Test Bank Essentials of Mathematical Statistics 1st Edition by Brian Albright

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Test Bank Essentials of Mathematical Statistics 1st Edition by Brian Albright

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Test Bank Essentials of Mathematical Statistics 1st Edition by Brian Albright

File:  chap01, Chapter 1

 

Section 1.2

  1. You roll a pair of fair four-sided dice (with faces labeled 1, 2, 3, and 4) and add the numbers on the sides that land face down.

 

  1. Give the elements of the sample space.
  2. Describe the distribution of the sum.
  3. Find the probability that the sum of the dice is greater than 5.
  4. Find the probability that the sum of the dice is odd.

 

Ans:

  1. {2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 8}
  2.  

x

2

3

4

5

6

7

8

P(x)

1/16

2/16

3/16

4/16

3/16

2/16

1/16

 

  1. 3/8
  2. 1/2

 

 

  1. You roll a pair of fair four-sided dice (with faces labeled 1, 2, 3, and 4) and multiply the numbers on the sides that land face down.

 

  1. Give the elements of the sample space in the form of a table.
  2. Describe the distribution of the product.
  3. Calculate the probability the product is greater than 4 or the first die is odd.

 

Ans:

A.

 

 

First Die

Second Die

1

2

3

4

1

1

2

3

4

2

2

4

6

8

3

3

6

9

12

4

4

8

12

16

B.

x

1

2

3

4

6

8

9

12

16

P(x)

1/16

2/16

2/16

3/16

2/16

2/16

1/16

2/16

1/16

C.13/16

 

 

  1. A number, call it X, is randomly chosen from the interval . Calculate the following probabilities:
  2.  
  3.  
  4. Is X a discrete or continuous random variable? Explain.

 

Ans:

  1. 5/11
  2. 2/11
  3. X is continuous because it can take any value between -5 and 6

 

 

 

  1. Bob is going to ask several different girls for a date. He figures that since each girl can either say yes or no, the probability that each girl says yes is 0.5 so about half of the girls should say yes. Explain what is wrong with this reasoning.

 

Ans: There are only 2 outcomes, but they are not equally likely.

 

 

 

  1. An experiment consists of selecting 2 cubes with replacement from a bag consisting of 1 red, 1 blue, and 1 green cube.

 

  1. List the elements of the sample space.
  2. Assuming all outcomes are equally likely, find the probability of getting 2 cubes of different colors.
  3. Let the random variable X denote the number of green cubes in the 2 cubes selected. Is X a discrete or continuous random variable? Explain.

 

Ans:

  1.  

 

1st Cube

2nd Cube

R

B

G

R

RR

RB

RG

B

BR

BB

BG

G

GR

GB

GG

  1. 6/9
  2. X is discrete because it can take values of only 0, 1, and 2.

 

 

 

  1. A point is randomly selected from inside a circle with radius r. Find the probability the point is closer to the center than to the perimeter.

 

Ans: The point is closer to the center than to the perimeter if the point is chosen from a circle of radius  centered at the center of the larger circle. The probability of this occurring is

                                                              

 

 

 

  1. Suppose that a gambler simulates 20 plays of a game, wins 1 time, and estimates P(winning) = 1/20 = 0.05. He then simulates 5,000 plays of the game, wins 450 times, and estimates P(winning) = 450/5,000 = 0.09. Which estimate, 0.05 or 0.09 is probably closer to the true theoretical probability and why?

 

Ans: The number 0.09 is probably closer because it is based on more trials. The Law of Large Numbers says that the more trials, the closer the relative frequency is to the theoretical probability.

 

 

 

  1. A gambler uses theory to calculate the probability of winning a card game and gets P(winning) = 0.10. Which of these options best describes the meaning of this probability?

 

  1. He is guaranteed to win exactly 10% of the time.
  2. In the long run he will win approximately 10% of the time
  3. He will win 10 times.
  4. He will win on every 10th play.

 

Ans: B

 

 

 

  1. Suppose you guess an answer to a math problem (not a multiple choice question). Is the probability that you guess correctly equal to 1/2? Why?

 

  1. Yes, because you have a 50-50 chance of getting it right.
  2. No, because this random experiment does not have a sample space.
  3. No, because the two outcomes of right and wrong are not equally likely.
  4. Yes, because the answer is either right or wrong.

 

Ans: C

 

 

 

Section 1.3

 

  1. If 8 boats enter a race, how many possibilities are there for the first through third place finishers?

 

Ans:  

 

 

 

  1. Five soccer players, three football players, and two baseball players are going to sit in a row of chairs. In how many ways can these athletes be arranged?

 

Ans:  

 

 

 

  1. Cindy must write evaluation reports on three hospitals and two health clinics as part of her degree program in public health. If there are six hospitals and five clinics in her area, how many different reports could she write?

 

Ans:  

 

 

 

  1. The manager of a baseball team has filled the first, fifth, and seventh slots in the nine-player starting lineup. If he has 12 players left to choose from, how many different starting lineups are possible?

 

Ans:  

 

 

 

  1. Find the coefficient of the term in the expansion of  

 

Ans:  

 

 

 

Section 1.4

 

  1. Consider a random experiment with the sample space and let A and B be events. State which of the axiom(s) of probability justifies each statement below:

 

  1.  
  2.  
  3. If , then
  4.  

