Exam 12: Modeling Randomness

There are ten Academic Senate faculty in the Statistics Department at the University of California, Irvine: three females and seven males. Two faculty are to be selected without replacement to serve on a committee where one of those selected will serve as the head of the committee. -Now, instead of recording the sex and role of the selected committee members, you only record the number of females selected. How many possible outcomes are in this new sample space?

Suppose half of all newborns are girls and half are boys. Hospital A, a large city hospital, records an average of 50 births a day. Hospital B, a small, rural hospital, records an average of 10 births per day. On a particular day, which hospital is less likely to record 80% or more female births?

Suppose X is a random variable with E(X)=5 and σ_X²=4. Define Y=2X+8. -Calculate the expected value of Y.

The probability of an event is always equal to the number of outcomes in the event divided by the number of outcomes in the sample space.

Suppose that only 8% of a large population has a certain disease. A diagnostic test has been developed which is 90% accurate for people with the disease (90% of people with the disease test positive), and 85% accurate for people without the disease (85% of people without the disease test negative). Define the following events: A = person has the disease B = person tests positive on the diagnostic test -Fill in the values below to create a hypothetical table of 1,000 randomly selected individuals in this population.

A soft drink company holds a contest in which a prize may be revealed on the inside of the bottle cap. The probability that each bottle cap reveals a prize is 0.39, and winning is independent from one bottle to the next. You buy six bottles. Let X be the number of prizes you win. -Calculate P(X>1).

Consider a game in which a fair die is thrown. The player pays $5 to play and wins $2 for each dot that appears on the roll. Define X = number on which the die lands, and Y = player's net profit (amount won - amount paid to play). -Calculate E(X).

Four friends are contemplating joining a local bowling league. Let X₁,X₂,X₃,X₄ be the score of the first, second, third, and fourth friend, respectively, on a randomly chosen game. From past experience, the friends know that: E(X₁ )=110, E(X₂ )=125, E(X₃ )=113, and E(X₄ )=140. Additionally, σ₁=7, σ₁=13, σ₁=10, and σ₁=20. Define their total score on a randomly chosen game as Y=X₁+X₂+X₃+X₄. Assume the four players' scores are independent. -Calculate σ_Y.

Let X be the amount in claims (in dollars) that a randomly chosen policyholder collects from an insurance company this year. From past data, the insurance company has determined that E(X) = $72, and σ_X = $60. Suppose the insurance company decides to offer a discount to attract new customers. They will pay the new customer $50 for joining, and offer a 5% "cash back" offer for all claims paid. Let Y be the amount in claims (in dollars) for a randomly chosen new customer. Then Y=50+1.05X. -Find E(Y).

According to Krantz in his book What the Odds Are (1992, p. 161), the probability that a randomly selected American will be injured by lightning in a given year is 1/685000. What does this probability mean?

Which of the following has a larger expected loss? Option 1: A sure loss of $740. Option 2: A 25% chance to lose nothing, and a 75% chance of losing $1000.

In a large population of college students, 56% live in a campus residence hall, 62% participate in a campus meal program, and 42% do both. Define events R = student lives in a campus residence hall M = student participates in a campus meal program -Use the addition rule to find P(R U M).

The volume in a can of soda is normally distributed with mean 358 mililiters (ml) and standard deviation 6 ml. -Find the z-score for a can filled with 349 ml of soda.

In a large population of college students, 56% live in a campus residence hall, 62% participate in a campus meal program, and 42% do both. Define events R = student lives in a campus residence hall M = student participates in a campus meal program -Find P(M^C).

A soft drink company holds a contest in which a prize may be revealed on the inside of the bottle cap. The probability that each bottle cap reveals a prize is 0.26, and winning is independent from one bottle to the next. You plan to keep buying bottles until you win a prize. Let X be the number of bottles you purchase. -What is the expected value of the number of bottles you will need to purchase until you win a prize?

Suppose that only 8% of a large population has a certain disease. A diagnostic test has been developed which is 90% accurate for people with the disease (90% of people with the disease test positive), and 85% accurate for people without the disease (85% of people without the disease test negative). Define the following events: A = person has the disease B = person tests positive on the diagnostic test -How would you express the probability 0.90 in terms of A and B?

A soft drink company holds a contest in which a prize may be revealed on the inside of the bottle cap. The probability that each bottle cap reveals a prize is 0.26, and winning is independent from one bottle to the next. You plan to keep buying bottles until you win a prize. Let X be the number of bottles you purchase. -Identify the parameter value for the distribution of X. π =

Define Y as in question 90. Find E(Y).

Which of the following variables is a discrete random variable?

Consider a game in which a fair die is thrown. The player pays $5 to play and wins $2 for each dot that appears on the roll. Define X = number on which the die lands, and Y = player's net profit (amount won - amount paid to play). -Express Y as a linear transformation of X. Y = ___(1)____ X + ____(2)_____