Question 1

Patients arriving at an outpatient clinic follow an exponential distribution at a rate of 1 patient per hour.What is the probability that a randomly chosen arrival to be more than 2.5 hours?

Accepted Answer

The answer of Patients arriving at an outpatient clinic follow...

Question 2

Patients arriving at an outpatient clinic follow an exponential distribution at a rate of 15 patients per hour.What is the probability that a randomly chosen arrival to be more than 5 minutes?

Accepted Answer

The answer of Patients arriving at an outpatient clinic follow...

Question 3

Patients arriving at an outpatient clinic follow an exponential distribution with mean 15 minutes.What is the probability that a randomly chosen arrival to be less than 15 minutes?

Accepted Answer

The answer of Patients arriving at an outpatient clinic follow...

Question 4

Times spent using a tablet or smartphone every week by primary school students follow an exponential distribution with mean 10 hours.The probability that a given primary school student spends between 10 and 15 hours using a tablet or smartphone is _______

Accepted Answer

The answer of Times spent using a tablet or smartphone...

Question 5

Patients arriving at an outpatient clinic follow an exponential distribution with mean 15 minutes.What is the average number of arrivals per minute?

Accepted Answer

The answer of Patients arriving at an outpatient clinic follow...

Question 6

The Townsville International Airport (TIA)has been criticised for the waiting times associated with departing flights.While the critics acknowledge that many flights have little or no waiting times,their complaints deal more specifically with the longer waits attributed to some flights.The critics are interested in showing,mathematically,exactly what the problems are.Which type of distribution would best model the waiting times of the departing flights at TIA?
A) Binomial distribution.
B) Exponential distribution.
C) Uniform distribution.
D) Normal distribution.

Accepted Answer

The exponential distribution would best model the waiting times of the departing flights at TIA. This distribution is frequently used to model waiting times or times between events, and is suitable when the time between events is random and the probability of an event occurring in any given time period is constant.

Question 7

Let X represent the amount of time it takes a student to park in the car park at the university.If we know that the distribution of parking times can be modelled using an exponential distribution with a mean of 4 minutes,find the probability that it will take a randomly selected student more than 10 minutes to park in the car park.
A) 0.329680
B) 0.670320
C) 0.917915
D) 0.082085

Accepted Answer

We know that the mean of the exponential distribution is 4 minutes, which means that the rate parameter λ is equal to 1/4.

We can use the cumulative distribution function (CDF) of the exponential distribution to find the probability that a randomly selected student will take more than 10 minutes to park:

P(X > 10) = 1 - P(X ≤ 10)
            = 1 - F(10)
where F(x) is the CDF of the exponential distribution with rate parameter λ = 1/4:

F(x) = 1 - e^(-λx)
       = 1 - e^(-x/4)

Substituting x = 10, we get:

P(X > 10) = 1 - F(10)
            = 1 - (1 - e^(-10/4))
            = e^(-10/4)
            ≈ 0.082

Therefore, the answer is D: 0.082085.

Question 8

Let X represent the amount of time it takes a student to park in the car park at the university.If we know that the distribution of parking times can be modelled using an exponential distribution with a mean of 4 minutes,find the probability that it will take a randomly selected student between 2 and 12 minutes to park in the car park.
A) 0.656318
B) 0.556744
C) 0.606531
D) 0.049787

Accepted Answer

The answer of Let X represent the amount of time...

Question 9

Times spent watching TV every week by primary school students follow an exponential distribution with mean 10 hours.The probability that a given primary school student spends less than 20 hours watching TV is _______

Accepted Answer

The answer of Times spent watching TV every week by...

Question 10

Instruction 6.7
The interval between consecutive hits at a website is assumed to follow an exponential distribution with an average of 40 hits per minute.
-Referring to Instruction 6.7,what is the average time between consecutive hits?

Accepted Answer

The answer of Instruction 6.7
The interval between consecutive hits at...

Question 11

Patients arriving at an outpatient clinic follow an exponential distribution at a rate of 1.5 patients per hour.What is the probability that a randomly chosen arrival to be less than 10 minutes?

Accepted Answer

The answer of Patients arriving at an outpatient clinic follow...

Question 12

Patients arriving at an outpatient clinic follow an exponential distribution at a rate of 1 patient per hour.What is the probability that a randomly chosen arrival to be more than 1 hour?

Accepted Answer

The answer of Patients arriving at an outpatient clinic follow...

Question 13

Patients arriving at an outpatient clinic follow an exponential distribution at a rate of 15 patients per hour.What is the probability that a randomly chosen arrival to be between 5 minutes and 15 minutes?

Accepted Answer

The answer of Patients arriving at an outpatient clinic follow...

Question 14

Times spent using a tablet or smartphone every week by primary school students follow an exponential distribution with mean 10 hours.The probability that a given primary school student spends more than 5 hours using a tablet or smartphone is _______

Accepted Answer

The answer of Times spent using a tablet or smartphone...

