erroneously endorse the diet when, in fact, it is useless in treating high serum cholesterol? [Hint:If the diet is ineffective, a person’s cholesterol level is equally likely to go up or down.]

 

 

______________

 

ANS:

The number of persons in the study who significantly lowered their serum cholesterol while on the diet.; 0.006

 

PTS:   1

 

10. The local Ford dealership claims 40% of their customers are return buyers, i.e., the customers previously purchased a Ford vehicle. Let the random variable x be the number of return buyers in a random sample of 10 recent customers.

 

 

Find  = ______________

 

 

Find P(x > 6) = ______________

 

What would you conclude about the accuracy of the dealership’s claim if you found 9 repeat buyers in the sample? Explain your reasoning.

 

 

________________________________________________________

 

ANS:

0.627; 0.166; There is reason to doubt the dealership’s claim because P(x = 9) = 0.01 if p = 0.40.

 

PTS:   1

 

11. An applicant must score at least 80 points on a particular psychological test to be eligible for the Peace Corps. From nearly 30 years experience, it is known 60% of the applicants meet this requirement. Let the random variable x be the number of applicants who meet this requirement out of a (randomly selected) group of 25 applicants.

 

Find the mean of x.

 

 

______________

 

Find the standard deviation of x.

 

 

______________

 

Using the Empirical Rule, within what limits would you expect most (say, 95%) of the measurements to be?

 

 

________________________________________________________

 

ANS:

15; 2.45; We would expect most of the measurements to be between 10.1 and 19.9 which, since x is discrete, translates as between 11 and 19.

 

PTS:   1

 

12. A politician claims 55% of the voters will vote for him in an upcoming election. An independent watchdog organization queried a random sample of 500 likely voters, of whom only 249 said they would vote for the politician in question.

 

Do the results of this sample support the politician’s claim? [Hint: Binomial distributions with p near 50% are quite mound-shaped.]

 

 

______________

 

Explain.

 

 

________________________________________________________

 

ANS:

No; 249 is more than 2 standard deviations below the mean. Sample results like this would occur approximately 2.5% of the time if the politician’s claim of 55% were true.

 

PTS:   1

 

13. An American Medical Association study showed 20% of all Americans suffer from high blood pressure. Suppose we randomly sample ten Americans to determine the number in the sample who have high blood pressure.

 

What is the probability at most two persons have high blood pressure?

 

 

______________

 

If, instead, we sampled 25 Americans, give a reason why the following statement is true: “The probability between one and nine individuals from a sample of 25 would have high blood pressure is at least 0.75.” [Hint: For n and p of approximately the sizes given, the distribution would not be particularly symmetric.]

 

 

________________________________________________________

 

ANS:

0.678; Tchebysheff’s Theorem says at least 75% of the observations lie within two standard deviations of the mean

 

PTS:   1

 

14. It is known that 70% of the customers in a sporting goods store purchase a pair of running shoes. A random sample of 25 customers is selected. Assume that customers’ purchases are made independently, and let x represent number of customers who purchase running shoes.

 

What is the probability that exactly 18 customers purchase running shoes?

 

 

______________

 

What is the probability that no more than 19 customers purchase running shoes?

 

 

______________

 

What is the probability that at least 17 customers purchase running shoes?

 

 

______________

 

What is the probability that between 17 and 21 customers, inclusively, purchase running shoes?

 

 

______________

 

ANS:

0.171; 0.807; 0.677; 0.644

 

PTS:   1

 

15. A Statistics Department is contacting alumni by telephone asking for donations to help fund a new computer laboratory. Past history shows that 80% of the alumni contacted in this manner will make a contribution of at least $50. A random sample of 20 alumni is selected. Let x represent number of alumni that makes at contribution of at least $50.

 

What is the probability that exactly 15 alumni will make a contribution of at least $50?

 

 

______________

 

What is the probability that between 14 and 18 alumni, inclusively, will make a contribution of at least $50?

 

 

______________

 

What is the probability that less than 17 alumni will make a contribution of at least $50?

 

 

______________

 

What is the probability that more than 15 alumni will make a contribution of at least $50?

 

 

______________

 

How many alumni would you expect to make a contribution of at least $50?

 

 

______________

 

ANS:

0.174; 0.844; 0.589; 0.630; 16

 

PTS:   1

 

16. Let x be a binomial random variable with n = 15 and p = 0.20.

 

Calculate P(x  4) using the binomial formula.

 

 

______________

 

Calculate P(x  4) using the cumulative binomial probabilities table in Appendix I of your book.

 

 

______________

 

Are the results of your two previous answers very different, or very similar?

 

 

______________

 

Calculate the mean of the random variable x.

 

 

______________

 

Calculate the standard deviation of the random variable x.

 

 

______________

 

Fill in the table below:

 

	probability
	______________
	______________
	______________

 

Are the results consistent with Tchebysheff’s Theorem?

 

 

______________

 

With the Empirical Rule?

 

 

______________

 

ANS:

.835767; .836; Very similar; 3; 1.549; .669; .982; .996; Yes; Yes

 

PTS:   1

 

17. In 1996, the average combined SAT score (math + verbal) for students in the United States was 1013, and 41% of all high school graduates took this test. Suppose that 500 students are randomly selected throughout the United States.

 

Which of the following random variables has an approximate binomial distribution? The number of students who took the SAT.

 

 

____________________________

 

The scores of the 500 students on the SAT.

