Exam 6: Introduction to Continuous Probability Distributions

One example of a difference between discrete random variables and continuous random variables is that in a discrete distribution P(x > 2)= P(x ≥ 3)while in a continuous distribution P(x > 2)is treated the same as P(x ≥ 2).

What is the difference between a normal distribution and the standard normal distribution?

A class takes an exam where the average time to complete the exam is normally distributed with a time of 40 minutes and standard deviation of 9 minutes.If the class lasts 1 hour,what percent of the students will have turned in the exam after 60 minutes?

For a standardized normal distribution,determine a value,say z₀,so that P(-z₀ ≤ z < 0)= 0.45.

For a normal distribution,the probability of a value being between a positive z-value and its population mean is the same as that of a value being between a negative z-value and its population mean.

The money spent by people at an amusement park,after paying to get in the gate,is thought to be uniformly distributed between $5.00 and $25.00.Based on this,what is the probability that someone will spend between $8.00 and $12.00?

A random variable,x,has a normal distribution with μ = 13.6 and σ = 2.90.Determine a value,x₀,so that P(μ - x₀ ≤ x ≤ μ + x₀)= 0.95.

The manager at a local movie theater has collected data for a long period of time and has concluded that the revenue from concession sales during the first show each evening is normally distributed with a mean equal to $336.25 and a variance equal to 1,456.Based on this information,what are the chances that the revenue on the first show will exceed $800?

An electronics repair shop has determined that the time between failures for a particular electronic component part is exponentially distributed with a mean time between failures of 200 hours.Based on this information,the probability that a part will not fail in the first 200 hours is 0.50.

A randomly selected value from a normal distribution is found to be 2.1 standard deviations above its mean.What is the probability that a randomly selected value from the distribution will be less than 2.1 standard deviations from the mean?

The time it takes a parent to assemble a children's bicycle has been shown to be normally distributed with a mean equal to 295 minutes with a standard deviation equal to 45 minutes.Given this information,the probability that it will take a randomly selected parent more than 220 minutes is about 0.0475.

A random variable,x,has a normal distribution with μ = 13.6 and σ = 2.90.Determine a value,x₀,so that P(x ≤ x₀)= 0.975.

It is assumed that the time customers spend in a record store is uniformly distributed between 3 and 12 minutes.Based on this information,what is the probability that a customer will be exactly 7.50 minutes in the record store?

A major cell phone service provider has determined that the number of minutes that its customers use their phone per month is normally distributed with a mean equal to 445.5 minutes with a standard deviation equal to 177.8 minutes.The company is thinking of changing its fee structure so that anyone who uses the phone less than 250 minutes during a given month will pay a reduced monthly fee.Based on the available information,what percentage of current customers would be eligible for the reduced fee?

If the time it takes for a customer to be served at a fast-food chain business is thought to be uniformly distributed between 3 and 8 minutes,then the probability that the time it takes for a randomly selected customer to be served will be less than 5 minutes is 0.40.

Watersports Rental at Flathead Lake rents jet skis and power boats for day use.Each piece of equipment has a clock that records the time that it was actually in use while rented.The company has observed over time that the distribution of time used is normally distributed with a mean of 3.6 hours and a standard deviation equal to 1.2 hours.Watersports management has decided to give a rebate to customers who use the equipment for only a short amount of time.They wish to grant a rebate to no more than 10 percent of all customers.Based on the information provided,the amount of time that should be set as the cut-off between getting the rebate and not getting the rebate is approximately 2.06 hours.

In comparing a uniform distribution with a normal distribution where both distributions have the same mean and the same range,explain which distribution will have the larger standard deviation.

Which of the following probability distributions would most likely be used to describe the time between failures for electronic components?

The transportation manager for the State of New Jersey has determined that the time between arrivals at a toll booth on the state's turnpike is exponentially distributed with λ = 4 cars per minute.Based on this information,what is the probability that the time between any two cars arriving will exceed 11 seconds?

A seafood shop sells salmon fillets where the weight of each fillet is normally distributed with a mean of 1.6 pounds and a standard deviation of 0.3 pounds.Based on this information we can conclude that 90 percent of the fillets weight more than 1.0 pound.