Question 1

For an exponential distribution with a lambda ()equal to 20, the mean equal to _______.
A)20
B).05
C)4.474
D)1
E)2.11

Accepted Answer

The mean of an exponential distribution is given by 1/λ, so with λ = 20, the mean is 1/20 = 0.05.

Question 2

The probability that someone will prefer Coke over Pepsi is 0.56.A researcher asked 90 people which soda pop they preferred, and used the normal approximation to the binomial with a correction for continuity, what would be the mean used in the calculation?
A)0.56
B)90
C)45.0
D)50.4
E)106.7

Accepted Answer

The mean ($\mu$) in a binomial distribution is calculated as $n \times p$, where $n$ is the number of trials (in this case, 90 people) and $p$ is the probability of success (preferring Coke over Pepsi, 0.56). Therefore, the mean is $90 \times 0.56 = 50.4$.

Question 3

The probability that someone will prefer Coke over Pepsi is 0.56.A researcher asked 90 people which soda pop they preferred, and used the normal approximation to the binomial with a correction for continuity, what would be the standard deviation used in the calculation?
A)4.7
B)22.2
C)50.4
D)39.6
E)22.5

Accepted Answer

The standard deviation ($\sigma$) for a binomial distribution is calculated using the formula $\sigma = \sqrt{n \cdot p \cdot (1-p)}$, where $n$ is the number of trials and $p$ is the probability of success. Here, $n = 90$ and $p = 0.56$, so $\sigma = \sqrt{90 \cdot 0.56 \cdot (1-0.56)} = \sqrt{90 \cdot 0.56 \cdot 0.44} = \sqrt{22.176} \approx 4.7$.

Question 4

Participants in a 2 day biking event, cross the finish line at a rate of 10 bike riders per fifteen minute interval.On average, how much time, in minutes, elapses between bike riders?
A)1.50
B).0667
C).1667
D)1.00
E)2.50

Accepted Answer

1.50If 10 bike riders cross the finish line every 15 minutes, then the time between each bike rider is 15 minutes divided by 10 riders, which equals 1.5 minutes per rider.

Question 5

At a certain workstation in an assembly line, the time required to assemble a component is exponentially distributed with a mean time of 10 minutes.Find the probability that a component is assembled in 3 to 7 minutes?
A)0.5034
B)0.2592
C)0.2442
D)0.2942
E)0.5084

Accepted Answer

The exponential distribution has a mean of 10 minutes, so the rate parameter λ is 1/10. The probability that a component is assembled in 3 to 7 minutes is the difference between the cumulative distribution function (CDF) values at 7 and 3 minutes: P(3 < X < 7) = F(7) - F(3) = (1 - e^(-7/10)) - (1 - e^(-3/10)) = e^(-3/10) - e^(-7/10) ≈ 0.2442.

Question 6

Participants in a 2 day biking event, cross the finish line at a rate of 10 bike riders per fifteen minute interval.The probability that at least 2 minutes will elapse between bike riders is closest to _____________.
A).0000
B).0498
C).2635
D).1353
E).4647

Accepted Answer

The probability that at least 2 minutes will elapse between bike riders can be calculated using the Poisson distribution formula, given the average rate of occurrence. With 10 riders crossing the finish line every 15 minutes, the rate is 10 riders per 15 minutes, or 2/3 rider per minute. The probability of no riders in 2 minutes is e^(-4/3), and the probability of exactly one rider in 2 minutes is (4/3)*e^(-4/3). The probability of at least 2 minutes elapsing between riders is 1 minus these probabilities, which is approximately 0.2635.

Question 7

The average time between phone calls arriving at a call center is 30 seconds.Assuming that the time between calls is exponentially distributed, find the probability that less than two minutes elapse between calls.
A)0.018
B)0.064
C)0.936
D)0.982
E)1.000

Accepted Answer

The probability that less than two minutes elapse between calls is the same as the probability that the time between calls is less than two minutes. The time between calls is exponentially distributed with a mean of 30 seconds, so the probability that the time between calls is less than two minutes is:P(X < 120) = 1 - e^(-120/30) = 0.982Therefore, the correct answer is D.

Question 8

The average time between phone calls arriving at a call center is 30 seconds.Assuming that the time between calls is exponentially distributed, find the probability that more than a minute elapses between calls.
A)0.135
B)0.368
C)0.865
D)0.607
E)0.709

Accepted Answer

The probability that more than a minute elapses between calls is equal to the probability that the time between calls is greater than 60 seconds. The time between calls is exponentially distributed with a mean of 30 seconds, so the probability that the time between calls is greater than 60 seconds is given by:P(X > 60) = e^(-60/30) = e^(-2) = 0.135Therefore, the probability that more than a minute elapses between calls is 0.135.