 

 

____________________________

 

The number of students who scored above average on the SAT.

 

 

____________________________

 

The amount of time it took each student to complete the SAT.

 

 

____________________________

 

ANS:

Binomial distribution; Not a binomial distribution; Binomial distribution; Not a binomial distribution

 

PTS:   1

 

18. Car color preferences change over the years and according to the particular model that the customer selects. In a recent year, 20% of all luxury cars sold were white. Assume that 25 cars of that year and type are randomly selected. Find the probabilities listed below:

 

Define x to be the number of cars that are white. Then p = P(white) = 0.20 and n = 25. Use the cumulative binomial probabilities table in Appendix I of your book.

 

At least five cars are white.

 

 

______________

 

At most six cars are white.

 

 

______________

 

Exactly four cars are white.

 

 

______________

 

More than three cars are white.

 

 

______________

 

Between three and five cars (inclusive) are white.

 

 

______________

 

More than 20 cars are not white.

 

 

______________

 

ANS:

0.579; 0.780; 0.187; 0.766; 0.519; 0.421

 

PTS:   1

 

19. A new surgical procedure is said to be successful 90% of the time. Suppose the operation is performed five times and the results are assumed to be independent of one another. Find the probabilities listed below:

 

Define x to be the number of successful operations. Then p = P(success) = 0.9 and n = 5. We will use the binomial formula. However, you may also use the cumulative binomial probabilities table in Appendix I of your book to calculate the necessary probabilities.

 

All five operations are successful.

 

 

______________

 

Exactly four are successful.

 

 

______________

 

Less than two are successful.

 

 

______________

 

ANS:

0.59049; 0.32805; 0.000460

 

PTS:   1

 

20. Four in ten Americans who travel by car look for gas and food outlets that are close to or visible from the highway. Suppose a random sample of n = 20 Americans who travel by car are asked how they determine where to stop for food and gas. Let x be the number in the sample who respond that they look for gas and food outlets that are close to or visible from the highway.

 

Calculate the mean of x.

 

 

______________

 

Calculate the variance of x.

 

 

______________

 

Calculate the interval .

 

 

______________

 

What values of the binomial random variable x fall into this interval?

 

 

______________

 

Find .

 

 

______________

 

How does this compare with the fraction in the interval  for any distribution? For mound-shaped distributions?

 

 

________________________________________________________

 

ANS:

8; 4.8; (3.61822, 12.3818); (4, 12); 0.963; This value agrees with Tchebysheff’s Theorem (at least 3 / 4 of the measurements in this interval) and also with the Empirical Rule (approximately 95% of the measurements in this interval.

 

PTS:   1

 

21. Let x be a binomial random variable with n = 20 and p = 0.05.

 

Calculate  using the cumulative binomial probabilities table in Appendix I of your book.

 

 

______________

 

Use the Poisson approximation to calculate .

 

 

______________

 

Compare your results. Is the approximation accurate?

 

 

______________

 

ANS:

0.925; 0.9197; Yes

 

PTS:   1

 

22. A psychiatrist believes that 90% of all people who visit doctors have problems of a psychosomatic nature. She decides to select 20 patients at random to test her theory.

 

Assuming that the psychiatrist’s theory is true, what is the expected value of x, the number of the 20 patients who have psychosomatic problems?

 

 

______________

 

What is the variance of x, assuming that the theory is true?

 

 

______________

 

Find , assuming that the theory is true.

 

 

______________

 

Based on the probability in part (c), if only 14 of the 20 sampled had psychosomatic problems, what conclusions would you make about the psychiatrist’s theory?

 

 

______________

 

Explain.

 

 

________________________________________________________

 

ANS:

18; 1.8; 0.011; The psychiatrist is INCORRECT; Assuming that the psychiatrist is correct, the probability of observing x = 14 or the more unlikely values, x = 0, 1, 2,..,13 is very unlikely. Hence, one of two conclusions can be drawn. Either we have observed a very unlikely event, or the psychiatrist is incorrect and p is actually less than 0.9. We would probably conclude that the psychiatrist is incorrect. The probability that we have made an incorrect decision is 0.011, which is quite small.

 

PTS:   1

 

23. About 80% of Florida residents believe that Florida is a nice or very nice place to live. Suppose that five randomly selected Florida residents are interviewed.

 

Find the probability that all five residents think that Florida is a nice or very nice place to live.

 

 

______________

 

What is the probability that at least one resident does not think that Florida is a nice or very nice place to live?

 

 

______________

 

What is the probability that exactly one resident does not think that Florida is a nice or very nice place to live?

 

 

______________

 

ANS:

0.3277; 0.6723; 0.4096

 

PTS:   1

 

MBB.IntroProb13.ch05sec1-2

 

TRUE/FALSE

 

1. A binomial random variable is an example of a discrete random variable.

 

ANS:  T                    PTS:   1

 

2. In a binomial experiment, the probability of success is the same on every trial.

 

ANS:  T                    PTS:   1

 

3. A coin toss experiment represents a binomial experiment only if the coin is balanced, i.e., p = 0.5.

 

ANS:  F                    PTS:   1

 

4. A jug contains 5 black marbles and 5 white marbles well mixed. A marble is removed and its color noted. A second marble is removed, without replacing the first marble, and its color is also noted. If x is the total number of black marbles in the two draws, then x has a binomial distribution.

 

ANS:  F                    PTS:   1

 

5. As a rule of thumb, if the sample size n is large relative to the population size N in particular, if n / N  0.05 then the resulting experiment will not be binomial.

