Exam 4: Discrete Random Variables

Classify the following random variable according to whether it is discrete or continuous. The number of cups of coffee sold in a cafeteria during lunch

Compute $\left( \begin{array} { l } 6 \\2\end{array} \right) ( .3 ) ^ { 2 } ( .7 ) ^ { 6 - 2 }$

Classify the following random variable according to whether it is discrete or continuous. The speed of a car on a Los Angeles freeway during rush hour traffic

A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20 eligible voters were randomly selected from the population of all eligible voters. How many of the sampled voters do we expect to vote in the next presidential election?

The hypergeometric random variable x counts the number of successes in the draw of 5 elements from a set of 10 elements containing 2 successes. List the possible values of x.

Compute $\frac { \lambda ^ { x } e ^ { - \lambda } } { x ! } \text { for } \lambda = 5 \text { and } x = 7$ for λ 5 and x 7.

Compute $\frac { 7 ! } { 3 ! ( 7 - 3 ) ! }$

The binomial distribution can be used to model the number of rare events that occur over a given time period.

A bottle contains 16 ounces of water. The variable x represents the volume, in ounces, of water remaining in the bottle after the first drink is taken. What are the natural bounds for the values of x? Is x discrete or continuous? Explain.

Which binomial probability is represented on the screen below? A) $P ( x \leq 4 )$ B) $P ( x < 4 )$ C) $P ( x = 4 )$ D) $P ( x > 4 )$

Suppose a man has ordered twelve 1-gallon paint cans of a particular color (lilac) from the local paint store in order to paint his mother's house. Unknown to the man, three of these cans contains an incorrect mix of paint. For this weekend's big project, the man randomly selects four of these 1-gallon cans to paint his mother's living room. Let x = the number of the paint cans selected that are defective. Unknown to the man, x follows a hypergeometric distribution. Find the probability that none of the four cans selected contains an incorrect mix of paint.

A coin is flipped 6 times. The variable x represents the number of tails obtained. List the possible values of x. Is x discrete or continuous? Explain.

The hypergeometric random variable x counts the number of successes in the draw of 5 elements from a set of 12 elements containing 7 successes. The numbers 0, 1, 2, 3, 4, 5, 6, and 7 are all possible values of x.

Suppose the number of babies born each hour at a hospital follows a Poisson distribution with a mean of 6. Some people believe that the presence of a full moon increases the number of births that take place. Suppose during the presence of a full moon, the hospital experienced eight consecutive hours with more than seven births each hour. Based on this fact, comment on the belief that the full moon increases the number of births.

A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20 eligible voters were randomly selected from the population of all eligible voters. Use a binomial probability table to find the probability that more than 10 but fewer than 16 of the 20 eligible voters sampled will vote in the next presidential election.

Given that x is a hypergeometric random variable with N = 10, n = 5, and r = 6, compute the mean of x.

For a binomial distribution, which probability is not equal to the probability of 1 success in 5 trials where the probability of success is .4?

A recent article in the paper claims that business ethics are at an all-time low. Reporting on a recent sample, the paper claims that 44% of all employees believe their company president possesses low ethical standards. Suppose 20 of a company's employees are randomly and independently sampled. Assuming the paper's claim is correct, find the probability that more than eight but fewer than 12 of the 20 sampled believe the company's president possesses low ethical standards. Round to six decimal places.

A recent survey found that 63% of all adults over 50 wear glasses for driving. In a random sample of 20 adults over 50, what is the mean and standard deviation of the number who wear glasses? Round to the nearest hundredth when necessary.

We believe that 82% of the population of all Business Statistics students consider statistics to be an exciting subject. Suppose we randomly and independently selected 39 students from the population. If the true percentage is really 82%, find the probability of observing 38 or more students who consider statistics to be an exciting subject. Round to six decimal places.