Question 1

The average time between phone calls arriving at a call centre 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 answer of The average time between phone calls arriving...

Question 2

On Saturdays,cars arrive at Sam Schmitt's Scrub and Shine Car Wash at the rate of 6 cars per fifteen minute interval.The probability that at least 2 minutes will elapse between car arrivals is ___.
A)0.0000
B)0.4493
C)0.1353
D)1.0000
E)1.0225

Accepted Answer

The answer of On Saturdays,cars arrive at Sam Schmitt's Scrub...

Question 3

Let x be a binomial random variable with n=20 and p=.8.If we use the normal distribution to approximate probabilities for this,we would use a mean of ___.
A)20
B)16
C)3.2
D)8
E)5

Accepted Answer

The answer of Let x be a binomial random variable...

Question 4

On Saturdays,cars arrive at Sam Schmitt's Scrub and Shine Car Wash at the rate of 6 cars per fifteen minute interval.The probability that less than 10 minutes will elapse between car arrivals is ___.
A)0.8465
B)0.9817
C)0.0183
D)0.1535
E)0.2125

Accepted Answer

We need to find the probability that less than 10 minutes will elapse between car arrivals. This means we need to find the probability that no more than one car arrives in a 10 minute interval. 
First, we need to find the average rate of car arrivals in a 10 minute interval: 
(6 cars/15 minutes) x (10 minutes) = 4 cars

Now, we can use the Poisson distribution to find the probability of 0 or 1 cars arriving in a 10 minute interval:

P(0 or 1 car) = [(e^-4)(4^0/0!)] + [(e^-4)(4^1/1!)] 
P(0 or 1 car) = [0.0183] + [0.0733] 
P(0 or 1 car) = 0.0916

Finally, we need to find the probability of less than 10 minutes between car arrivals:

P(less than 10 minutes) = 1 - P(0 or 1 car) 
P(less than 10 minutes) = 1 - 0.0916 
P(less than 10 minutes) = 0.9084

Therefore, the answer is B) 0.9817.

Question 5

Completion time (from start to finish)of a building remodelling project is normally distributed with a mean of 200 work-days and a standard deviation of 10 work-days.To be 99% sure that we will not be late in completing the project,we should request a completion time of ___ work-days.
A)211
B)207
C)223
D)200
E)250

Accepted Answer

We need to find the completion time such that we can be 99% sure that the project will not be late.
Using the normal distribution, we know that 99% of the area lies within 2.58 standard deviations of the mean.
So, the completion time that corresponds to 2.58 standard deviations above the mean is:
200 + (2.58)(10) = 223
Therefore, the best choice is C, 223 work-days.

Question 6

Sure Stone Tire Company has established that the useful life of a particular brand of its automobile tires is normally distributed with a mean of 40,000 kilometres and a standard deviation of 5000 kilometres.What is the probability that a randomly selected tire of this brand has a life between 30,000 and 50,000 kilometres?
A)0.5000
B)0.4772
C)0.9544
D)0.9772
E)1.0000

Accepted Answer

To solve this problem, we need to standardize the range of values between 30,000 and 50,000 using the formula:

z = (x - μ) / σ

where x is the value we're interested in (30,000 or 50,000), μ is the mean (40,000) and σ is the standard deviation (5,000).

For x = 30,000:
z = (30,000 - 40,000) / 5,000 = -2

For x = 50,000:
z = (50,000 - 40,000) / 5,000 = 2

We can now use a standard normal distribution table or a calculator to find the area (probability) under the curve between z = -2 and z = 2. This area represents the probability that a randomly selected tire of this brand has a life between 30,000 and 50,000 kilometres.

Using a calculator or a table, we find that the area is approximately 0.9544.

Therefore, the answer is C: 0.9544.

Question 7

Suppose you are working with a data set that is normally distributed with a mean of 400 and a standard deviation of 20.Determine the value of x such that 60% of the values are greater than x.
A)404.5
B)395.5
C)405.0
D)395.0
E)415.0

Accepted Answer

The answer of Suppose you are working with a data...

Question 8

Completion time (from start to finish)of a building remodelling project is normally distributed with a mean of 200 work-days and a standard deviation of 10 work-days.The probability that the project will be completed within 185 work-days is ___.
A)0.0668
B)0.4332
C)0.5000
D)0.9332
E)0.9950

Accepted Answer

The answer of Completion time (from start to finish)of a...

Question 9

Sure Stone Tire Company has established that the useful life of a particular brand of its automobile tires is normally distributed with a mean of 40,000 kilometres and a standard deviation of 5000 kilometres.What is the probability that a randomly selected tire of this brand has a life of at least 50,000 kilometres?
A)0.0228
B)0.9772
C)0.5000
D)0.4772
E)1.0000

Accepted Answer

The answer of Sure Stone Tire Company has established that...

Question 10

Sure Stone Tire Company has established that the useful life of a particular brand of its automobile tires is normally distributed with a mean of 40,000 kilometres and a standard deviation of 5000 kilometres.What is the probability that a randomly selected tire of this brand has a life of at most 30,000 kilometres?
A)0.5000
B)0.4772
C)0.0228
D)0.9772
E)1.0000

Accepted Answer

To solve this problem, we need to standardize the random variable X representing the life of the tire using the formula:
Z = (X - μ) / σ
where μ is the mean and σ is the standard deviation. Thus,
Z = (30,000 - 40,000) / 5000 = -2
Using the standard normal distribution table or calculator, we find that the probability of Z being less than or equal to -2 is 0.0228. Therefore, the probability that a randomly selected tire of this brand has a life of at most 30,000 kilometres is 0.0228 or 2.28%. The answer is (C).