 

ANS:  T                    PTS:   1

 

6. The binomial random variable is the number of successes that occur in a certain period of time or space.

 

ANS:  T                    PTS:   1

 

7. If x is a binomial random variable with n = 20, and p = 0.5, then P(x = 20) = 1.0.

 

ANS:  F                    PTS:   1

 

8. The binomial probability distribution is an example of discrete probability distributions.

 

ANS:  T                    PTS:   1

 

9. A binomial probability distribution shows the probabilities associated with possible values of a discrete random variable that are generated by a binomial experiment.

 

ANS:  T                    PTS:   1

 

10. A binomial experiment is a sequence of n identical trials such that each trial (1) produces one of two outcomes that are conventionally called success and failure and (2) is independent of any other trial so that the probability of success or failure is constant from trial to trial.

 

ANS:  T                    PTS:   1

 

11. The number of successes observed during the n trials of a binomial experiment is called the binomial random variable.

 

ANS:  T                    PTS:   1

 

12. The binomial experiment requires that the successes and failure probabilities be constant from one trial to the next and also that these two probabilities be equal to each other.

 

ANS:  F                    PTS:   1

 

13. The binomial distribution could be used to describe the spread of tennis balls when the players are serving.

 

ANS:  F                    PTS:   1

 

14. A life insurance salesperson makes 15 sales calls daily. The chance of making a sale on each call is 0.40. The probability that he will make at most 2 sales is less than 0.10.

 

ANS:  F                    PTS:   1

 

15. The distribution of the number of phone calls to a doctor’s office in a one-hour time period is likely to be described by a binomial distribution.

 

ANS:  F                    PTS:   1

 

16. The number of defects in a random sample of 200 parts produced by a machine is binomially distributed with p = .03. Based on this information, the expected number of defects in the sample is 6.

 

ANS:  T                    PTS:   1

 

17. The number of defects in a random sample of 200 parts produced by a machine is binomially distributed with p = .03. Based on this information, the standard deviation of the number of defects in the sample is 5.82.

 

ANS:  T                    PTS:   1

 

18. The binomial distribution is used to describe continuous random variables.

 

ANS:  F                    PTS:   1

 

MULTIPLE CHOICE

 

1. Which of the following is a characteristic of a binomial problem?

 

a.	There are n identical trials, and all trials are independent
b.	Each trial has two possible outcomes which are traditionally labeled “failure” and “success” and the probability of success p is the same on each trial.
c.	We are interested in x, the number of successes observed during the n trials.
d.	All of these are characteristics of a binomial experiment.
e.	None of these

 

 

ANS:  D                    PTS:   1

 

2. Which of the following statements regarding a binomial experiment is false, where n is the number of trials, and p is the probability of success in each trial?

 

a.	The n trials are independent.
b.	The standard deviation is np(1 – p).
c.	The mean is np.
d.	There are only two possible outcomes.
e.	The n trials are independent and the standard deviation is np(1 – p).

 

 

ANS:  B                    PTS:   1

 

3. For a binomial experiment with n trials,  p is the probability of success, q is the probability of failure, and x is the number of successes in n trials. Which one of the following statements is correct?

 

a.	p + q = 1
b.	p(x) = 1 for x = 0, 1, . . ., n
c.	P(x = 0) =
d.	p + q = 1 and p(x) = 1 for x = 0, 1, . . ., n
e.	all of these

 

 

ANS:  E                    PTS:   1

 

4. A telephone survey of American families is conducted to determine the number of children in the average American family. Past experience has shown that 30% of the families who are telephoned will refuse to respond to the survey. Which of the following statements is not a binomial random variable?

 

a.	The number of families out of 50 who respond to the survey.
b.	The number of families out of 50 who refuse to respond to the survey.
c.	The number of children in a family which responds to the survey.
d.	All of these are binomial random variables.
e.	None of these is a binomial random variable.

 

 

ANS:  C                    PTS:   1

 

5. Which of the following is an example of a binomial experiment?

 

a.	A shopping mall is interested in the income level of its customers and is taking a survey to gather information.
b.	A business firm introducing a new product wants to know how many purchases its clients will make each year.
c.	A sociologist is researching an area in an effort to determine the proportion of households with a male head of household.
d.	A study is concerned with the average number of hours worked by high school students.
e.	All of these.

 

 

ANS:  C                    PTS:   1

 

6. The standard deviation of a binomial distribution for which n = 50 and p = 0.15 is:

 

a.	50.15
b.	7.082
c.	6.375
d.	2.525
e.	0.15

 

 

ANS:  D                    PTS:   1

 

7. The National Rifle Association chapter in a university town claims 60% of the students at the university favor gun control. A social scientist at the university conducts a survey of 20 randomly chosen students and finds 9 who favor gun control. Which of the following is a reasonable conclusion?

 

a.	There is no reason to doubt the claim.
b.	The survey results constitute a rare event.
c.	The NRA’s claim is incorrect.
d.	The true percentage of students who favor gun control must be 45%.
e.	None of these.

