**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 6% 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 9 major bridges in the city, at least 3 will have an inspection rating of 4 or below in 2020.

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

**b.** Suppose that you actually observe 3 or more of the sample of 9 bridges with inspection rating of 4 or below in 2020. What inference can you make? Why?

Choose correct answers:

A. Since the probability of this observation occurring is so large, it can be concluded that the forecast of 6% is too small. There would probably be more than 6%.

B. Since the probability of this observation occurring is so small, it can be concluded that the forecast of 6% is too large. There would be probably be less than 6%.

C. Since the probability of this observation occurring is so large, it can be concluded that the forecast of 6% is too large. There would be less than 6%.

D. Since the probability of this observation occurring is so small, it can be concluded that the forecast of 6% is too small. There would be more than 6%.

**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 associationâ€™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 10% 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 = 0.6 b. P(x) for n = 15, p = 0.3

c. P(x) for n = 5, p = 0.5 d. P(x) for n = 15, p = 0.9

e. P(x) for n = 25, p = 0,9 f. P(x = 3) for n = 20, p = 0.4

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

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

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

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

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

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

(Please, help me to know how to solve these questions by steps)

**Question 4**: The chances of a tax return being audited are about 14 in 1,000 of an income is less than $1,000 and 37 in 1,000 if an income is $100,000 or more. Complete parts **a** through **e**

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) = ? (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) = ? (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.