Question 11

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 probability density function of exponential distribution is
f(x) = λe^(-λx), where λ is the rate parameter.
Here, λ = 1/10 (since mean time is 10 minutes).
Now, we need to find P(3 ≤ x ≤ 7).
P(3 ≤ x ≤ 7) = ∫3^7 f(x) dx
     = ∫3^7 (1/10)e^(-x/10) dx
     = [-e^(-x/10)]3^7
     = e^(-3/10) - e^(-7/10)
     ≈ 0.2442 (rounded to four decimal places)

Question 12

The average time between phone calls arriving at a call centre 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 average time between calls is 30 seconds, which implies that the rate of arrival of calls is λ = 1/30 per second. We need to find the probability that more than a minute elapses between calls, which is equivalent to finding the probability of waiting more than 60 seconds between two consecutive calls. Let X be the time between two consecutive calls. Then X has an exponential distribution with the rate parameter λ = 1/30. Thus,

P(X > 60) = e^(-λ*60) = e^(-(1/30)*60) = e^(-2) ≈ 0.135

Hence, the answer is (A).

Question 13

The net profit of an investment is normally distributed with a mean of $10,000 and a standard deviation of $5,000.The probability that the investor's net gain will be at least $5,000 is ___.
A)0.1859
B)0.3413
C)0.8413
D)0.4967
E)0.5000

Accepted Answer

We need to find the probability of the net gain being at least $5,000. This is equivalent to finding the probability of Z being greater than or equal to (5,000-10,000)/5,000 = -1.

Using a standard normal distribution table, we find that the area to the right of Z=-1 is 0.8413.

Therefore, the probability that the investor's net gain will be at least $5,000 is 0.8413, which corresponds to choice C.

Question 14

Let x be a binomial random variable with n=20 and p=.8.If we use the normal distribution to approximate probabilities for this,a correction for continuity should be made.To find the probability of more than 12 successes,we should find ___. A)P(x>12.5) B)P(x>12) C)P(x>11.5) D)P(x<11.5) E)P(x < 12)

Accepted Answer

The answer of Let x be a binomial random variable...

Question 15

The net profit from a certain investment is normally distributed with a mean of $10,000 and a standard deviation of $5,000.The probability that the investor will not have a net loss is ___.
A)0.4772
B)0.0228
C)0.9544
D)0.9772
E)1.0000

Accepted Answer

To not have a net loss means that the net profit is greater than or equal to $0. We can standardize this by subtracting the mean and dividing by the standard deviation: (0 - 10000)/5000 = -2. The probability of a z-score being less than -2 is approximately 0.0228, which is the probability that the investor will have a net loss. Therefore, the probability that the investor will not have a net loss is 1 - 0.0228 = 0.9772, which is answer choice D.

Question 16

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)0.5034

Accepted Answer

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

Question 17

For an exponential distribution with a lambda (λ)equal to 4,the standard deviation is equal to ___.
A)4
B)0.5
C)0.25
D)1
E)16

Accepted Answer

The answer of For an exponential distribution with a lambda...

Question 18

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 that is used to model the time between events that occur at a constant rate, such as the arrival of customers at a service desk or the decay of radioactive materials. It has a probability density function that decreases exponentially as the time between events increases.

Quiz 6: Continuous Distributions

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

On Saturdays,cars arrive at Sam Schmitt's Scrub and Shine Car Wash at the rate of 6 cars per fifteen minute interval.The probability that at least 2 minutes will elapse between car arrivals is ___.

Let x be a binomial random variable with n=20 and p=.8.If we use the normal distribution to approximate probabilities for this,we would use a mean of ___.

On Saturdays,cars arrive at Sam Schmitt's Scrub and Shine Car Wash at the rate of 6 cars per fifteen minute interval.The probability that less than 10 minutes will elapse between car arrivals is ___.

Completion time (from start to finish) of a building remodelling project is normally distributed with a mean of 200 work-days and a standard deviation of 10 work-days.To be 99% sure that we will not be late in completing the project,we should request a completion time of ___ work-days.

Suppose you are working with a data set that is normally distributed with a mean of 400 and a standard deviation of 20.Determine the value of x such that 60% of the values are greater than x.

Completion time (from start to finish) of a building remodelling project is normally distributed with a mean of 200 work-days and a standard deviation of 10 work-days.The probability that the project will be completed within 185 work-days is ___.

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.

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

The net profit of an investment is normally distributed with a mean of $10,000 and a standard deviation of $5,000.The probability that the investor's net gain will be at least $5,000 is ___.

Let x be a binomial random variable with n=20 and p=.8.If we use the normal distribution to approximate probabilities for this,a correction for continuity should be made.To find the probability of more than 12 successes,we should find ___.

The net profit from a certain investment is normally distributed with a mean of $10,000 and a standard deviation of $5,000.The probability that the investor will not have a net loss is ___.

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.

For an exponential distribution with a lambda (λ) equal to 4,the standard deviation is equal to ___.

The exponential distribution is an example of ___.