Question 9

Inquiries arrive at a record message device according to a Poisson process at a rate of 15 inquiries per minute.The probability that it takes more than 12 seconds for the first inquiry to arrive is approximately _________
A)0.05
B)0.75
C)0.25
D)0.27
E)0.73

Accepted Answer

The answer of Inquiries arrive at a record message device...

Question 10

Participants in a 2 day biking event, cross the finish line at a rate of 10 bike riders per fifteen minute interval.The probability that less than 10 minutes will elapse between car arrivals is _____________.
A).0001
B).9987
C).0013
D).6667
E).1667

Accepted Answer

Given the rate of 10 bike riders per fifteen minutes, we can calculate the probability of an event happening in a shorter interval using the Poisson distribution. The average rate (λ) of bike riders crossing the finish line per minute is 10 riders / 15 minutes = 2/3 riders per minute. For less than 10 minutes between arrivals, we're interested in the probability of at least one arrival in a 10-minute window. The probability of exactly zero events in a given interval in a Poisson process is given by P(0) = e^(-λt), where λ is the average rate and t is the time interval. Here, λ = 2/3 and t = 10, so λt = 20/3. Thus, P(0) = e^(-20/3). The probability of at least one arrival is 1 - P(0), which is 1 - e^(-20/3). This calculation results in a value close to 0.9987, indicating a very high probability of less than 10 minutes elapsing between bike rider arrivals.

Question 11

In a busy gas station, an average of 2.4 cars arrive every minute.What is the average time between arrivals?
A)0.42 minutes
B)2.08 minutes
C)5.24 minutes
D)2.40 minutes
E)5.00 minutes

Accepted Answer

The average time between arrivals is the reciprocal of the arrival rate, so 1 / 2.4 = 0.4167 minutes, which rounds to approximately 0.42 minutes.

Question 12

Based on the caterer's experience, 38% of attendees to events will prefer chicken for the main dish.As the caterer plans for an event attended by 780 individuals, the normal approximation will be used for the binomial with a correction for continuity.In this case, what is the probability that less than 300 will prefer chicken?
A)0.604
B)0.380
C)0.590
D)0.620
E)0.618

Accepted Answer

The normal approximation to the binomial distribution is used here. The mean (μ) is np = 780 * 0.38 = 296.4, and the standard deviation (σ) is √(np(1-p)) = √(780 * 0.38 * 0.62) ≈ 13.67. For P(X < 300), use the continuity correction: P(X < 299.5). Convert to a z-score: z = (299.5 - 296.4) / 13.67 ≈ 0.227. Using standard normal distribution tables or a calculator, P(Z < 0.227) ≈ 0.590.

Question 13

Based on the caterer's experience, 38% of attendees to events will prefer chicken for the main dish.As the caterer plans for an event attended by 780 individuals, the normal approximation will be used for the binomial with a correction for continuity.In this case, what is the standard deviation of the number that she would expect to prefer chicken, when determining the probability that at less than 300 will prefer chicken?
A)114.0
B)300.0
C)183.8
D)296.4
E)13.6

Accepted Answer

The standard deviation for a binomial distribution is calculated using the formula √(n * p * (1-p)), where n is the number of trials and p is the probability of success. Here, n = 780 and p = 0.38, so the standard deviation is √(780 * 0.38 * 0.62) ≈ 13.6.

Question 14

At a certain workstation in an assembly line, the time required to assemble a component is exponentially distributed with a mean time of 10 minutes.Find the probability that a component is assembled in 7 minutes or less?
A)0.349
B)0.591
C)0.286
D)0.714
E)0.503

Accepted Answer

The answer of At a certain workstation in an assembly...

Question 15

If variable x is normally distributed with mean 0 and standard deviation 1, x ~ N(0,1), then the probability that x is exactly 0 is _________.
A)0.05
B)0.04
C)0.02
D)0.01
E)0.00

Accepted Answer

In a continuous probability distribution like the normal distribution, the probability of the variable taking on any single, exact value is 0. This is because there are infinitely many possible values the variable can take, making the probability of any specific value effectively zero.

Question 16

If x is uniformly distributed over some interval.The mean value (μ)of x is 2 and its variance (σ²)is 1/3.The probability that x is between 2 and 2.5, P(2 $\le$ x $\le$ 2.5), is _________. A)0.45 B)0.40 C)0.35 D)0.33 E)0.25

Accepted Answer

For a uniform distribution, the probability is proportional to the length of the interval. The total interval length is 2√3, and the interval from 2 to 2.5 is 0.5. Thus, the probability is 0.5 / 2√3 = 0.25.

