About 60% of U.S full-time college students drank alcohol within a one-month period. You randomly select six U.S. full-time students. Find the probability that the number of U.S. full-time college students who drank alcohol within one-month period isa.Exactly twob.At least threec.Less than fourd.Assume, we sampled 5000 students. What expected number of U.S. full-time drank alcohol within a one-month period?e.From part (d), find the variance and the standard deviation.

Consider 2-bit input functions that has either constant output (0 or 1) or balanced output (the number of {0,1} outputs are equal to each other). First a class is selected with1/2 probability and then one of the functions that belong to the chosen class is selected uniform randomly. We would like to guess the class of the function with minimum function evaluations.

• What is the average number of guesses before we identify the chosen class using classical bits?

• Draw the quantum circuit that can predict the class using only one function evaluation.

Facebook recently examined all active Facebook users and determined that the average user has 183 Facebook friends. Suppose that the population of number of Facebook friends of a randomly selected user has a standard deviation of σ = 241. Facebook draws a SRS of 250 Facebook users. What is the probability that the average number of Facebook friends is less than 175? Apply the Central Limit Theorem.

Caution: do not round at your intermediate steps. Hint: you may need to use Excel for this calculation.

Group of answer choices

0.2998

0.5000

0.6873

0.9235

0.3714

Suppose the DNA bases in a gene sequence follow the distribution:

In an experiment, the number of observed bases that are “A” or “C” in a gene sequence is x, and the number of observed bases that are “G” or “T” is y. The EM method is used to find the best value for the parameter θ. Describe the Expectation step for computing the expected numbers of A, C, G, and T bases and the Maximization step for estimating θ. Give formulas for the estimations and detailed steps about how to obtain the formulas.

Six names will be drawn from a group of H+2 freshmen, F+L sophomores, 2*(H-1) juniors, and 2*H seniors. Calculate (rounding your answers to three decimal places) the probability that:

a. freshmen, sophomores and juniors are equally represented. Ans.___________

2. the number of freshmen drawn equates the number of sophomores and juniors drawn.Ans.___________

3. the ratio of drawn seniors to drawn sophomores is 3:1. Ans.___________

4. the ratio of drawn juniors to drawn freshmen is at least 3:1. Ans.___________

H= 6

T= 21

F=6

L =3

S=46

Mr. Mario Mendoza is a famous baseball player (Shortstop) in the Major League Baseball. Usually, he can successfully hit 2 out of 10 at bats. Therefore, the Mendoza line is named after him.

1. In a particular season, he attempted 100 at bats. Define X as the number of hits he will have. Write down how X is distributed.
2. What is the probability that he makes exactly 25 hits? (There are two ways to do this part, either way is fine.)  
3. Construct a 90% Confidence Interval for the average number of hits he will have in 100 at bats

5. A hand of five cards is drawn without replacement from a standard deck.

(a) Compute the probability that the hand contains both the king of hearts and the king of spades.

(b) Let X = the number of kings in the hand. Compute the expected value E(X). Hint: consider certain random variables X1, . . . , X4.

(c) Let Y = the number of “face” cards in the hand. Given is that E(Y ) = 15/13. Find the variance V ar(Y ). Hint: consider certain random variables Y1, . . . , Y12 and use your result from part (a).

2. Suppose a hurricane hits south Florida in any given year with probability 0.2. Describe the distribution of the following random variables, including both the name of the distribution and the parameters (such as X~Bernoulli(0.4)). (2 points each)

a.Let X be the number of years until the next hurricane hits South Florida.

b.Let X be the number of hurricanes that will hit south Florida in the next 10 years.

c.Let X indicate whether a hurricane will hit south Florida in 2020.

d.Let X indicate whether south Florida will avoid getting a hurricane in 2020.

Use 5% level of significance. At the fifth hockey game of the season at a certain arena, 200 people were selected at random and asked as to how many of the previous four games they had attended. The results are given in the table below.

Test the hypothesis that these  200  observed values can be regarded as a random sample from a  binomial distribution,where  ‘θ’  is the unknown parameter representing the probability of success that a person will attend a game.

An automated machine costs $1,000 and has a 20% probability of breaking irreparably at the end of each year (assuming it was working in the previous year). The machine has a maximum five-year life and will be disposed of with zero value at the end of five years. The machine produces $400 of cashflow at the end of each year and the discount rate is 10% per year. What is the most likely number of years the machine will last and what would the machines value be?

What is the expected number of years the machine will last and what would the value of themachine be? What is the NPV of the machine?