Math 533.

*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.

μ= (Round to the nearest tenth as needed).

Find the standard deviation of the binomial distribution.

σ= (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 15 14 113

Totals 112 39 32 183

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?

Meals Service good Service poor Totals

Lunch 24 39 63

Dinner 41 39 80

Totals 65 78 143

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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