Question 17

The probability that someone will prefer Coke over Pepsi is 0.56.A researcher asked 90 people which soda pop they preferred, and used the normal approximation to the binomial with a correction for continuity, what would be the probability at least 50 preferring Coke?
A)0.615
B)0.507
C)0.560
D)0.900
E)0.576

Accepted Answer

The answer of The probability that someone will prefer Coke...

Question 18

In a busy gas station, an average of 2.4 cars arrive every 5 minutes.What is the average time between arrivals?
A)0.42 minutes
B)2.08 minutes
C)5.24 minutes
D)2.40 minutes
E)5.00 minutes

Accepted Answer

The average time between arrivals can be calculated by dividing the total time period by the average number of arrivals. So, 5 minutes / 2.4 cars = 2.08 minutes.

Question 19

The exponential distribution is an example of _______.
A)a discrete distribution
B)a continuous distribution
C)a bimodal distribution
D)a normal distribution
E)a symmetrical distribution

Accepted Answer

The exponential distribution is a continuous distribution, which is used to model the time between events in a Poisson process.

Question 20

Based on the caterer's experience, 38% of attendees to events will prefer chicken for the main dish.As the caterer plans for an event attended by 780 individuals, the normal approximation will be used for the binomial with a correction for continuity.In this case, what is the average number that she would expect to prefer chicken, when determining the probability that less than 300 will prefer chicken?
A)296.4
B)114.0
C)483.6
D)13.6
E)183.8

Accepted Answer

The average number of attendees expected to prefer chicken is calculated by multiplying the total number of attendees (780) by the probability of preferring chicken (38%), which is 780 * 0.38 = 296.4.

Quiz 6: Continuous Distributions

For an exponential distribution with a lambda () equal to 20, the mean equal to _______.

The probability that someone will prefer Coke over Pepsi is 0.56.A researcher asked 90 people which soda pop they preferred, and used the normal approximation to the binomial with a correction for continuity, what would be the mean used in the calculation?

The probability that someone will prefer Coke over Pepsi is 0.56.A researcher asked 90 people which soda pop they preferred, and used the normal approximation to the binomial with a correction for continuity, what would be the standard deviation used in the calculation?

Participants in a 2 day biking event, cross the finish line at a rate of 10 bike riders per fifteen minute interval.On average, how much time, in minutes, elapses between bike riders?

At a certain workstation in an assembly line, the time required to assemble a component is exponentially distributed with a mean time of 10 minutes.Find the probability that a component is assembled in 3 to 7 minutes?

Participants in a 2 day biking event, cross the finish line at a rate of 10 bike riders per fifteen minute interval.The probability that at least 2 minutes will elapse between bike riders is closest to _____________.

The average time between phone calls arriving at a call center is 30 seconds.Assuming that the time between calls is exponentially distributed, find the probability that less than two minutes elapse between calls.

The average time between phone calls arriving at a call center is 30 seconds.Assuming that the time between calls is exponentially distributed, find the probability that more than a minute elapses between calls.

Inquiries arrive at a record message device according to a Poisson process at a rate of 15 inquiries per minute.The probability that it takes more than 12 seconds for the first inquiry to arrive is approximately _________

Participants in a 2 day biking event, cross the finish line at a rate of 10 bike riders per fifteen minute interval.The probability that less than 10 minutes will elapse between car arrivals is _____________.

In a busy gas station, an average of 2.4 cars arrive every minute.What is the average time between arrivals?

At a certain workstation in an assembly line, the time required to assemble a component is exponentially distributed with a mean time of 10 minutes.Find the probability that a component is assembled in 7 minutes or less?

If variable x is normally distributed with mean 0 and standard deviation 1, x ~ N(0,1) , then the probability that x is exactly 0 is _________.

If x is uniformly distributed over some interval.The mean value (μ) of x is 2 and its variance (σ²) is 1/3.The probability that x is between 2 and 2.5, P(2 $\le$ x $\le$ 2.5) , is _________.

The probability that someone will prefer Coke over Pepsi is 0.56.A researcher asked 90 people which soda pop they preferred, and used the normal approximation to the binomial with a correction for continuity, what would be the probability at least 50 preferring Coke?

In a busy gas station, an average of 2.4 cars arrive every 5 minutes.What is the average time between arrivals?

The exponential distribution is an example of _______.