Question 1

The shape of the exponential distribution is completely described by only one parameter, µ.

Accepted Answer

The exponential distribution is a continuous probability distribution that describes the time between events in a Poisson process. It is completely described by a single parameter, µ, which is the mean of the distribution.

Question 2

Bananas are sold in bunches at a grocery store and typically consist of 4-8 bananas per bunch. Suppose the weight of these bunches follows a normal distribution with a mean of 3.54 pounds and a standard deviation of 0.63 pounds. The interval around the mean that contains 99.7% of the bunch weights is --------.

Accepted Answer

The interval around the mean that contains 99.7% of the bunch weights is 1.65, 5.43. This is because the z-score for 99.7% of the data is 3, which means that the interval is 3 standard deviations from the mean.

Question 3

According to the MIT Airline Data Project, American Airlines controlled 15.5% of the domestic market during a recent year. A random sample of 125 domestic passengers that year was selected. Using the normal approximation to the binomial distribution, what is the probability that 10 or fewer passengers from this sample were on American Airline flights?

Accepted Answer

The answer of According to the MIT Airline Data Project,...

Question 4

For continuous distributions, the probability that x is less than or equal to a value is the same as the probability that x is less than that value.

Accepted Answer

In continuous distributions, the probability of any single point is zero, so P(X ≤ x) = P(X < x).

Question 5

The mathematical expression that describes the shape of normal curves is known as the normal probability density function.

Accepted Answer

The normal probability density function mathematically describes the shape of normal (Gaussian) curves, which are symmetric and bell-shaped, detailing how the values of a variable are distributed.

Question 6

The z-score in a normal probability distribution determines the number of standard deviations that a particular value, x, is from the mean.

Accepted Answer

A z-score measures exactly how many standard deviations an element x is from the mean of a distribution, indicating its relative position within that distribution.

Question 7

The z-score follows a normal distribution with µ = 1 and a = 0, which is known as the standard normal distribution.

Accepted Answer

The standard normal distribution has a mean (µ) of 0 and a standard deviation (σ) of 1, not µ = 1 and σ = 0.

Question 8

During the 3 PM to 5 PM time period, cars arrive at a bank's drive-through window at an average rate of 15 customers per hour. Assume that the time between arrivals follows the exponential distribution. What is the standard deviation for this probability distribution?

Accepted Answer

The standard deviation for an exponential distribution is the inverse of the rate (λ), which in this case is 15 customers per hour or 1/15 of an hour per customer. Converting hours to minutes gives us 60/15 = 4 minutes. However, the correct calculation for the standard deviation in this context is actually the inverse of the rate without needing to convert into minutes, leading to a misunderstanding in the explanation. The correct interpretation should be that for an exponential distribution, the standard deviation is the inverse of the rate, λ, which when dealing with time between arrivals, directly translates to the same value as the mean. Since the arrivals are 15 customers per hour, the mean time between arrivals is 1/15 hours. To find the standard deviation in minutes, we convert this to minutes: (1/15) * 60 = 4 minutes, making the correct answer 4 minutes, not 2 minutes as previously stated.

Question 9

A normal probability distribution's standard deviation (a)completely describes its shape.

Accepted Answer

The shape of a normal probability distribution is described by both its mean (which determines its center) and its standard deviation (which determines its spread), not just the standard deviation alone.

Question 10

Assume that the amount of time that it takes an employee to service a car at an oil change facility follows the uniform probability distribution between 21 and 38 minutes. What is the probability that a randomly selected car will require exactly 30 minutes to service?

Accepted Answer

In a uniform distribution, the probability of observing any single specific value, such as exactly 30 minutes, is 0 because the distribution is continuous.

Question 11

The standard deviation of the continuous uniform distribution is equal to the mean of this distribution.

Accepted Answer

The standard deviation of a continuous uniform distribution is not equal to its mean. The mean of a continuous uniform distribution defined between a and b is (a+b)/2, while the standard deviation is (b-a)/√12, which are not equivalent expressions.

Question 12

With the continuous uniform probability distribution, the probability of any interval in the distribution is equal to any other interval with the same width.

Accepted Answer

In a continuous uniform probability distribution, all intervals of the same length have the same probability because the distribution is uniform, meaning the probability density is constant across the entire range of the distribution.

Question 13

The area under the curve of the normal probability distribution is always equal to 1.0.

Accepted Answer

The total area under the curve of a normal probability distribution represents the entirety of possible outcomes, which is always equal to 1.0, signifying 100% of the probability distribution.

Question 14

The mathematical expression that describes the shape of the curve for the exponential distribution is known as the exponential probability density function.

Accepted Answer

The exponential probability density function indeed describes the shape of the curve for the exponential distribution, specifying the likelihood of different outcomes in a continuous random variable scenario.

Question 15

The time that customers wait to be served at the delicatessen for a grocery store follows the uniform distribution between 0 and 7 minutes. What is the probability that a randomly selected customer will wait will be exactly 5 minutes at the deli?

Accepted Answer

In a continuous uniform distribution, the probability of observing any exact value is 0 because there are infinitely many possible values within any interval.

Question 16

The time between customer arrivals is a common application for the uniform probability distribution.

Accepted Answer

The time between customer arrivals is commonly modeled using the exponential distribution, not the uniform probability distribution, because it better represents the random nature of arrival times.

Question 17

The mathematical expression that describes the shape of the uniform distribution is known as the continuous uniform probability density function.

Accepted Answer

The continuous uniform probability density function describes the shape of the uniform distribution, indicating that any interval of numbers of the distribution has an equal probability of occurring.

Question 18

Assume that the flight time for U.S. Airways flight 1267 from Philadelphia to Dallas follows the uniform probability distribution between 210 and 250 minutes. What is the standard deviation of this distribution?

Accepted Answer

The answer of Assume that the flight time for U.S....

Question 19

Assume that the amount of time that it takes an employee to service a car at an oil change facility follows the uniform probability distribution between 21 and 38 minutes. What is the standard deviation of this distribution?

Accepted Answer

The standard deviation ($\sigma$) of a uniform distribution is calculated using the formula $\sigma = \frac{b-a}{\sqrt{12}}$, where $a$ and $b$ are the minimum and maximum values of the distribution, respectively. Here, $a = 21$ and $b = 38$, so $\sigma = \frac{38-21}{\sqrt{12}} = \frac{17}{\sqrt{12}} \approx 4.91$.

Question 20

Bob's golf score at his local course follows the normal distribution with a mean of 92.1 and a standard deviation of 3.8. The interval around the mean that contains 68% of Bob's golf scores is -----.

Accepted Answer

The interval that contains 68% of the scores in a normal distribution is defined by one standard deviation above and below the mean. Therefore, we calculate the interval as mean ± standard deviation = 92.1 ± 3.8, which gives us the interval 88.3 to 95.9.

Quiz 6: Addiction and Drug Abuse