Chapter 6: Continuous Probability Distributions
The time it takes to ring up a customer at the grocery store follows an exponential distribution with a mean of 3.5 minutes. Which of the following expressions gives the probability that it takes from 1 to 2 minutes to ring up a customer?
(1-e^-2/3.5)-(1-e^1/3.)
The newest model of smart car is supposed to get excellent gas mileage. A thorough study showed that gas mileage (measured in miles per gallon) is normally distributed with a mean of 75 miles per gallon and a standard deviation of 10 miles per gallon. What is the probability that, if driven normally, the car will get at least 100 miles per gallon?
.006 =1-NORM.DIST(100,75,10,TRUE)
Fast food restaurants pride themselves in being able to fill orders quickly. A study was done at a local fast food restaurant to determine how long it took customers to receive their order at the drive thru. It was discovered that the time it takes for orders to be filled is exponentially distributed with a mean of 1.5 minutes. What is the probability that it takes from 2 to 3 minutes to fill an order?
.1283 =EXPON.DIST(3,1/1.5,TRUE)-EXPON.DIST(2,1/1.5,TRUE)
The time it takes to ring up a customer at the grocery store follows an exponential distribution with a mean of 3.5 minutes. What is the probability that it takes more than 5 minutes to ring up a customer?
.2397 =1-Expon.Dist(5,1/3.5,true)
The height of the probability density function for a uniform distribution ranging between 2 and 6 is:
.25 =1/(6-2)
Fast food restaurants pride themselves in being able to fill orders quickly. A study was done at a local fast food restaurant to determine how long it took customers to receive their order at the drive thru. It was discovered that the time it takes for orders to be filled is exponentially distributed with a mean of 1.5 minutes. What is the probability that it takes less than one minute to fill an order?
.4866 =EXPON.DIST(1,1/1.5,TRUE)
For the standard normal probability distribution, the area to the left of the mean is:
.50
The random variable x is known to be uniformly distributed between 2 and 12. Compute P(x > 6).
.60 =(12-6)/(12-2)
The random variable x is known to be uniformly distributed between 2 and 12. Compute P(x = 6).
.60 P(x ≥ 6) = (width)(height) = (6)(1/10) = .60. See Section 6.1, Uniform Probability Distribution.
In a standard normal distribution, what z-score corresponds to the 75th percentile?
.67 (use z tables)
If z is a standard normal random variable, then compute P(-1.5 < z < 1.5).
.87 =NORM.S.DIST(1.5,TRUE)-NORM.S.DIST(-1.5,TRUE)
If z is a standard normal random variable, then compute P(z > -1.75).
.96 =1-NORM.S.DIST(-1.75,TRUE)
A health conscious student faithfully wears a device that tracks his steps. Suppose that the number of steps he takes is normally distributed with a mean of 10,000 and a standard deviation of 1500 steps. What is the probability that he takes less than 15,000 steps in a given day?
.9996 =NORM.DIST(15000,10000,1500,TRUE)
Fast food restaurants pride themselves in being able to fill orders quickly. A study was done at a local fast food restaurant to determine how long it took customers to receive their order at the drive thru. It was discovered that the time it takes for orders to be filled is exponentially distributed with a mean of 1.5 minutes. What is the standard deviation of this distribution?
1.5 The expected value of an exponential random variable X with rate parameter λ is given by; Therefore, the standard deviation is equal to the mean.
A health conscious student faithfully wears a device that tracks his steps. Suppose that the number of steps he takes is normally distributed with a mean of 10,000 and a standard deviation of 1500 steps. How many steps would he have to take to make the cut for the top 5% for his distribution?
12,467 =NORM.INV(0.95,10000,1500)
The time it takes to ring up a customer at the grocery store follows an exponential distribution with a mean of 3.5 minutes. What is the variance of this distribution?
12.25 A property of the exponential distribution is that the mean of the distribution and the standard deviation of the distribution are equal, so the standard deviation is 3.5. The variance is the square of the standard deviation, so the variance = 3.5^2 = 12.25.
Fast food restaurants pride themselves in being able to fill orders quickly. A study was done at a local fast food restaurant to determine how long it took customers to receive their order at the drive thru. It was discovered that the time it takes for orders to be filled is exponentially distributed with a mean of 1.5 minutes. What is the variance of this distribution?
2.25 A property of the exponential distribution is that the mean of the distribution and the standard deviation of the distribution are equal, so the standard deviation is 1.5. The variance is the square of the standard deviation, so the variance = 1.5^2 = 2.25.
A health conscious student faithfully wears a device that tracks his steps. Suppose that the number of steps he takes is normally distributed with a mean of 10,000 and a standard deviation of 1500 steps. What percent of the time will he exceed 13,000 steps?
2.28% =1-norm.dist(13000,10000,1500,true)
The random variable x is known to be uniformly distributed between 2 and 12. Compute the standard deviation of x.
2.887 =(12-2)/SQRT(12)
Suppose that a basketball player scored, on average, 15 points per game. Also suppose that the distribution of points scored by this player was normal. If he scores 20 points or more 4.78% of the time, what is his standard deviation?
3 =(20-15)/NORM.S.INV(1-0.0478)
The time it takes to ring up a customer at the grocery store follows an exponential distribution with a mean of 3.5 minutes. What is the standard deviation of this distribution?
3.5 The expected value of an exponential random variable X with rate parameter λ is given by; Therefore, the standard deviation is equal to the mean.
The random variable x is known to be uniformly distributed between 2 and 12. Compute E(x).
7 Mean of 2 variables = (2+12)/2
The newest model of smart car is supposed to get excellent gas mileage. A thorough study showed that gas mileage (measured in miles per gallon) is normally distributed with a mean of 75 miles per gallon and a standard deviation of 10 miles per gallon. What value represents the 50th percentile of this distribution?
75 =Norm.Inv(0.5,75,10)
The drying time of one particular paint is approximately normally distributed with a mean of 45 minutes. It is also known that 15% of walls painted with this paint need more than 55 minutes to dry completely. Find the standard deviation of this distribution.
9.62 =(55-45)/NORM.S.INV(1-0.15)
uniform probability distribution
A continuous probability distribution for which the probability that the random variable will assume a value in any interval is the same for each interval of equal length.
exponential probability distribution
A continuous probability distribution that is useful in describing the time to complete a task.
normal probability distribution
A continuous probability distribution. Its probability density function is bell-shaped and determined by its mean and standard deviation .
probability density function
A function used to compute probabilities for a continuous random variable. The area under the graph of a probability density function over an interval represents probability.
Which of the following is not a characteristic of the normal probability distribution? The standard deviation must be one. The distribution is symmetrical. The mean of the distribution can be negative, zero, or positive. The mean, median, and mode are equal.
The standard deviation must be one.
Standard normal probability distribution
a normal distribution with a mean of zero and a standard deviation of one
A normal distribution with a mean of zero and a standard deviation of one is called:
a standard normal probability distribution.
There is a lower limit but no upper limit for a random variable that follows the:
exponential probability distribution
A continuous probability distribution that is useful in describing the time, or space, between occurrences of an event is a(n):
exponential probability distribution.
If arrivals follow a Poisson probability distribution, the time between successive arrivals must follow a(n):
exponential probability distribution.
For a uniform probability density function, the height of the function:
is the same for each value of x.
The center of a normal curve is:
the mean of the distribution.
A negative value of z indicates that:
the number of standard deviations of an observation is below the mean.