IRS data indicates that the tax refunds it issued this year follow the normal distribution with μ = 1,200 and σ = 200. Based on this information calculate the following probabilities.

1. Probability of selecting a tax return, the refund for which will fall between $1,170 and $1,200:
   
   
   
2. Probability of selecting a tax return, the refund for which will be less than $1,406:
   
   
   
3. Probability of selecting a tax return, the refund for which will be more than $1,598:
   
   
   
4. Probability of selecting a tax return, the refund for which will fall between $1,132 and $1,354:

b.The lead engineer on the design team has requested that you match the reliability of the parallel system with a series system. All three of the valves in series will have the same probability of functioning correctly. What does this probability need to be to equal the probability of the parallel system?

c. Comment on the advantages and disadvantages of using parallel systems in aircraft design.

Based on the experimental data, it was determined that the probability of the three valves functioning correctly are:

Valve 1: 95% • Valve 2: 94% • Valve 3: 92%

a) What is the probability that a hand of 13 cards contains four of a kind (e.g., four 5’s, four Kings, four aces, etc.)?

b) A single card is randomly drawn from a thoroughly shuffled deck of 52 cards. What is the probability that the drawn card will be either a diamond or a queen?

c) The probability that the events A and B both occur is 0.3. The individual probabilities of the events A and B are 0.7 and 0.5. What is the probability that neither event A nor event B occurs?

Assume that​ women's heights are normally distributed with a mean given by mu equals 62.2 in​, and a standard deviation given by sigma equals 1.9 in.

Complete parts a and b.

a. If 1 woman is randomly​ selected, find the probability that her height is between 61.9 in and 62.9 in. The probability is approximately nothing. ​(Round to four decimal places as​ needed.)

b. If 14 women are randomly​ selected, find the probability that they have a mean height between 61.9 in and 62.9 in. The probability is approximately nothing. ​(Round to four decimal places as​ needed.)

Ultimate frisbee players are so poor they don’t own coins. So, team captains decide which team will play offense first by flipping frisbees before the start of the game. Rather than flip one frisbee and call a side, each team captain flips a frisbee and one captain calls whether the two frisbees will land on the same side, or on different sides. Presumably, they do this instead of just flipping one frisbee because a frisbee is not obviously a fair coin - the probability of one side seems likely to be different from the probability of the other side.

1. Suppose you flip two fair coins. What is the probability they show different sides?
2. Suppose two captains flip frisbees. Assume the probability that a frisbee lands convex side up is pp. Compute the probability (in terms of pp) that the two frisbees match.
3. Make a graph of the probability of a match in terms of pp.
4. One Reddit user flipped a frisbee 800 times and found that in practice, the convex side lands up 45% of the time. When captains flip, what is the probability of “same”? What is the probability of “different”?
5. What advice would you give to an ultimate frisbee team captain?
6. Is the two frisbee flip better than a single frisbee flip for deciding the offense?

A small University in Florida offers STEM (science, technology, engineering, and mathematics) internships to men in STEM majors at the university. A man must be 20 years old or older to meet the age requirement for the internships. The table below shows the probability distribution of the ages of the men in STEM majors at the university.

Ages 17 18 19 20 21 22 23 or older

Probability 0.005 0.107 0.111 0.252 0.249 0.213 0.063

a. Suppose one man is selected at random from the men in STEM majors at the university. What is the probability that the man selected will not meet the age requirement for the internships? Show your work and label your answer with appropriate probability notation.

The university will select a simple random sample of 10 men in STEM majors to participate in a focus group about the internships.

(b) What is the probability that exactly 2 of the 10 men selected will not meet the age requirement for the internships? Show your work, round your answer to four decimal places, and label your answer with appropriate probability notation.

(c) What is the probability that more than 2 of the 10 men selected will not meet the age requirement for the internships? Show your work, round your answer to four decimal places, and label your answer with appropriate probability notation.

John finds a bill on his desk. He has three options: ignore it and leave it on his own desk, move the bill over to his wife Mary's desk, or pay the bill immediately. The probability that he leaves it on his own desk is 0.6. The probability that he moves it to Mary's desk is 0.3. The probability that he pays the bill immediately is 0.1.

