# Math 533 Assignment Problems 2015

Question

Question 1: A national standard requires that public bridge over 200 feet in length

must be inspected and rated every 2 years. The rating scale ranges from 0 (poorest

rating to 0 (highest rating). A group of engineers used a probabilistic model to

forecast the inspection ratings of all major brides in a city. For the year 2020, the

engineers forecast that 9% of all major bridges in that city will have rating of 4 or

below. Complete parts a and b.

a. Use the forecast to find the probability that in a random sample of 10 major

bridges in the city, at least 3 will have an inspection rating of 4 or below in 2020.

P(x ?3 ) = ? (Round to five decimal places as needed)

b. Suppose that you actually observe 3 or more of the sample of 10 bridges with

inspection rating of 4 or below in 2020. What inference can you make? Why?

Question 2: According to a certain golf association, the weight of the golf ball ball

shall not be greater than 1.620 ounces (45.93 grams). The diameter of the ball shall

not be less than 1.680 inches. The velocity of the ball shall not be greater than 250

feet per second. The golf association periodically checks the specifications of golf

balls using random sampling. Three dozen of each kind are sampled, and if more

than three do not meet size or velocity requirements, that kind of ball ir removed

from the golf association’s approved list. Complete parts a and b.

a. What assumption must be made in order to use the binomial probability

distribution to calculate the probability that a particular kind of golf ball will be

removed?

A. The experiment consists of n identical trials. The number of outcomes can vary.

The probability of success can change. The trials are independent.

B. The experiment consists of n identical trials. There are only two possible

outcomes on each trial. The probability of success remains the same from trial to

trial. The trials are independent.

C. The experiment consists of n identical trials. There are only two possible

outcomes on each trial. The probability of success can change from trial to trial. The

trials are dependent.

What information must be known in order to use the binomial probability distribution

to calculate the probability that a particular kind of golf ball will be removed?

A. The percentage of that kind of golf ball that meets the velocity requirements.

B. The percentage of that kind of golf ball that meets all the requirements.

C. The percentage of that kind of golf ball that meets the size requirements.

D. The percentage of that kind of golf ball that fails to meet both size and velocity

requirements.

b. Suppose 15% of all balls produced by a particular manufacturer are less than 1.680

inches in diameter, and assume that the number of such balls, x, in a sample of three

dozen balls can be adequately characterized by a binomial probability distribution.

Find each mean of the binomial distribution.

?=¿ (Round to the nearest tenth as needed).

Find the standard deviation of the binomial distribution.

? =¿ (Round to the nearest thousandth as needed).

Question 3: If x is a binomial random variable, use the binomial probability table to

find the probabilities below.

a. P(x=3) for n = 10, p = 0.3

b. P(x ? 6 ) for n = 15, p = 0.5

c. P(x ¿ 1 ) for n = 5, p = 0.6

d. P(x ¿ 6 ) for n = 15, p =

0.8

e. P(x ?14 ) for n = 25, p = 0,8

f. P(x = 3) for n = 20, p = 0.1

a. P(x = 3) = ? (Round to three decimal places as needed)

b. P(x ? 6 ) = ? (Round to three decimal places as needed)

c. P(x ¿ 1 ) = ? (Round to three decimal places as needed)

d. P(x ¿ 6 ) = ? (Round to three decimal places as needed)

e. P(x ?14 ) = ? (Round to three decimal places as needed)

f. P(x = 3) =? (Round to three decimal places as needed)

Question 4: The chances of a tax return being audited are about 22 in 1,000 of an

income is less than $1,000 and 34 in 1,000 if an income is $100,000 or more.

Complete parts a through e

a . What is the probability that a taxpayer with income less than $100,000 will be

audited? With income of $100,000 or more?

P(taxpayer with income less than $100,000 is audited) = ?

(Type an integer or a decimal)

What is the probability that a taxpayer with income of $100,000 or more will be

audited?

P(taxpayer with income of $100,000 or higher is audited) = ?

(Type an integer or a decimal.)

b. If three taxpayers with income under $100,000 are randomly selected, what is the

probability that exactly one will be audited? That more than one will be audited?

P(x=1) = ? (Round to four decimal places as needed)

What is the probability that more than one will be audited?

P(x ¿ 1 ) = ? (Round to four decimal places as needed)

c. Repeat part b assuming that three taxpayers with income of $100,000 or more are

randomly selected.

P(x=1) = ? (Round to four decimal places as needed)

What is the probability that more than one will be audited?

P(x ¿ 1 ) = ? (Round to four decimal places as needed)

d. If two taxpayers with incomes under $100,000 are randomly selected and two

with incomes more than $100,000 are randomly selected, what is the probability

that none of these taxpayers will be audited?

P(none of the taxpayers will be audited) = ? (Round to four decimal places as

needed)

e. What assumption did you have to make in order to answer these questions?

A. We must assume that the variables are binomial random variables. We must

assume that the trials are identical, the probability of success is the same from trial

to trial, and that the trials are independent.

B. We must assume that the variables are binomial random variables. We must

assume that the trials are identical and dependent.

C. We must assume that the variables are random variables. We must assume that

the trials are identical, and the probability of success varies from trial to trial.

