Question 1

Completion time (from start to finish)of a building remodeling 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

To be 99% sure that we will not be late in completing the project, we need to find the completion time at which only 1% of the population is expected to exceed. We need to find the z-score for the 99th percentile, which is 2.33 (using a z-table or calculator). Then we use the formula:

z = (x - mu) / sigma

where x is the completion time we are looking for, mu is the mean, and sigma is the standard deviation. Solving for x:

2.33 = (x - 200) / 10
x - 200 = 23.3
x = 223.3

Rounding up to the nearest whole number gives a completion time of 223 work-days. Therefore, the best choice is C) 223.

Question 2

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 3

Let x be a binomial random variable with n=35 and p=.20.If we use the normal distribution to approximate probabilities for this,we would use a mean of _______.
A)35
B)20
C)70
D)7
E)3.5

Accepted Answer

The mean ($\mu$) of a binomial distribution is calculated as $\mu = np$, where $n$ is the number of trials and $p$ is the probability of success on each trial. For $n=35$ and $p=0.20$, the mean is $35 \times 0.20 = 7$.

Question 4

The net profit from a certain investment is normally distributed with a mean of $2,500 and a standard deviation of $1,000.The probability that the investor will not have a net loss is _____________.
A)0.4938
B)0.0062
C)0.9938
D)0.5062
E)0.0000

Accepted Answer

To find the probability that the investor will not have a net loss, we need to calculate the probability that the net profit is greater than or equal to $0. Using the Z-score formula, Z = (X - μ) / σ, where X is the value of interest ($0 in this case), μ is the mean ($2,500), and σ is the standard deviation ($1,000), we find Z = (0 - 2500) / 1000 = -2.5. Looking up the Z-score of -2.5 in the standard normal distribution table or using a calculator, we find that the probability of having a value less than -2.5 is approximately 0.0062. Therefore, the probability of not having a net loss (or having a net profit) is 1 - 0.0062 = 0.9938.

Question 5

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 answer of The average time between phone calls arriving...

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 miles and a standard deviation of 5000 miles.What is the probability that a randomly selected tire of this brand has a life of at most 30,000 miles?
A)0.5000
B)0.4772
C)0.0228
D)0.9772
E)1.0000

Accepted Answer

We need to find P(X ≤ 30000) where X is the random variable representing the useful life of the tire. We can calculate the standard score (z-score) for 30000 using the formula z = (x - mu) / sigma = (30000 - 40000) / 5000 = -2. Using a standard normal table or calculator, we can find the probability that a standard normal variable is less than or equal to -2, which is approximately 0.0228. Therefore, the probability that a randomly selected tire of this brand has a life of at most 30,000 miles is 0.0228 or 2.28%.

Question 7

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

Accepted Answer

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

Question 8

The net profit from a certain investment is normally distributed with a mean of $2,500 and a standard deviation of $1,000..The probability that the investor's net gain will be at least $2,000 is _____________.
A)0.0000
B)0.3413
C)0.0005
D)0.0500
E)0.5000

Accepted Answer

The answer of The net profit from a certain investment...

Question 9

Assuming an equal chance of a new baby being a boy or a girl (that is,p= 0.5),we would like to find the probability of 40 or more of the next 100 births at a local hospital will be boys.Using the normal approximation for binomial with a correction for continuity,we should use the z-score _______
A)0.4
B)-2.1
C)0.6
D)2
E)-1.7

Accepted Answer

The answer of Assuming an equal chance of a new...

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 miles and a standard deviation of 5000 miles.What is the probability that a randomly selected tire of this brand has a life between 30,000 and 50,000 miles?
A)0.5000
B)0.4772
C)0.9544
D)0.9772
E)1.0000

Accepted Answer

We need to find the probability that a tire has a life between 30,000 and 50,000 miles, which is the same as finding the probability that a z-score is between (-2 and 2) using the standard normal distribution. To find this probability, we can use the z-score formula:

z = (x - μ) / σ

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

For 30,000:

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

For 50,000:

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

Using a standard normal distribution table or calculator, we can find that the probability of a z-score being between -2 and 2 is approximately 0.9544. Therefore, the probability that a randomly selected tire of this brand has a life between 30,000 and 50,000 miles is 0.9544, or 95.44%. The answer is C.

Question 11

The expected (mean)life of a particular type of light bulb is 1,000 hours with a standard deviation of 50 hours.The life of this bulb is normally distributed.What is the probability that a randomly selected bulb would last fewer than 940 hours?
A)0.3849
B)0.8849
C)0.1151
D)0.6151
E)0.6563

Accepted Answer

We need to find the probability that a randomly selected bulb would last fewer than 940 hours. Using the standard normal distribution, we can calculate the z-score as:
z = (x - μ) / σ
where x is the value we want to find the probability for, μ is the mean and σ is the standard deviation.
In this case, x = 940, μ = 1000 and σ = 50.
z = (940 - 1000) / 50 = -1.2
Now, we can use a standard normal distribution table or calculator to find the probability of getting a z-score of -1.2 or less. The probability is 0.1151, which corresponds to the area to the left of the z-score on the standard normal distribution curve. Therefore, the answer is C.

