Question 1

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

We can use the cumulative distribution function (CDF) of exponential distribution to solve this problem:
P(3≤X≤7) = F(7) - F(3) 
where F(x) is the CDF of exponential distribution with mean time of 10 minutes:
F(x) = 1 - e^(-x/10)
Therefore, 
P(3≤X≤7) = (1 - e^(-7/10)) - (1 - e^(-3/10))
           = e^(-3/10) - e^(-7/10)
           ≈ 0.2442
Therefore, the correct choice is C.

Question 2

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 3

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

Let's first find the mean and standard deviation of the binomial distribution:
mean = n * p = 100 * 0.8 = 80
standard deviation = sqrt(n * p * (1-p)) = sqrt(100 * 0.8 * 0.2) = 4

To use the normal approximation, we need to standardize the variable:
z = (x - mean) / standard deviation
where x is the number of calls answered within 10 seconds.

Now we need to find the probability that at least 75 calls are answered within 10 seconds. This is equivalent to finding the probability that x >= 75. We can standardize this as follows:
z = (75 - 80) / 4 = -1.25

Using a standard normal table, we can find the probability that z is less than -1.25, and then subtract that from 1 to get the probability that z is greater than or equal to -1.25:
P(z >= -1.25) = 0.8944
So the probability that at least 75 calls are answered within 10 seconds is approximately 0.8944. However, we need to apply a correction for continuity:
P(x >= 75) = P(z >= -1.25 + 0.5/n) = P(z >= -1.195)
Using the standard normal table again, we find that P(z >= -1.195) = 0.9154.
Therefore, the answer is D) 0.9154.

Question 4

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 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 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 6

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 7

Let x be a binomial random variable with n=50 and p=.3. The probability of less than or equal to13 successes,when using the normal approximation for binomial is ____ ____
A) -.6172
B).3086
C)3.240
D).2676
E)-.23224

Accepted Answer

We need to use the normal approximation for binomial distribution, where the mean is np=15 and the standard deviation is sqrt(npq)=2.872.
We need to find P(x<=13) which is equivalent to finding P(z<=(13-15)/2.872) since we standardize the distribution using z-scores.
This gives us P(z<=-0.698) which we can find using a standard normal distribution table or calculator to be 0.2437.
Therefore the probability of less than or equal to 13 successes is 0.2437 or approximately 0.2676 (rounded to four decimal places).

Question 8

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

We need to find the completion time that ensures that we will not be late 99% of the time. Since we are dealing with a normal distribution, we can use the z-score formula to find the z-score that corresponds to the 99th percentile:

z = invNorm(0.99) = 2.33

Then we can use the formula for a z-score to find the corresponding completion time:

z = (x - mu) / sigma
2.33 = (x - 200) / 10
x - 200 = 23.3
x = 223.3

Rounding up to the nearest integer, we get an answer of 223 work-days.

Question 9

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 means it applies to variables that can take on any value within a range, including decimal values.

Question 10

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...

Question 11

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 12

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 13

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 14

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

When using the normal distribution to approximate binomial probabilities, we use the mean and variance of the binomial distribution and apply the continuity correction. The mean of the binomial distribution is μ = np, where n=35 and p=.20, so μ = 35(.20) = 7. Therefore, the answer is D) 7.

Question 15

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 16

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 17

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

To find the probability that a randomly selected tire of this brand has a life between 30,000 and 50,000 miles, we need to calculate the z-scores for both values first:

z1 = (30,000 - 40,000) / 5000 = -2
z2 = (50,000 - 40,000) / 5000 = 2

Then, we can use a standard normal distribution table or calculator to find the area under the curve between these z-scores:

P(-2 < Z < 2) = 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, which corresponds to answer choice C.

Question 18

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 19

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 Z-score for $2,000 is (2000-2500)/1000 = -0.5. The probability of a Z-score greater than -0.5 is 0.6915. Since the normal distribution is symmetric, the probability of a net gain of at least $2,000 is 0.6915, which is closest to 0.5000 among the given options.

Question 20

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 want to find the probability that a randomly selected tire has a life of at most 30,000 miles, which means we need to find the area to the left of 30,000 miles on the normal distribution curve. We can standardize the variable X (tire life in miles) using the formula:
z = (X - μ) / σ
where μ = 40,000 miles and σ = 5,000 miles. 
Plugging in the values, we get:
z = (30,000 - 40,000) / 5,000 = -2
Using a standard normal distribution table or calculator, we can find the area to the left of z = -2, which is 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 approximately 2.28%. 
The correct answer is C.

Quiz 6: Continuous Distributions

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 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 _____________.

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

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?

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?

Let x be a binomial random variable with n=50 and p=.3. The probability of less than or equal to13 successes,when using the normal approximation for binomial is

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.

The exponential distribution is an example of _______.

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.

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 _______.

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=35 and p=.20. If we use the normal distribution to approximate probabilities for this,we would use a mean of _______.

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 _______

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.

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.

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 _____________.