 

Ans: 3, 2, 1, 3

 

 

 

  1. Determine whether or not it is reasonable to assume that each pair of events is disjoint. Explain your reasoning.

 

  1. A student goes to the store to purchase a single calculator. Let A be the event she purchases a Texas Instruments and B be the event she purchases a Hewlett Packard.
  2. A hiker hikes through a forest. Let A be the event he sees a bird and B be the event he crosses a stream.

 

Ans:

  1. These events are disjoint because she can’t purchase both a Texas Instruments and a Hewlett Packard.
  2. These are not disjoint because he can see a bird and cross a stream only his hike.

 

 

 

  1. A parking lot contains a total of 45 vehicles each of which is either a pickup or blue in color. There are 30 pickups and 25 blue vehicles. Consider the random experiment of choosing a vehicle from this lot. Find the probability of choosing a blue pickup.

 

Ans: Let A denote the event of selecting a pickup and B denote the event of selecting a blue vehicle. Then , , and . Thus by the addition rule,

                                      

 

 

 

  1. A game is played by flipping a fair coin until the first tail appears. Find the probability the game lasts more than 3 flips.

 

Ans: We use complements and disjoint events:

                                

 

 

Section 1.5

  1. 100 people were given either a 10-mg pill of a new experimental drug or a placebo. They recorded whether or not they got a headache. The following table summarizes the results.

 

 

Received drug

Received placebo

Totals

Headache

15

65

80

No headache

17

3

20

Totals

32

68

100

 

  1. Find the probability of selecting someone who received the drug or got a headache.
  2. If 3 different subjects are randomly selected without replacement, find the probability that at least 1 received a placebo.
  3. If one subject is randomly selected, find P(Headache | Received drug).
  4. Are the events of getting a headache and receiving a placebo disjoint? Why?

 

Ans:

  1.  
  2.  
  3.  
  4. No they are not disjoint because 65 people received a placebo and got a headache.

 

 

 

 

  1. Suppose the student body at a certain university contains 1200 students and is 55% female and 45% male. Consider the experiment of randomly choosing 5 students and recording their genders.

 

  1. Treating the selections as dependent, find the probability that the first and fourth students are female.
  2. Repeat part a, but treat the selections as independent.
  3. Treating the selections as independent, find the probability that we choose exactly two females.
  4. If we were to select 200 students, would it be reasonable to treat the selections as independent? Why or why not?

 

Ans:

  1. There are a total of 660 females and 540 males at the university so that

                          

  1.  
  2. There are ways of choosing exactly two females, each of which has probability 0.027565. Thus the total probability is .
  3. Since 200 is more than 5% of the population, it would not be reasonable to treat the selections as independent.

 

 

  1. Suppose a bag contains 2 red cubes and 3 blue cubes. You pick 2 cubes without replacement. Find the probability that the first is blue and the second is red.

 

Ans:  

 

 

 

  1. If 5 people are randomly selected with replacement from a group of 15 people of which Joe is a member, find the probability that Joe is selected.

 

Ans:  

 

 

 

Section 1.6

 

 

 

  1. Health officials estimate that about 0.5% of the population has a certain disease. There is a test that when given to people who have the disease would correctly identify it 95% of the time. This test also gives a false positive about 3% of the time. A person is selected at random and gets a positive result when given this test. What is the conditional probability that the person actually has the disease?

 

Ans:

 

 

 

  1. Three toy boxes, call them boxes A, B, and C, contain 20, 50, and 30 stuffed animal, respectively. Of the animals in box A, 45% are bears; in box B, 35% are bears; and in box C, 80% are bears. A girl randomly chooses a stuffed animal from one of the boxes and gets a bear. Find the conditional probability the bear was chosen from box A.

 

Ans:  

 

 

 

  1. If events and  form a partition of the sample space, B is an event, , , , and , find.

 

Ans: By Bayes’ Theorem,  . Solving this for  yields

 

 

 

Section 1.7

 

  1. A fair six-sided die is rolled. Let A be the event the resulting number is even and B be the event the number is greater than 4. List the elements of both events and use the definition of independence to show that A and B are independent.

 

Ans:  

 

 

 

  1. A math professor has 10 algebra and 5 calculus books on her bookshelf. If she randomly selects 3 books, find the probability that

 

  1. they are all algebra books,
  2. the third book is the first calculus book selected, and
  3. at least 1 book is an algebra book.

 

Ans:

  1.  
  2.  
  3.  

                             

 

 

 

  1. Determine whether or not it is reasonable to assume that each pair of events is independent. Explain your reasoning.

 

  1. You go to the beach. Let A be the event you swim all day. Let B be the event you get a sunburn.
  2. A car is randomly chosen from a parking lot. Let A bet the event the car is gray and B be the event the car has an out-of-state license plate.

 

Ans:

  1. These events are not independent. If you swim all day, you will be in the sun the entire day, so you increase the probability of getting a sunburn.
  2. It is reasonable to assume these are independent.

 

 

 

  1. You roll a pair of fair five-sided dice (with faces 1, 2, 3, 4, and 5). Find the probability that the second die is greater than 2 or the first die is even.

 

Ans:

  

 

 

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