Question 12

Completion time (from start to finish)of a building remodeling 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

To find the probability that the project will be completed within 185 work-days, we use the Z-score formula: Z = (X - μ) / σ, where X is the value (185 days), μ is the mean (200 days), and σ is the standard deviation (10 days). Plugging in the values, we get Z = (185 - 200) / 10 = -15 / 10 = -1.5. Looking up the Z-score of -1.5 in the standard normal distribution table gives us a probability of approximately 0.0668.

Question 13

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 miles and a standard deviation of 5000 miles.What is the probability that a randomly selected tire of this brand has a life of at least 50,000 miles?
A)0.0228
B)0.9772
C)0.5000
D)0.4772
E)1.0000

Accepted Answer

To find the probability that a tire has a life of at least 50,000 miles, we calculate the Z-score for 50,000 miles using the formula Z = (X - μ) / σ, where X is the value of interest (50,000 miles), μ is the mean (40,000 miles), and σ is the standard deviation (5,000 miles). Thus, Z = (50,000 - 40,000) / 5,000 = 2. The probability of a Z-score being greater than 2 (to the right of 2 on the standard normal distribution) is approximately 0.0228.

Question 14

The probability that a call to an emergency help line is answered in less than 10 seconds is 0.8.Assume that the calls are independent of each other.Using the normal approximation for binomial with a correction for continuity,the probability that at least 75 of 100 calls are answered within 10 seconds is approximately _______
A)0.8
B)0.1313
C)0.5235
D)0.9154
E)0.8687

Accepted Answer

The number of emergency calls answered within 10 seconds follows a binomial distribution with n=100 and p=0.8. We need to find P(X≥75), where X is the number of calls answered within 10 seconds.
Using the normal approximation with a correction for continuity, we have:

μ = np = 100 x 0.8 = 80
σ = sqrt(np(1-p)) = sqrt(100 x 0.8 x 0.2) = 4

Z = (X - μ + 0.5) / σ (continuity correction)

P(X≥75) = P(Z ≥ (75 - 80 + 0.5) / 4) = P(Z ≥ -1.375) = 0.9154 (from standard normal table)

Therefore, the answer is D) 0.9154.

Question 15

Let x be a binomial random variable with n=50 and p=.3.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 15 successes,we should find _______. A)P(x>15.5) B)P(x>15) C)P(x>14.5) D)P(x<14.5) E)P(x < 15)

Accepted Answer

When approximating a binomial distribution with a normal distribution, a continuity correction is applied because the binomial distribution is discrete while the normal distribution is continuous. For "more than 15 successes," we adjust to P(x > 15.5) to account for the fact that in a discrete distribution, more than 15 includes 16 and above, which starts at 15.5 in a continuous approximation.

Question 16

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 17

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 describes the time between events in a Poisson process.

Question 18

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 19

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

Since the time required to assemble a component is exponentially distributed with a mean time of 10 minutes, the rate parameter λ is given by λ = 1/mean = 1/10 = 0.1.
Let X be the time required to assemble a component. Then, X ~ Exp(0.1).
We need to find P(3 ≤ X ≤ 7).
P(3 ≤ X ≤ 7) = ∫₃ ⁰ ⁹exp(-0.1x) dx - ∫₇ ⁰ ⁹exp(-0.1x) dx
= [-exp(-0.1x)]₃⁹ - [-exp(-0.1x)]₇⁹
= (exp(-0.3) - exp(-0.9)) - (exp(-0.7) - exp(-0.9))
= 0.2442 (rounded to four decimal places)
Therefore, the correct choice is C.

Question 20

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 answer of The average time between phone calls arriving...

Quiz 6: Continuous Distributions

Completion time (from start to finish) of a building remodeling 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.

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?

Let x be a binomial random variable with n=35 and p=.20.If we use the normal distribution to approximate probabilities for this,we would use a mean of _______.

The net profit from a certain investment is normally distributed with a mean of $2,500 and a standard deviation of $1,000.The probability that the investor will not have a net loss is _____________.

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.

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

The net profit from a certain investment is normally distributed with a mean of $2,500 and a standard deviation of $1,000..The probability that the investor's net gain will be at least $2,000 is _____________.

Assuming an equal chance of a new baby being a boy or a girl (that is,p= 0.5) ,we would like to find the probability of 40 or more of the next 100 births at a local hospital will be boys.Using the normal approximation for binomial with a correction for continuity,we should use the z-score _______

The expected (mean) life of a particular type of light bulb is 1,000 hours with a standard deviation of 50 hours.The life of this bulb is normally distributed.What is the probability that a randomly selected bulb would last fewer than 940 hours?

Completion time (from start to finish) of a building remodeling 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 ______.

Let x be a binomial random variable with n=50 and p=.3.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 15 successes,we should find _______.

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?

The exponential distribution is an example of _______.

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.

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 center is 30 seconds.Assuming that the time between calls is exponentially distributed,find the probability that less than two minutes elapse between calls.