 

 

ANS:  A                    PTS:   1

 

8. The expected number of heads in 200 tosses of an unbiased coin is:

 

a.	100
b.	50
c.	75
d.	125
e.	200

 

 

ANS:  A                    PTS:   1

 

9. The expected value, E(X), of a binomial probability distribution with n trials and probability p of success is:

 

a.	n / p
b.	np(1 –  p)
c.	np
d.	np – 1
e.	n(1 – p)

 

 

ANS:  C                    PTS:   1

 

10. Which of the following best describes a binomial probability distribution?

 

a.	It is a probability distribution that shows the probabilities associated with possible values of a discrete random variable that are generated by a binomial experiment.
b.	It is a probability distribution that shows, for each of a random variable’s possible values, the probability of the variable being less than or equal to that value.
c.	It is a probability distribution that shows the probabilities associated with possible values of a discrete random variable when these values are generated by sampling without replacement and the probability of success, therefore, changes from one trial to the next.
d.	It is a probability distribution that shows the probabilities associated with possible values of a discrete random variable when these values equal the number of occurrences of a specified event within a specified time or space.
e.	It is a probability distribution that shows, for each of a random variable’s possible values, the probability of the variable is positive.

 

 

ANS:  A                    PTS:   1

 

11. Which of the following best describes the binomial probability distribution?

 

a.	If is a probability distribution that shows the probabilities associated with possible values of a discrete random variable when these values are generated by sampling without replacement and the probability of success, therefore, changes from one trial to the next.
b.	It is a probability distribution that shows the probabilities associated with possible values of a discrete random variable when these values equal the number of occurrences of a specified event within a specified time or space.
c.	It is a probability distribution that shows the probabilities associated with possible values of a discrete random variable that are generated by a binomial experiment.
d.	It is a probability distribution that shows the probabilities associated with possible values of a continuous random variable that are generated by a binomial experiment.
e.	None of these.

 

 

ANS:  C                    PTS:   1

 

12. The binomial distribution formula:

 

a.	computes the number of permutations of x successes (and, therefore, n – x failures) that can be achieved in n trials of a random experiment that satisfies the conditions of the binomial experiment
b.	computes the probability of x successes in n trials of a random experiment that satisfies the conditions of a binomial experiment
c.	produces one of two possible outcomes that are conventionally called success and failure
d.	computes the probability of x successes when a random sample of size n is drawn without replacement from a population of size N within which M units have the characteristic that denotes success
e.	does all of these

 

 

ANS:  B                    PTS:   1

 

13. Given that n is the number of trials and p is the probability of success in any one trial of a random experiment, the expected value of a binomial random variable equals:

 

a.	n
b.	p
c.	np
d.	n/p
e.	n/p(1 – p)

 

 

ANS:  C                    PTS:   1

 

14. A manufacturer of tennis balls uses a production process that produces 5 percent defective balls. A quality inspector takes samples of a week’s output with replacement. Using the cumulative binomial probability table available in your text, the inspector can determine which of the following probabilities?

 

a.	If 5 units are inspected, the probability of zero units being defective is .774.
b.	If 10 units are inspected, the probability of 2 units being defective is .374.
c.	If 15 units are inspected, the probability of 4 units being defective is .999.
d.	If 20 units are inspected, the probability of 6 units being defective is .0569.
e.	If 20 units are inspected, the probability of 6 units being defective is .5.

 

 

ANS:  A                    PTS:   1

 

15. A manufacturer of golf balls uses a production process that produces 10 percent defective balls. A quality inspector takes samples of a week’s output with replacement. Using the cumulative binomial probability table available in your text, the inspector can determine which of the following probabilities?

 

a.	If 5 units are inspected, the probability of at most 3 of these units being defective is .984.
b.	If 10 units are inspected, the probability of 5 or 6 of these units being defective is .002.
c.	If 15 units are inspected, the probability of at least 10 of these units being defective is .547.
d.	If 20 units are inspected, the probability of at least 19 of these units being defective is .0009.

 

 

ANS:  B                    PTS:   1

 

16. It has been alleged that 40 percent of all college students favor Dell computers. If this were true, and we took a random sample of 50 students, the binomial probability table for cumulative values of x available in your text, would reveal which of the following probabilities?

 

a.	The probability of 10 or fewer students in favor is .586.
b.	The probability of fewer than 20 students in favor is 1.000.
c.	The probability of more than 15 students in favor is .013.
d.	All of these.
e.	None of these.

 

 

ANS:  D                    PTS:   1

 

17. Which of the following is not a characteristic of a binomial experiment?

 

a.	The experiment consists of n identical and independent trials.
b.	The probability of success on a single trial remains constant from trial to trial.
c.	The standard deviation of the binomial random variable is the square root of the mean.
d.	Each trial results in one of two outcomes.
e.	All of these.

 

 

ANS:  C                    PTS:   1

 

18. If the random variable x is binomially distributed with n = 10 and p = .05, then P(x = 2) is:

 

a.	0.599
b.	0.914
c.	0.074
d.	0.55
e.	1.00

 

 

ANS:  C                    PTS:   1

 

19. Which of the following statements is true with respect to the binomial distribution?

 

a.	The binomial distribution tends to be more symmetric as the probability of success p approaches 0.5.
b.	As the number of trials increase, the expected value of the random variable decreases.
c.	As the number of trials increase for a given probability of success, the binomial distribution becomes more skewed.
d.	The binomial distribution tends to be more symmetric as the probability of success p approaches 1.0.
e.	None of these.

 

 

ANS:  A                    PTS:   1

 

PROBLEM

 

1. A fellow student tells you he is working with a discrete random variable, x, which takes on integer values from 0 to 50 and has a mean of 40 and a variance of 10.

 

Is x a binomial random variable?