D. We must assume that the variables are binomial random variables. We must

assume that the trials are identical, the probability of success varies from trial to

trial, and that trials are dependent.

Question 5: According to a consumer survey of young adults (18-24 years of age)

who shop online, 17% own a mobile phone with internet access. In a random sample

of 200 young adults who shop online, let x be the number who own a mobile phone

with internet access.

a. Explain why x is a binomial random variable (to a reasonable degree of

approximation). Choose the correct explanation below.

A. The experiment consists of n identical, dependent trials, where there are only two

possible outcomes, S (for Success) and F (for Failure).

B. The experiment consists of n identical, dependent trials, with more than two

possible outcomes. The probability that an event occurs varies from trial to trial.

C. The experiment consists of n identical, independent trials, where there are only

two possible outcomes, S (for Success) and F (for Failure). The probability of S

remains the same from trial to trial. The variable x is the number of S’s in n trials.

D. The experiment consists of n identical, independent trials, where there are only

two possible outcomes, S (for Success) and F(for Failure). The probability of S varies

from trial to trial. The variable x is the number of F’s in n trials.

b. What is the value of p? Interpret this value.

p = ? (Type an integer or a decimal)

Choose correct interpretation of p below.

A. For any young adult, the probability that they own a mobile phone with internet

access is 1 – p.

B. For any young adult, the probability that they own a mobile phone with internet

access is np.

C. For any young adult, the probability that they do not own a mobile phone with

internet access is p.

D. For any young adult, the probability that they own a mobile phone with internet

access is p.

c. What is expected value of x? Interpret this value.

E(x) = ?

Choose correct interpretation below.

A. In a random sample of 200 young adult E(x) is the average number of young

people surveyed that will not own mobile phones with internet access.

B. In a random sample of 200 young adults E(x) will always the number of young

people surveyed that will own mobile phones with internet access.

C. In a random sample of 200 young adults E(x) is the average number of young

people surveyed that will own mobile phones with internet access.

D. In a random sample of 200 young adults E(x) is the largest possible number of

young people surveyed that will own mobile phones with internet access.

Question 6: A country’s government has devoted considerable funding to missile

defense research over the past 20 years. The latest development is the Space-Based

Infrared System (SBIRS), which uses satellite imagery to detect and track missiles.

The probability that an intruding object (e.g., a missile) will be detected on a flight

track by SBIRS is 0.6. Consider a sample of 10 simulated tracks, each with an

intruding object. Let x equal the number of these tracks where SBIRS detects the

object. Complete parts a through d.

a. Give the values of p and n for the binomial distribution.

p = ? (Round to one decimal place as needed)

n=?

b. Find P(x=3), the probability that SBIRS will detect the object on exactly 3 trakcs.

P(x=3) = ? (Round to three decimal places as needed)

c. Find P(x ?3 ), the probability that SBIRS will detect the object on at least 3

tracks.

P(x ?3 ) = ? (Round to three decimal places as needed)

d. Find E(x) and interpret the result.

A. E(x) = 4. For every 10 intruding object, SBIRS will detect an average of 4.

B. E(x) = 3. For every 10 intruding objects, SBIRS will detect an average of 3.

C. E(x) = 6. For every 10 intruding objects, SBIRS will detect an average of 6.

D. E(x) = 2.4. For every 10 intruding objects, SBIRS will detect an average of 2.4

E. E(x) = 5. For every 10 intruding objects, SBIRS will detect an average of 5

Question 7: Zoologists investigated the likelihood of fallow deer bucks fighting

during the mating season. During a 270-hour observation period, the researchers

recorded 205 encounters between two bucks. Of these, 183 involved one buck

clearly initiating the encounter with the other. In these 183 initiated encounters, the

zoologists kept track of whether or not a physical contact fight occurred and

whether the initiator ultimately won or lost the encounter. (The buck that is driven

away by the other is considered the losers). Suppose we select one of these 183

encounters and note the outcome (fight status and winner).

Initiator Wins No Clear Winner Initiator Loses Totals

Fight

28

24

18

70

No Fight 84

Totals 112

15

39

14

32

113

183

a. What is the probability that a fight occurs and the initiator wins?

The probability is = ? (Round to the nearest thousandth as needed)

b. What is the probability that there is no fight occurs?

The probability is = ? (Round to the nearest thousandth as needed)

c. What is the probability that there is no clear winner?

The probability is = ? (Round to the nearest thousandth as needed)

d. What is the probability that a fight occurs or initiator loses?

The probability is = ? (Round to the nearest thousandth as needed)

e. Are the events “no clear winner” and “initiator loses” mutually exclusive?

A. Yes

B. No

Question 8: A table classifying sample of 143 patrons of a restaurant according to

type of meal and their rating of the service is shown below. Suppose we select, at

random, one of the 143 patrons. Give that the meal was lunch, what is the

probability that the service was poor?

Service good

Meals

Lunch 24

Dinner 41

Totals 65

Service poor Totals

39

39

78

63

80

143

Given that the meal was lunch, the probability that the service was poor is = ?

(Type an integer or a simplified fraction).

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