Exam 6: Continuous Probability Distributions

You work in marketing for a company that produces work boots. Quality control has sent you a memo detailing the length of time before the boots wear out under heavy use. They find that the boots wear out in an average of 208 days, but the exact amount of time varies, following a normal distribution with a standard deviation of 14 days. For an upcoming ad campaign, you need to know the percent of the pairs that last longer than six months-that is, 180 days. Use the empirical rule to approximate this percent.

If an exponential distribution has the rate parameter λ = 5, what is its variance?

You are planning a May camping trip to Denali National Park in Alaska and want to make sure your sleeping bag is warm enough. The average low temperature in the park for May follows a normal distribution with a mean of 32°F and a standard deviation of 8°F. Above what temperature must the sleeping bag be suited such that the temperature will be too cold only 5% of the time?

It is known that the length of a certain product X is normally distributed with μ = 20 inches and σ = 4 inches. How is the probability P(X > 28) related to P(X < 16)?

Given normally distributed random variable X with a mean of 10 and a variance of 4, find the following probabilities. A) P(8 < X ≤ 10) B) P(6 ≤ X < 8) C) P(X < 13) D) P(X > 4)

If an exponential distribution has the rate parameter λ = 5, what is its expected value?

The probability P(Z < −1.28) is closest to ________.

The probability density function for a continuous uniform distribution is positive for all values between -∞ and +∞.

The exponential distribution is based entirely on one parameter called the ________ parameter.

Which of the following does not represent a continuous random variable?

Find the mean of the lognormal variable if the mean and standard deviation of the underlying normal variable are 2 and 0.8, respectively.

The time for a professor to grade a student's homework in statistics is normally distributed with a mean of 12.6 minutes and a standard deviation of 2.5 minutes. What is the probability that randomly selected homework will require between 11 and 15 minutes to grade?

The time of a call to a technical support line is uniformly distributed between 2 and 10 minutes. What are the mean and variance of this distribution?

An analyst is forecasting net income for Excellence Corporation for the next fiscal year. Her low-end estimate of net income is $250,000, and her high-end estimate is $350,000. Prior research allows her to assume that net income follows a continuous uniform distribution. The probability that net income will be greater than or equal to $337,500 is ________.

The time to complete the construction of a soapbox derby car is normally distributed with a mean of three hours and a standard deviation of one hour. Find the probability that it would take between 2.5 and 3.5 hours to construct a soapbox derby car.

The probability P(Z > 1.28) is closest to ________.

The time you must wait for an Orange Line train of the Massachusetts Bay Transit Authority follows a uniform distribution with a lower bound of 0 minutes and an upper bound of 8 minutes. Jonathan is running to catch the train to get to a meeting. He knows that the train needs to arrive within five minutes or else he will be late. What is the probability that he will be late to his meeting?

A soft drink company fills two-liter bottles on several different lines of production equipment. The fill volumes are normally distributed with a mean of 1.97 liters and a variance of 0.04 (liter)². A) Find the probability that a randomly selected two-liter bottle would contain more than 1.92 liters. B) If X is the fill volume of a randomly selected two-liter bottle, find the value of x for which P(X < x) = 0.6293.

The normal probability distribution is symmetric and bell-shaped.

Excel's function NORMAL.DIST can be used to compute probabilities for the normal distribution.