 

 

______________

 

Explain.

 

 

________________________________________________________

 

ANS:

No; If x is binomial, it has n = 50 trials, since x = 0, 1, 2, . . ., 50. The binomial mean is np, so if x is binomial, np = 50p = 40, so p = 0.8. If x is binomial with n = 50 and p = 0.8, the variance of x is npq. However, npq = (50) .(0.8) .(0.2) = 8, not 10 as stated in the problem. Thus the random variable x is not binomially distributed.

 

PTS:   1

 

2. The taste test for PTC (phenylthiourea) is a common class demonstration in the study of genetics. It is known 70% of the American people are “tasters” and 30% are “non-tasters.” Suppose a genetics class of size 20 does the test to see if they match the U.S. percentage of “tasters” and “non-tasters.” (Assume the assignment of students to classes constitutes a random process.)

 

For the random variable x, the number of “non-tasters” in the class,

 

Find P(3 < x < 9).

 

 

______________

 

Find the mean of x.

 

 

______________

 

Find the variance of x.

 

 

______________

 

ANS:

0.78; 6; 4.2

 

PTS:   1

 

3. An oil firm plans to drill 20 wells, each having probability 0.2 of striking oil. Each well costs $20,000 to drill; a well which strikes oil will bring in $750,000 in revenue. Find the expected gain from the 20 wells.

 

 

______________

 

ANS:

$2,600,000

 

PTS:   1

 

4. From past experience, it is known 90% of one-year-old children can distinguish their mother’s voice from the voice of a similar sounding female. A random sample of 20 one-year-olds are given this voice recognition test.

 

Find the probability at least 3 children do not recognize their mother’s voice.

 

 

______________

 

Find the probability all 20 children recognize their mother’s voice.

 

 

______________

 

Let the random variable x denote the number of children who do not recognize their mother’s voice. Find the mean of x.

 

 

______________

 

Let the random variable x denote the number of children who do not recognize their mother’s voice. Find the variance of x.

 

 

______________

 

Find the probability that at most 4 children do not recognize their mother’s voice.

 

 

______________

 

ANS:

0.323; 0.122; 2; 1.8; 0.957

 

PTS:   1

 

5. A quiz consists of 15 multiple choice questions. Each question has 5 choices, with exactly one correct choice. A student, totally unprepared for the quiz, guesses on each of the 15 questions.

 

How many questions should the student expect to answer correctly?

 

 

______________

 

What is the standard deviation of the number of questions answered correctly?

 

 

______________

 

If at least 9 questions must be answered correctly to pass the quiz, what is the chance the student passes?

 

 

______________

 

ANS:

3; 1.549; 0.01

 

PTS:   1

 

6. Suppose 40% of the TV sets in use in the U.S. on a particular night were tuned in to game 7 of the World Series.

 

If we were to take a sample of six in-use TV sets that night, what is the probability exactly three are tuned to the World Series?

 

 

______________

 

If, instead, the sample consisted of 15 in-use TVs, what is the probability five or more are tuned to the World Series?

 

 

______________

 

ANS:

0.277; 0.783

 

PTS:   1

 

7. In an effort to persuade customers to reduce their electricity consumption, special discounts have been offered for low consumption customers. In Florida, the EPA estimates that 70% of the residences have qualified for these discounts.

 

If we randomly select four residences in Florida, what is the probability exactly two of the four will have qualified for the discounts?

 

 

______________

 

If, instead, a sample of 20 residences is selected, what is the probability 12 or more will have qualified for the discounts?

 

 

______________

 

ANS:

0.264; 0.887

 

PTS:   1

 

8. Hotels, like airlines, often overbook, counting on the fact that some people with reservations will cancel at the last minute. A certain hotel chain finds 20% of the reservations will not be used.

 

If we randomly selected 15 reservations, what is the probability more than 8 but less than 12 reservations will be used?

 

 

______________

 

If four reservations are made, what is the chance fewer than two will cancel?

 

 

______________

 

ANS:

0.334; 0.8192

 

PTS:   1

 

9. To determine the effectiveness of a diet to reduce the amount of serum cholesterol, 20 people were randomly selected to try the diet. After they had been on the diet a sufficient amount of time, their blood was tested to determine the level of serum cholesterol. The clinic running the experiment will approve and endorse the diet only if at least 16 of the 20 people lower their cholesterol significantly.

 

What is the random variable of interest?

 

 

________________________________________________________

 

What is the probability the clinic will erroneously endorse the diet when, in fact, it is useless in treating high serum cholesterol? [Hint:If the diet is ineffective, a person’s cholesterol level is equally likely to go up or down.]

 

 

______________

 

ANS:

The number of persons in the study who significantly lowered their serum cholesterol while on the diet.; 0.006

 

PTS:   1

 

10. The local Ford dealership claims 40% of their customers are return buyers, i.e., the customers previously purchased a Ford vehicle. Let the random variable x be the number of return buyers in a random sample of 10 recent customers.

 

 

Find  = ______________

 

 

Find P(x > 6) = ______________

 

What would you conclude about the accuracy of the dealership’s claim if you found 9 repeat buyers in the sample? Explain your reasoning.

 

 

________________________________________________________

 

ANS:

0.627; 0.166; There is reason to doubt the dealership’s claim because P(x = 9) = 0.01 if p = 0.40.

