21. A random sample of size n = 250 yields 80

successes. Calculate the 95% confidence interval for p.

22. A random sample of size n = 452 yields 113

successes. Calculate the 95% confidence interval for p.

23. George enjoys throwing horse shoes. Last week

he tossed 150 shoes and obtained 36 ringers.

(Ringers are good.) Next week he plans to

throw 250 shoes. Assume that George's tosses

satisfy the assumptions of Bernoulli trials.

(a) Calculate the point prediction of the number of ringers that George will obtain next

week.

(b) Calculate the 90% prediction interval for

the number of ringers George will obtain

next week.

(c) It turns out that next week George obtains

62 ringers. Given this information, comment on your answers in parts (a) and (b)

1. A full house in poker is a hand where three cards share one rank and two cards share

another rank. How many ways are there to get a full-house? What is the probability of

getting a full-house?

2. 20 politicians are having a tea party, 6 Democrats and 14 Republicans. To prepare,

they need to choose:

3 people to set the table, 2 people to boil the water, 6 people to make the scones.

Each person can only do 1 task. (Note that this doesn't add up to 20. The rest of the

people don't help.)

(a) In how many different ways can they choose which people perform these tasks?

(b) Suppose that the Democrats all hate tea. If they only give tea to 10 of the 20 people,

what is the probability that they only give tea to Republicans?

(c) If they only give tea to 10 of the 20 people, what is the probability that they give tea

to 9 Republicans and 1 Democrat?

3. Let C and D be two events with P(C) = 0.25, P(D) = 0.45, and P(C ? D) = 0.1.

What is P(Cc ? D)?

4. More cards! Suppose you want to divide a 52 card deck into four hands with 13 cards

each. What is the probability that each hand has a king?

5. Corrupted by their power, the judges running the popular game show America's Next

Top Mathematician have been taking bribes from many of the contestants. Each episode,

a given contestant is either allowed to stay on the show or is kicked off.

If the contestant has been bribing the judges she will be allowed to stay with probability 1.

If the contestant has not been bribing the judges, she will be allowed to stay with probability

1/3.

Suppose that 1/4 of the contestants have been bribing the judges. The same contestants

bribe the judges in both rounds, i.e., if a contestant bribes them in the first round, she

bribes them in the second round too (and vice versa).

(a) If you pick a random contestant who was allowed to stay during the first episode, what

is the probability that she was bribing the judges?

(b) If you pick a random contestant, what is the probability that she is allowed to stay

during both of the first two episodes?

(c) If you pick random contestant who was allowed to stay during the first episode, what

is the probability that she gets kicked off during the second episode?

6. There is a screening test for prostate cancer that looks at the level of PSA (prostatespecific antigen) in the blood. There are a number of reasons besides prostate cancer that

a man can have elevated PSA levels. In addition, many types of prostate cancer develop

so slowly that that they are never a problem. Unfortunately there is currently no test

6. Suppose that the probability of being tutored in Statistics is 0.55, while the probability of being tutored in Physics is 0.32, and the probability of being tutored in both Statistics and Physics is 0.09. a) What is the probability of being tutored in Statistics or Physics? b) What is the probability of not being tutored in Statistics nor Physics? c) Are the events "being tutored in Statistics" and "being tutored in Physics" mutually exclusive? Explain.Machines M and N manufacture components. The probability that a component is of an acceptable standard is 0.9 when manufactured by machine M and 0.8 when manufactured by machine N. Machine M supplies 65 % of the components and machine / supplies the rest. (a) Calculate the probability that a component picked at random is of an acceptable standard. (b) A component is not of an acceptable standard. Calculate the pro- bability that it is made by machine N. (c) Two components are picked at random. Calculate the probability that they are made by different machines.6- (15%) Assume that X= The final grade of a student in Stochastic Modeling class and Y= The final grade of a student in Structural Analysis class The joint probability of the grades earned by a student is P(X, Y) which is as follows. Structural Analysis Grades Stochastic Modeling 95 80 Grades 90 0.2 0.4 75 0.3 0.1 How the grades of Stochastic and Structural Analysis classes are related to each other? (Positive. Negative or no relationship?) Hint: No need for the final value of p. Do necessary calculations to show how X and Y are related to each other (Positive, Negative or no relationship)Problem 4 Each year, ratings are compiled concerning the performance of new cars during the first 90 days of use. Suppose that the cars have been categorized according to whether the car needs warranty-related repair (yes or no) and the country in which the company manufacturing the car is based (United States or not United States). Based on the data collected, the probability that the new car needs a warranty repair is 0.04, the probability that it was manufactured by a U.S-based company is 0.60, and the probability that the new car needs a warranty repair and was manufactured by a U.S. - based company is 0.025. a. Suppose you know that a company based in the United States manufactures a particular car. What is the probability that the car needs warranty repair