Exam 12: Modeling Randomness

Which of the following has a higher expected earning? Option 1: A gift of $240, guaranteed. Option 2: A 25% chance to win $1,000, and a 75% chance of getting nothing.

A tire manufacturer designed a new tread pattern for its all-weather tires. Repeated tests were conducted on cars of approximately the same weight traveling at 60 miles per hour. The tests showed that the new tread pattern enables the cars to stop completely in an average distance of 125 feet with a standard deviation of 6.5 feet and that the stopping distances are approximately normally distributed. -Use the Normal Probability Calculator applet to find the probability that one randomly selected car in the study will stop completely in a distance that is greater than 135 feet.

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 the variance of their total score on a randomly chosen game.

An "instant lottery" is played by buying a ticket and scratching off a coating to reveal whether or not you have won a prize, and if so, how much. Suppose an instant lottery pays $5 with probability 0.05 and $100 with probability 0.006. Otherwise it pays nothing. Define X = amount won for a single ticket. (You can ignore the cost of the ticket.) -Determine the probability distribution of X:

A tire manufacturer designed a new tread pattern for its all-weather tires. Repeated tests were conducted on cars of approximately the same weight traveling at 60 miles per hour. The tests showed that the new tread pattern enables the cars to stop completely in an average distance of 125 feet with a standard deviation of 6.5 feet and that the stopping distances are approximately normally distributed. -Fill in the two blanks: About 95% of cars of approximately the same weight traveling at 60 miles per hour will stop completely in a distance of between ____(1)_____ and _____(2)_____ feet.

The volume in a can of soda is normally distributed with mean 358 mililiters (ml) and standard deviation 6 ml. -If the company wanted to ensure that 90% of soda cans were filled to the volume advertised on the can or more, what volume should they advertise? Round to the next highest integer.

There are ten Academic Senate faculty in the UCI Statistics Department, three females and seven males. Two faculty are to be selected without replacement to serve on a committee and the X = the number of females will be recorded. Is X a binomial random variable?

Consider a random process that involves repeatedly rolling a rubber pig. If the pig lands on its feet, a person scores 10 points. Otherwise, a person scores zero points. Which of the following is a binomial random variable?

A lightbulb manufacturing plant's production line has determined that the probability a lightbulb will be defective is 0.04. The quality control team decides to take a random sample of 500 lightbulbs and measure the proportion of those lightbulbs that are defective. -Use the Normal Probability Calculator applet to approximate the probability the quality control team sees at least 25 defective items in their sample.

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. -The events in the sample space are all equally likely.

Based on her past experience, a professor knows that the probability distribution function for X = number of students who come to her office hours on any given Wednesday is given below. -What is the value of P(X = 4) (the question mark in the table)?

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. -The friends decide it would only be worth it to join the bowling league if they could average a total score of 500. Should they join the league?

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. -How could you simulate this random process?

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 σ_Y.

Your brother says he is better than you at playing some video game. You play 6 games and he wins 4 of them. He says that this proves he is a better player. You have studied statistics and you want to determine the probability of anyone winning at least 4 games out of 6 just by chance if you are both equally as skilled. Which of the following would provide an accurate estimate of that probability?

An "instant lottery" is played by buying a ticket and scratching off a coating to reveal whether or not you have won a prize, and if so, how much. Suppose an instant lottery pays $5 with probability 0.05 and $100 with probability 0.006. Otherwise it pays nothing. Define X = amount won for a single ticket. (You can ignore the cost of the ticket.) -Let $S be the standard deviation of X. Which of the following is a correct interpretation of this value?

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 -What is the probability that a randomly selected person tests positive on the diagnostic test?

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). -Find σ_Y.

An online psychic (psi) ability or extrasensory perception (ESP) test shows the participant five face-down cards on the screen and asks the participant to "click on the one card that has a picture on the other side." After clicking a card, the test shows you the card to determine if your guess was correct. -How many possible outcomes are in this sample space?

Math SAT scores for students admitted to University A are bell-shaped with a mean of 520 and a standard deviation of 60. Math SAT scores for students admitted to University B are also bell-shaped, but with a mean of 580 and a standard deviation of 45. Suppose you plan to take a simple random sample of 15 students from University A and 25 students from University B. Let x_B-x_A be the difference in sample mean math SAT scores between the two samples (B - A). -Fill in the blanks: Approximately 95% of differences in sample mean math SAT scores between the two universities will be between ____(1)____ and ____(2)____ points.