 

PTS:   1

 

11. An applicant must score at least 80 points on a particular psychological test to be eligible for the Peace Corps. From nearly 30 years experience, it is known 60% of the applicants meet this requirement. Let the random variable x be the number of applicants who meet this requirement out of a (randomly selected) group of 25 applicants.

 

Find the mean of x.

 

 

______________

 

Find the standard deviation of x.

 

 

______________

 

Using the Empirical Rule, within what limits would you expect most (say, 95%) of the measurements to be?

 

 

________________________________________________________

 

ANS:

15; 2.45; We would expect most of the measurements to be between 10.1 and 19.9 which, since x is discrete, translates as between 11 and 19.

 

PTS:   1

 

12. A politician claims 55% of the voters will vote for him in an upcoming election. An independent watchdog organization queried a random sample of 500 likely voters, of whom only 249 said they would vote for the politician in question.

 

Do the results of this sample support the politician’s claim? [Hint: Binomial distributions with p near 50% are quite mound-shaped.]

 

 

______________

 

Explain.

 

 

________________________________________________________

 

ANS:

No; 249 is more than 2 standard deviations below the mean. Sample results like this would occur approximately 2.5% of the time if the politician’s claim of 55% were true.

 

PTS:   1

 

13. An American Medical Association study showed 20% of all Americans suffer from high blood pressure. Suppose we randomly sample ten Americans to determine the number in the sample who have high blood pressure.

 

What is the probability at most two persons have high blood pressure?

 

 

______________

 

If, instead, we sampled 25 Americans, give a reason why the following statement is true: “The probability between one and nine individuals from a sample of 25 would have high blood pressure is at least 0.75.” [Hint: For n and p of approximately the sizes given, the distribution would not be particularly symmetric.]

 

 

________________________________________________________

 

ANS:

0.678; Tchebysheff’s Theorem says at least 75% of the observations lie within two standard deviations of the mean

 

PTS:   1

 

14. It is known that 70% of the customers in a sporting goods store purchase a pair of running shoes. A random sample of 25 customers is selected. Assume that customers’ purchases are made independently, and let x represent number of customers who purchase running shoes.

 

What is the probability that exactly 18 customers purchase running shoes?

 

 

______________

 

What is the probability that no more than 19 customers purchase running shoes?

 

 

______________

 

What is the probability that at least 17 customers purchase running shoes?

 

 

______________

 

What is the probability that between 17 and 21 customers, inclusively, purchase running shoes?

 

 

______________

 

ANS:

0.171; 0.807; 0.677; 0.644

 

PTS:   1

 

15. A Statistics Department is contacting alumni by telephone asking for donations to help fund a new computer laboratory. Past history shows that 80% of the alumni contacted in this manner will make a contribution of at least $50. A random sample of 20 alumni is selected. Let x represent number of alumni that makes at contribution of at least $50.

 

What is the probability that exactly 15 alumni will make a contribution of at least $50?

 

 

______________

 

What is the probability that between 14 and 18 alumni, inclusively, will make a contribution of at least $50?

 

 

______________

 

What is the probability that less than 17 alumni will make a contribution of at least $50?

 

 

______________

 

What is the probability that more than 15 alumni will make a contribution of at least $50?

 

 

______________

 

How many alumni would you expect to make a contribution of at least $50?

 

 

______________

 

ANS:

0.174; 0.844; 0.589; 0.630; 16

 

PTS:   1

 

16. Let x be a binomial random variable with n = 15 and p = 0.20.

 

Calculate P(x  4) using the binomial formula.

 

 

______________

 

Calculate P(x  4) using the cumulative binomial probabilities table in Appendix I of your book.

 

 

______________

 

Are the results of your two previous answers very different, or very similar?

 

 

______________

 

Calculate the mean of the random variable x.

 

 

______________

 

Calculate the standard deviation of the random variable x.

 

 

______________

 

Fill in the table below:

 

	probability
	______________
	______________
	______________

 

Are the results consistent with Tchebysheff’s Theorem?

 

 

______________

 

With the Empirical Rule?

 

 

______________

 

ANS:

.835767; .836; Very similar; 3; 1.549; .669; .982; .996; Yes; Yes

 

PTS:   1

 

17. In 1996, the average combined SAT score (math + verbal) for students in the United States was 1013, and 41% of all high school graduates took this test. Suppose that 500 students are randomly selected throughout the United States.

 

Which of the following random variables has an approximate binomial distribution? The number of students who took the SAT.

 

 

____________________________

 

The scores of the 500 students on the SAT.

 

 

____________________________

 

The number of students who scored above average on the SAT.

 

 

____________________________

 

The amount of time it took each student to complete the SAT.

 

 

____________________________

 

ANS:

Binomial distribution; Not a binomial distribution; Binomial distribution; Not a binomial distribution

 

PTS:   1

 

18. Car color preferences change over the years and according to the particular model that the customer selects. In a recent year, 20% of all luxury cars sold were white. Assume that 25 cars of that year and type are randomly selected. Find the probabilities listed below:

 

Define x to be the number of cars that are white. Then p = P(white) = 0.20 and n = 25. Use the cumulative binomial probabilities table in Appendix I of your book.

 

At least five cars are white.

 

 

______________

 

At most six cars are white.

 

 

______________

 

Exactly four cars are white.

 

 

______________

 

More than three cars are white.

 

 

______________

 

Between three and five cars (inclusive) are white.

 

 

______________

 

More than 20 cars are not white.

