Exam 5: Continuous Random Variables

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. What proportion of the paint cans contain less than 5.54 ml of the dye?

Find a value of the standard normal random variable z, called z0, such that P(-z0 ≤ z ≤ z0) = 0.98.

The time (in years) until the first critical-part failure for a certain car is exponentially distributed with a mean of 3.4 years. Find the probability that the time until the first critical-part failure is 5 years or more.

Determine if it is appropriate to use the normal distribution to approximate a binomial distribution when n = 36 and p = 0.3.

Suppose that x has an exponential distribution with $\theta = 2.5 . \text { Find } P ( x \geq 4 )$

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

Suppose that 67% of the employees of a company participate in the company's medical savings program. Let x be the number of employees who participate in the program in a random sample of 50 employees. Find the mean and standard deviation of x.

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 2600 miles. If the manufacturer guarantees the tread life of the tires for the first 56,880 miles, what proportion of the tires will need to be replaced under warranty?

Find a value of the standard normal random variable z, called z0, such that P(z ≥ z0) = 0.70.

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. Find the approximate probability that more than 17 and less than 26 of the sampled students had at least one tattoo.

The price of a gallon of milk follows a normal distribution with a mean of $3.20 and a standard deviation of $0.10. What proportion of the milk vendors had prices that were less than $3.075 per gallon?

Suppose that 88% of the stocks listed on a particular exchange increased in value yesterday. Let x be the number of stocks that increased in value yesterday in a random of 72 stocks listed on the exchange. Find the mean and standard deviation of x.

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 69°F to 99°F. Find the temperature which is exceeded by the high temperatures on 90% of the days in August.

After a particular heavy snowstorm, the depth of snow reported in a mountain village followed a uniform distribution over the interval from 15 to 22 inches of snow. Find the standard deviation of the snowfall amounts.

The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 3.5 to 5.5 millimeters. What is the probability that a randomly selected ball bearing has a diameter greater than 4.6 millimeters?

The preventable monthly loss at a company has a normal distribution with a mean of $4700 and a standard deviation of $40. A new policy was put into place, and the preventable loss the next month was $4460. What inference can you make about the new policy?

Which geometric shape is used to represent areas for a uniform distribution?

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

Suppose a random variable x is best described by a normal distribution with μ = 60 and σ = 8. Find the z-score that corresponds to the value x = 30.

The time (in years) until the first critical-part failure for a certain car is exponentially distributed with a mean of 3.4 years. Find the probability that the time until the first critical-part failure is less than 1 year.