Exam 5: Continuous Random Variables

The rate of return for an investment can be described by a normal distribution with mean 40% and standard deviation 3%. What is the probability that the rate of return for the investment exceeds 46%?

Suppose a uniform random variable can be used to describe the outcome of an experiment with outcomes ranging from 40 to 90. What is the mean outcome of this experiment?

A study of college students stated that 25% of all college students have at least one tattoo. In a random sample of 80 college students, let x be the number of the students that have at least one Tattoo. Can the normal approximation be used to estimate the binomial distribution in this Problem?

A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a Mean of 440 seconds and a standard deviation of 60 seconds. Find the probability that a randomly Selected boy in secondary school can run the mile in less than 302 seconds.

The tread life of a particular brand of tire is a random variable best described by a normal distribution with a mean of 60,000 miles and a standard deviation of 2800 miles. What warranty Should the company use if they want 96% of the tires to outlast the warranty?

A paint machine dispenses dye into paint cans to create different shades of paint. The amount of dye dispensed into a can is known to have a normal distribution with a mean of 5 milliliters (ml) And a standard deviation of 0.4 ml. Answer the following questions based on this information. Find the dye amount that represents the 9th percentile of the distribution.

The exponential distribution is sometimes called the waiting-time distribution, because it is used to describe the length of time between occurrences of random events.

Find a value of the standard normal random variable $z$ , called $z _ { 0 }$ , such that $P \left( - z _ { 0 } \leq z \leq z _ { 0 } \right) = 0.98$ .

Suppose x is a uniform random variable with c = 10 and d = 90. Find the mean of the random variable x.

Assume that x is a binomial random variable with n = 400 and p = 0.30. Use a normal approximation to find $\mathrm { P } ( \mathrm { x } \geq 100 )$

It is against the law to discriminate against job applicants because of race, religion, sex, or age. Of the individuals who apply for an accountantʹs position in a large corporation, 43% are over 45 years old. If the company decides to choose 44 of a very large number of applicants for closer credential screening, claiming that the selection will be random and not age-biased, what is the z-value associated with fewer than 20 of those chosen being over 45 years old? (Assume that the applicant pool is large enough so that x, the number in the sample over 45 years old, has a binomial probability distribution.)

Suppose that $x$ has an exponential distribution with $\theta = 2$ . Find $P ( x < 1.5 )$ .

The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 8.5 to 10.5 millimeters. What is the mean diameter of ball Bearings produced in this manufacturing process?

Suppose a random variable $x$ is best described by a normal distribution with $\mu = 60$ and $\sigma = 4$ . Find the $z$ -score that corresponds to the value $x = 45$ .

Transportation officials tell us that 60% of drivers wear seat belts while driving. Find the probability that more than 409 drivers in a sample of 650 drivers wear seat belts.

Which of the following statements is not a property of the normal curve?

Use the standard normal distribution to find $P ( - 2.25 < z < 1.25 )$

A machine is set to pump cleanser into a process at the rate of 10 gallons per minute. Upon inspection, it is learned that the machine actually pumps cleanser at a rate described by the Uniform distribution over the interval 10.0 to 13.0 gallons per minute. Would you expect the Machine to pump more than 12.85 gallons per minute?

Suppose that the random variable $x$ has an exponential distribution with $\theta = 1.5$ . Find the mean and standard deviation of $x$ .

The time between arrivals at an ATM machine follows an exponential distribution with θ = 10 minutes. Find the probability that between 15 and 25 minutes will pass between arrivals.