 

 

______________

 

ANS:

0.579; 0.780; 0.187; 0.766; 0.519; 0.421

 

PTS:   1

 

19. A new surgical procedure is said to be successful 90% of the time. Suppose the operation is performed five times and the results are assumed to be independent of one another. Find the probabilities listed below:

 

Define x to be the number of successful operations. Then p = P(success) = 0.9 and n = 5. We will use the binomial formula. However, you may also use the cumulative binomial probabilities table in Appendix I of your book to calculate the necessary probabilities.

 

All five operations are successful.

 

 

______________

 

Exactly four are successful.

 

 

______________

 

Less than two are successful.

 

 

______________

 

ANS:

0.59049; 0.32805; 0.000460

 

PTS:   1

 

20. Four in ten Americans who travel by car look for gas and food outlets that are close to or visible from the highway. Suppose a random sample of n = 20 Americans who travel by car are asked how they determine where to stop for food and gas. Let x be the number in the sample who respond that they look for gas and food outlets that are close to or visible from the highway.

 

Calculate the mean of x.

 

 

______________

 

Calculate the variance of x.

 

 

______________

 

Calculate the interval .

 

 

______________

 

What values of the binomial random variable x fall into this interval?

 

 

______________

 

Find .

 

 

______________

 

How does this compare with the fraction in the interval  for any distribution? For mound-shaped distributions?

 

 

________________________________________________________

 

ANS:

8; 4.8; (3.61822, 12.3818); (4, 12); 0.963; This value agrees with Tchebysheff’s Theorem (at least 3 / 4 of the measurements in this interval) and also with the Empirical Rule (approximately 95% of the measurements in this interval.

 

PTS:   1

 

21. Let x be a binomial random variable with n = 20 and p = 0.05.

 

Calculate  using the cumulative binomial probabilities table in Appendix I of your book.

 

 

______________

 

Use the Poisson approximation to calculate .

 

 

______________

 

Compare your results. Is the approximation accurate?

 

 

______________

 

ANS:

0.925; 0.9197; Yes

 

PTS:   1

 

22. A psychiatrist believes that 90% of all people who visit doctors have problems of a psychosomatic nature. She decides to select 20 patients at random to test her theory.

 

Assuming that the psychiatrist’s theory is true, what is the expected value of x, the number of the 20 patients who have psychosomatic problems?

 

 

______________

 

What is the variance of x, assuming that the theory is true?

 

 

______________

 

Find , assuming that the theory is true.

 

 

______________

 

Based on the probability in part (c), if only 14 of the 20 sampled had psychosomatic problems, what conclusions would you make about the psychiatrist’s theory?

 

 

______________

 

Explain.

 

 

________________________________________________________

 

ANS:

18; 1.8; 0.011; The psychiatrist is INCORRECT; Assuming that the psychiatrist is correct, the probability of observing x = 14 or the more unlikely values, x = 0, 1, 2,..,13 is very unlikely. Hence, one of two conclusions can be drawn. Either we have observed a very unlikely event, or the psychiatrist is incorrect and p is actually less than 0.9. We would probably conclude that the psychiatrist is incorrect. The probability that we have made an incorrect decision is 0.011, which is quite small.

 

PTS:   1

 

23. About 80% of Florida residents believe that Florida is a nice or very nice place to live. Suppose that five randomly selected Florida residents are interviewed.

 

Find the probability that all five residents think that Florida is a nice or very nice place to live.

 

 

______________

 

What is the probability that at least one resident does not think that Florida is a nice or very nice place to live?

 

 

______________

 

What is the probability that exactly one resident does not think that Florida is a nice or very nice place to live?

 

 

______________

 

ANS:

0.3277; 0.6723; 0.4096

 

PTS:   1

 

MBB.IntroProb13.ch05sec4

 

TRUE/FALSE

 

1. Hypergeometric probability distributions is an example of continuous probability distribution.

 

ANS:  F                    PTS:   1

 

2. Hypergeometric probability distributions is an example of discrete probability distributions.

 

ANS:  T                    PTS:   1

 

3. The hypergeometric probability distribution formula calculates the probability of x successes when a random sample of size n is drawn without replacement from a population of size N within which M units have the characteristic that denotes success, and N – M units have the characteristic that denotes failure.

 

ANS:  T                    PTS:   1

 

4. The hypergeometric random variable is the number of successes achieved when a random sample of size n is drawn with replacement from a population of size N within which M units have the characteristic that denotes success.

 

ANS:  F                    PTS:   1

 

5. A hypergeometric probability distribution shows the probabilities associated with possible values of a discrete random variable when these values are generated by sampling with replacement and the probability of success, therefore, changes from one trial to the next.

 

ANS:  F                    PTS:   1

 

6. A warehouse contains six parts made by company A and ten parts made by company B. If four parts are selected at random from the warehouse, the probability that none of the four parts is from company A is approximately .1154.

 

ANS:  T                    PTS:   1

 

7. A warehouse contains six parts made by company A and ten parts made by company B. If four parts are selected at random from the warehouse, the probability that one of the four parts is from company B is approximately .1099.

 

ANS:  T                    PTS:   1

 

8. A warehouse contains six parts made by company A and ten parts made by company B. If four parts are selected at random from the warehouse, the probability that all four parts are from company B is approximately .008.