Question 15

A catalogue company that receives the majority of its orders by telephone conducted a study to determine how long customers were willing to wait on hold before ordering a product.The length of time was found to be a random variable best approximated by an exponential distribution with a mean equal to 2.8 minutes.
-What proportion of callers is put on hold longer than 2.8 minutes?
A) 0.60810
B) 0.50
C) 0.367879
D) 0.632121

Accepted Answer

We are told that the length of time customers are willing to wait on hold is best described by an exponential distribution with a mean of 2.8 minutes. The probability density function of an exponential distribution with mean 𝜇 is:

f(x) = (1/𝜇) * e^(-x/𝜇)

We want to find the proportion of callers who are put on hold longer than 2.8 minutes, which is the same as finding:

P(X > 2.8) = ∫2.8∞ f(x)dx

Plugging in the mean 𝜇 = 2.8, we get:

P(X > 2.8) = ∫2.8∞ (1/2.8) * e^(-x/2.8) dx

Using a software or a calculator, we can evaluate this integral to get:

P(X > 2.8) = 0.367879

Therefore, the answer is choice C.

Question 16

Patients arriving at an outpatient clinic follow an exponential distribution at a rate of 15 patients per hour.What is the probability that a randomly chosen arrival to be less than 15 minutes?

Accepted Answer

The answer of Patients arriving at an outpatient clinic follow...

Question 17

Patients arriving at an outpatient clinic follow an exponential distribution at a rate of 1 patient per hour.What is the probability that a randomly chosen arrival to be less than 20 minutes?

Accepted Answer

The answer of Patients arriving at an outpatient clinic follow...

Question 18

Patients arriving at an outpatient clinic follow an exponential distribution with mean 15 minutes.What is the probability that a randomly chosen arrival to be more than 18 minutes?

Accepted Answer

The answer of Patients arriving at an outpatient clinic follow...

Question 19

Patients arriving at an outpatient clinic follow an exponential distribution at a rate of 1.5 patients per hour.What is the probability that a randomly chosen arrival to be between 10 and 15 minutes?

Accepted Answer

The answer of Patients arriving at an outpatient clinic follow...

Question 20

A catalogue company that receives the majority of its orders by telephone conducted a study to determine how long customers were willing to wait on hold before ordering a product.The length of time was found to be a random variable best approximated by an exponential distribution with a mean equal to 2.8 minutes.
-What is the probability that a randomly selected caller is placed on hold fewer than 7 minutes?
A) 0.9990881
B) 0.917915
C) 0.0009119
D) 0.082085

Accepted Answer

To solve the problem, we need to use the cumulative distribution function (CDF) of the exponential distribution, which is:

F(x) = 1 - e^(-λx)

where λ is the rate parameter, which is equal to 1/mean for the exponential distribution.

In this case, we have mean = 2.8 minutes, so λ = 1/2.8 = 0.3571 (rounded to 4 decimal places).

Now we can calculate the probability that a randomly selected caller is placed on hold for less than 7 minutes:

P(X < 7) = F(7) = 1 - e^(-0.3571 * 7) = 0.917915 (rounded to 6 decimal places)

Therefore, the answer is B. The probability that a randomly selected caller is placed on hold fewer than 7 minutes is 0.917915.

Quiz 6: The Normal Distribution and Other Continuous Distributions

Patients arriving at an outpatient clinic follow an exponential distribution at a rate of 1 patient per hour.What is the probability that a randomly chosen arrival to be more than 2.5 hours?

Patients arriving at an outpatient clinic follow an exponential distribution at a rate of 15 patients per hour.What is the probability that a randomly chosen arrival to be more than 5 minutes?

Patients arriving at an outpatient clinic follow an exponential distribution with mean 15 minutes.What is the probability that a randomly chosen arrival to be less than 15 minutes?

Times spent using a tablet or smartphone every week by primary school students follow an exponential distribution with mean 10 hours.The probability that a given primary school student spends between 10 and 15 hours using a tablet or smartphone is _______

Patients arriving at an outpatient clinic follow an exponential distribution with mean 15 minutes.What is the average number of arrivals per minute?

Times spent watching TV every week by primary school students follow an exponential distribution with mean 10 hours.The probability that a given primary school student spends less than 20 hours watching TV is _______

Instruction 6.7 The interval between consecutive hits at a website is assumed to follow an exponential distribution with an average of 40 hits per minute. -Referring to Instruction 6.7,what is the average time between consecutive hits?

Patients arriving at an outpatient clinic follow an exponential distribution at a rate of 1.5 patients per hour.What is the probability that a randomly chosen arrival to be less than 10 minutes?

Patients arriving at an outpatient clinic follow an exponential distribution at a rate of 1 patient per hour.What is the probability that a randomly chosen arrival to be more than 1 hour?

Patients arriving at an outpatient clinic follow an exponential distribution at a rate of 15 patients per hour.What is the probability that a randomly chosen arrival to be between 5 minutes and 15 minutes?

Times spent using a tablet or smartphone every week by primary school students follow an exponential distribution with mean 10 hours.The probability that a given primary school student spends more than 5 hours using a tablet or smartphone is _______

Patients arriving at an outpatient clinic follow an exponential distribution at a rate of 15 patients per hour.What is the probability that a randomly chosen arrival to be less than 15 minutes?

Patients arriving at an outpatient clinic follow an exponential distribution at a rate of 1 patient per hour.What is the probability that a randomly chosen arrival to be less than 20 minutes?

Patients arriving at an outpatient clinic follow an exponential distribution with mean 15 minutes.What is the probability that a randomly chosen arrival to be more than 18 minutes?

Patients arriving at an outpatient clinic follow an exponential distribution at a rate of 1.5 patients per hour.What is the probability that a randomly chosen arrival to be between 10 and 15 minutes?