Similarly, if Mary finds a bill on her desk she can choose to leave it on her own desk, put it on John's desk, or pay it immediately. The probability that it remains on her desk is 0.6. The probability she moves it to John's desk is 0.1. The probability she pays it immediately is 0.3.

Once a bill is paid it will not return to either of the desks. In other words, there is a 0% chance that a bill will return to John's desk or Mary's desk once it goes to the mailbox.

Assume this is a Markov Chain process. Set up the transition matrix and use it to answer the following questions. (Hint: When determining what your matrix labels should be, think of the location of the bill, not the action done to it. For example, the label "moves to the other desk" would not be a valid label.)

(a) What is the probability that a bill currently on John's desk will be paid within two days?


(b) What is the probability that a bill currently on John's desk will be on Mary's desk 3 days later?

A survey found that 25% of consumers from a Country A are more likely to buy stock in a company based in Country​ A, or shop at its​ stores, if it is making an effort to publicly talk about how it is becoming more sustainable. Suppose you select a sample of 200 respondents from Country A. Complete parts​ (a) through​ (d) below.

1. ​ What is the probability that in the​ sample, fewer than 25% are more likely to buy stock in a company based in Country​ A, or shop at its​ stores, if it is making an effort to publicly talk about how it is becoming more​ sustainable?

The probability is __%?

(Round to two decimal places as needed)

2. What is the probability that in the​ sample, between 20​% and 30​% are more likely to buy stock in a company based in Country​ A, or shop at its​ stores, if it is making an effort to publicly talk about how it is becoming more​ sustainable?

The probability is __%?

(Round to two decimal places as needed)

3. What is the probability that in the​ sample more than 20​% are more likely to buy stock in a company based in Country​ A, or shop at its​ stores, if it is making an effort to publicly talk about how it is becoming more​ sustainable?

The probability is __%?

(Round to two decimal places as needed)

4. If a sample of 800 is taken, how does this change you answers to (a) through (c)

If a sample of 800 is taken, what is the probability that in the​ sample, fewer than 25% are more likely to buy stock in a company based in Country​ A, or shop at its​ stores, if it is making an effort to publicly talk about how it is becoming more​ sustainable?

The probability is __%?

(Round to two decimal places as needed)

If a sample of 800 is taken, what is the probability that in the​ sample, between 20​% and 30​% are more likely to buy stock in a company based in Country​ A, or shop at its​ stores, if it is making an effort to publicly talk about how it is becoming more​ sustainable?

The probability is __%?

(Round to two decimal places as needed)

If a sample of 800 is taken, what is the probability that in the​ sample more than 20​% are more likely to buy stock in a company based in Country​ A, or shop at its​ stores, if it is making an effort to publicly talk about how it is becoming more​ sustainable?

The probability is __%?

(Round to two decimal places as needed)

1. The method of tree ring dating gave the following years A.D. for an archaeological excavation site. Assume that the population of x values has an approximately normal distribution.

(a) Use a calculator with mean and standard deviation keys to find the sample mean year x and sample standard deviation s. (Round your answers to the nearest whole number.

(b) Find a 90% confidence interval for the mean of all tree ring dates from this archaeological site. (Round your answers to the nearest whole number.)

2.

Sherds of clay vessels were put together to reconstruct rim diameters of the original ceramic vessels at the Wind Mountain archaeological site†. A random sample of ceramic vessels gave the following rim diameters (in centimeters).

(a) Use a calculator with mean and sample standard deviation keys to find the sample mean x and sample standard deviation s. (Round your answers to one decimal place.)

(b) Compute a 98% confidence interval for the population mean μ of rim diameters for such ceramic vessels found at the Wind Mountain archaeological site. (Round your answers to one decimal place.)

3. Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 13 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with σ = 0.30 gram.

(a) Find an 80% confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error? (Round your answers to two decimal places.)

lower limit

upper limit

margin of error

(b)What conditions are necessary for your calculations? (Select all that apply.)

n is large

uniform distribution of weights

σ is unknownσ is known

normal distribution of weights

(c) Interpret your results in the context of this problem.

a. The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80.

b. The probability that this interval contains the true average weight of Allen's hummingbirds is 0.20.    

c. There is a 20% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region.

d. There is an 80% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region.

e. The probability to the true average weight of Allen's hummingbirds is equal to the sample mean.

(d) Find the sample size necessary for an 80% confidence level with a maximal margin of error E = 0.16 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)