 

ANS:  F                    PTS:   1

 

MULTIPLE CHOICE

 

1. When sampling without replacement, the appropriate probability distribution is:

 

a.	a binomial distribution
b.	a hypergeometric distribution
c.	a Poisson distribution
d.	all of these

 

 

ANS:  B                    PTS:   1

 

2. The hypergeometric probability distribution is identical to:

 

a.	the binomial distribution
b.	the Poisson distribution
c.	any continuous probability distribution
d.	none of these

 

 

ANS:  D                    PTS:   1

 

3. Using the hypergeometric formula, :

 

a.	we calculate the probability of k successes when a random sample of size n is drawn without replacement from a population of size N within which M units have the characteristic that denotes success
b.	we assume that k = 0, 1, 2, ….n or M (whichever is smaller)
c.	we assume that n  < N and M  < N
d.	all of these are true

 

 

ANS:  D                    PTS:   1

 

4. Given that n is the number of trials of a random experiment, N is population size, M is the number of population units with the “success” characteristic, and p is the probability of success in the first trial, the mean of the hypergeometric random variable’s probability distribution always equals:

 

a.	n
b.	np
c.	n(N/M)
d.	none of these

 

 

ANS:  B                    PTS:   1

 

5. The hypergeometric probability distribution:

 

a.	provides probabilities associated with possible values of a binomial random variable in situations in which these values are generated by sampling a finite population
b.	provides probabilities associated with possible values of a binomial random variable in situations in which sampling is done without replacement
c.	provides probabilities associated with possible values of a binomial random variable in situations in which the probability of success changes from one trial to the next
d.	is correctly described by all of these

 

 

ANS:  D                    PTS:   1

 

6. The hypergeometric probability distribution is used rather than the binomial distribution when the sampling is performed:

 

a.	with replacement from a finite population
b.	without replacement from a finite population of size N and that size N is small in relation to the sample size n namely, n / N .05
c.	without replacement from an infinite population
d.	with replacement from an infinite population

 

 

ANS:  B                    PTS:   1

 

7. A small community college in Ohio has four student organizations (A, B, C, and D). Organization A has 5 students, B has 8, C has 10, and D has 12. It is thought that new students have no preference for one of these organizations over the other. If seven new students are admitted to the college, what is the probability that one student will choose organization A, one will choose B, two will choose C, and three will choose D?

 

a.	approximately .059
b.	.200
c.	approximately .243
d.	Cannot be determined without additional information.

 

 

ANS:  A                    PTS:   1

 

PROBLEM

 

1. A professor has received a grant to travel to an archaeological dig site. The grant includes funding for five graduate students. If there are five male and four female graduate students eligible and equally qualified, what is the probability that the professor will select three male and two female graduate students to accompany her to the dig site?

 

 

______________

 

ANS:

0.4762

 

PTS:   1

 

2. Let x be a hypergeometric random variable with N = 12, n = 4, and M = 3.

 

Calculate p(0).

 

 

______________

 

Calculate p(1).

 

 

______________

 

Calculate p(2).

 

 

______________

 

Calculate p(3).

 

 

______________

 

Calculate the mean.

 

 

______________

 

Calculate the variance.

 

 

______________

 

What proportion of the population of measurements fall into the interval ?

 

 

______________

 

Into the interval ?

 

 

______________

 

Do these results agree with those given by Tchebysheff’s Theorem?

 

 

______________

 

ANS:

0.2545; 0.5091; 0.2182; 0.0182; 1.0; 0.5455; 0.9818; 1.0; Yes

 

PTS:   1

 

3. A company has five applicants for two positions: three women and two men. Suppose that the five applicants are equally qualified and that no preference is given for choosing either gender. Let x equal the number of men chosen to fill the two positions.

 

What is the mean of the probability distribution of x?

 

 

______________

 

What is the variance of the probability distribution of x?

 

 

______________

 

What is the standard deviation of the probability distribution of x?

 

 

______________

 

ANS:

0.8; 0.36; 0.6

 

PTS:   1

 

4. A student has decided to rent three movies for a three-day weekend. If there are 4 action movies and 6 romance movies that are of equal interest to the student, what is the probability that the student will select 1 romance movie and 2 action movies?

 

 

______________

 

ANS:

0.30

 

PTS:   1

 

5. A random sample of 4 units is taken from a group of 15 items in which 4 units are known to be defective. Assume that sampling occurs without replacement, and the random variable x represents the number of defective units found in the sample.

 

The mean of the random variable x is:

 

 

______________

 

The variance of the random variable x is:

 

 

______________

 

 

P(x = 0) = ______________

 

 

P(x = 1) = ______________

 

 

P(x = 2) = ______________

 

 

P(x = 3) = ______________

 

 

P(x = 4) = ______________

 

ANS:

1.067; 0.6146; 0.2418; 0.4835; 0.2418; 0.0322; 0.0007

 

PTS:   1

 

6. A college has seven applicants for three scholarships: four females and three males. Suppose that the seven applicants are equally qualified and that no preference is given by the selection committee for choosing either gender. Let x equal the number of female students chosen for the three scholarships.

 

What is the mean of the distribution of x?

 

 

______________

 

What is the variance of the distribution of x?

 

 

______________

 

What is the probability that only one female will receive a scholarship?

 

 

______________

 

What is the probability that two females will receive a scholarship?

 

 

______________

 

What is the probability that none of the three males will receive a scholarship?

 

 

______________

 

What is the probability that none of the four females will receive a scholarship?

 

 

______________

 

ANS:

1.714; 0.4898; 0.3429; 0.5143; 0.1143; 0.0286

 

PTS:   1