Exam 8: Random Variables

Use the following information for questions: Suppose that a student needs to buy 6 books for her history course. The number of books that she will be able to find used is a binomial random variable X with n = 6 and p = 0.30. In other words, the probability that she will find any given book used is 0.30, and is independent from one book to the next. -What is the variance for the number of used books she will find, V(X)?

The following probability distribution is for the random variable X = number of classes for which full time students at a university are enrolled in a semester: What is the mean number (expected value) of courses taken per student?

Use the following information for questions: Did high gas prices keep Americans from hitting the road this past summer? In a nationwide survey of adults, one variable measured was how many days vacationers spent driving on the road on their longest trip. Consider the following (partial) probability distribution for the random variable X = the number of days for the longest car trip. -What is the expected number of days for the longest trip? Include symbol, value, and units.

Verbal SAT scores have approximately a normal distribution with mean equal to 500 and standard deviation equal to 100. The 95^th percentile of z-scores is z = 1.65. What is the 95^th percentile of verbal SAT scores?

Use the following information for questions: Suppose X is a binomial random variable with n = 800 and p = 0.20. -Find the approximate the probability P(X ≥ 180) with the normal approximation with a continuity correction.

Use the following information for questions: Some managers of companies use employee rankings to laud the best and let go of the worst. Suppose the distribution of rankings of employees at a large company is normal with a mean of 65 points and a standard deviation of 6 points. -Managers at this large company were told to determine the top 20 percent, the bottom 10 percent and the remaining 70 percent in the middle, and then "weed out" (let go) those in that bottom tier. Using the provided model for rankings, what is the cut-off for an employee to be in the top 20 percent?

Use the following information for questions: Suppose the time to wait for placing an order at a drive-through window has a uniform distribution between 0 and 8 minutes. -What proportion of customers is expected to wait more than 6 minutes?

Use the following information for questions: Centerpieces are normally distributed with mean of 30 ounces and standard deviation of2 ounces. The weights of the shipping boxes are normally distributed with mean of 12 ounces and standard deviation of 1 ounce. Suppose that centerpieces are chosen at random and packed into randomly chosen shipping boxes. -What are the mean and standard deviation of the total weights of the packages?

Use the following information for questions: Suppose that a student needs to buy 6 books for her history course. The number of books that she will be able to find used is a binomial random variable X with n = 6 and p = 0.30. In other words, the probability that she will find any given book used is 0.30, and is independent from one book to the next. -What is the probability that she will find exactly 3 used books?

The probability is p = 0.80 that a patient with a certain disease will be successfully treated with a new medical treatment. Suppose that the treatment is used on 40 patients. What is the "expected value"of the number of patients who are successfully treated?

Use the following information for questions: The probability distribution for X = number of heads in 4 tosses of a fair coin is partially given in the table below. -What is the probability of getting 4 heads?

Use the following information for questions: Suppose that a student needs to buy 6 books for her history course. The number of books that she will be able to find used is a binomial random variable X with n = 6 and p = 0.30. In other words, the probability that she will find any given book used is 0.30, and is independent from one book to the next. -What is the probability that she will find exactly 2 used books?

Use the following information for questions: The time it takes a student to finish buying his/her text books, W, is a normal random variable. This variable W is the sum of two other normal variables, X and Y (W = X + Y), where X = the time to wait in line at the ATM machine to get cash, and Y = the time to wait in line at the cashier to buy the books. Assume that X and Y are independent normal random variables, with the following means and standard deviations: -What is the probability that a student has to wait no more than 10 minutes total to buy her books?

Use the following information for questions: The probability distribution for X = number of heads in 4 tosses of a fair coin is partially given in the table below. -What is the value of the cumulative distribution function at 3, i.e. P(X ? 3)?

Use the following information for questions: Find the requested probability for the standard normal random variable Z. -What is the probability that Z is between -1 and 1, P(-1 $\leq$ Z $\leq$ 1)?

Use the following information for questions: In an experiment, a person guesses which one of three different cards a researcher has randomly picked (and hidden from the person who guesses). This is repeated four times, replacing the cards each time. Let X = number of correct guesses in the four tries. The probability distribution for X, assuming the person is just guessing, is partially provided below. -Give the cumulative distribution function for the number of correct guesses.

Use the following information for questions: A child is observing squirrels in the park and notices that some are black and some are gray. For the next five squirrels she sees, she counts X = the number of black squirrels. Suppose X is a binomial random variable with n = 5 and p = 0.50. -What is the variance for the number of black squirrels she will see, V(X)?

Use the following information for questions: Ellen is taking 4 courses for the semester. She believes that the probability distribution function for X = the number of courses for which she will get an A grade is given below. -What is the variance for the number of A's she will get? (i.e. What is V(X)?)

Use the following information for questions: A bank offers free coffee to its customers. The coffee is brewed each morning in a large coffee urn, and the amount brewed varies slightly. It is independent from day to day, with mean of 125 ounces and standard deviation of 3 ounces. The amount customers drink from day to day is also random, with mean of 100 ounces and standard deviation of 5 ounces, and is independent of the amount brewed. -If the amount of coffee brewed and the amount of coffee customers drink are both normally distributed, what is the probability that amount of coffee remaining at the end of a day exceeds 40 ounces?

Use the following information for questions: Joan has noticed that the probability distribution for X = number of students in line to use the campus ATM machine when she shows up to use it is shown below. -What is the expected value of X, E(X)?