Question 1

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

Since the sample size is greater than 30, we can approximate the binomial distribution with a normal distribution. The mean of the normal distribution is equal to np where n is the sample size and p is the probability of success. 
So, the mean used in the calculation would be:
mean = np = 90*0.56 = 50.4
Therefore, the answer is D, 50.4. To use the normal approximation to the binomial, we need to calculate the mean and standard deviation of the binomial distribution. The mean of a binomial distribution is n*p, where n is the sample size and p is the probability of success. In this case, n=90 and p=0.56, so the mean is 90*0.56 = 50.4. Therefore, the best choice for the mean used in the calculation is D) 50.4.

Question 2

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 that occur randomly and independently at a constant rate. It has a continuous probability density function that is skewed to the right and ranges from 0 to infinity. The exponential distribution is a continuous distribution. It describes the time between events in a Poisson process, where events occur at a constant rate and independently of each other.

Question 3

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 4

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 probability that less than two minutes elapse between calls is the integral of the exponential distribution from 0 to 120 seconds. The exponential distribution is given by f(x) = lambda * e^(-lambda * x), where lambda is the average time between calls. In this case, lambda = 1/30 = 0.0333. So, the probability is:P(X < 120) = integral from 0 to 120 of 0.0333 * e^(-0.0333 * x) dx = 0.982

Question 5

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 6

For the normal approximation to be a good approximation for binomial distribution, n*p must be greater than ___________.
A)50
B)5
C)0.5
D)0.05
E)0

Accepted Answer

The rule of thumb for a good approximation is that n*p and n*(1-p) both must be at least 5. However, in most cases, if one of them is greater than or equal to 10, then the approximation is very good. Therefore, the answer is B as n*p should be greater than 5. According to the rule of thumb, for the normal approximation to be a good approximation for binomial distribution, np and n(1-p) must both be greater than 5. Therefore, the minimum value of np for the normal approximation to be a good approximation is 5.

Question 7

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 8

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 9

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 10

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 11

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 12

Based on the caterer's experience, 38% of attendees to events will prefer chicken for their 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 the caterer 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.

Question 13

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

We can use the normal approximation for binomial since np=100*0.8=80 and n(1-p)=100*0.2=20 are both greater than 10. With a correction for continuity, we want to find P(X≥75.5) where X~N(80, 4.0). Using the standard normal distribution, we have z=(75.5-80)/2=-2.25. Looking up the area to the right of z=−2.25 in the standard normal distribution table or calculator, we get 0.9878. However, we need to correct for continuity by subtracting 0.5 from 75.5 to get z=(75-80)/2. Then, the area to the right of z=−2.5 is 0.0062. Thus, the probability that at least 75 of 100 calls are answered in less than 10 seconds is approximately 1-0.0062=0.9938, which is closest to D) 0.9154. Let's define X as the number of calls answered within 10 seconds out of 100. Since each call can either be answered within 10 seconds or not, X follows a binomial distribution with parameters n=100 and p=0.8.

Now, we want to find the probability that at least 75 calls are answered within 10 seconds, which is equivalent to finding P(X>=75).

Using the normal approximation for binomial, we can approximate X as a normal distribution with mean = np = 100*0.8 = 80 and standard deviation = sqrt(np(1-p)) = sqrt(100*0.8*0.2) = 4.

To correct for continuity, we need to add or subtract 0.5 from our inequality to get P(X>=75) = P(X>74.5).

Using standardized normal distribution, we have:

P(Z > (74.5 - 80) / 4) = P(Z > -1.375) = 0.9154

Therefore, the answer is D.

Question 14

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

There are 288 fifteen minute intervals in 2 days (2 days x 24 hours x 60 minutes / 15 minutes = 288). Let X be the number of bike riders that cross the finish line in each interval. Then X follows a Poisson distribution with mean λ = 10.

We need to find the probability that at least 2 minutes will elapse between bike riders. Let Y be the time between consecutive riders. Then Y follows an exponential distribution with rate parameter λ' = 1/λ = 1/10.

The probability that at least 2 minutes will elapse between bike riders is equivalent to the probability that Y > 2 minutes.

P(Y > 2 minutes) = 1 - P(Y ≤ 2 minutes) = 1 - (1 - e^(-λ' * 2)) = 1 - e^(-2/10) ≈ 0.2635

Therefore, the closest answer choice is (C) 0.2635. To find the probability that at least 2 minutes will elapse between bike riders, we need to find the probability that there will be a gap of at least 2 minutes between any two successive bike riders. Let's assume that the time between two successive bike riders follows an exponential distribution with a rate of 40 riders per hour (i.e. 10 riders per 15-minute interval). Then the time between two successive bike riders follows an exponential distribution with a mean of 1/40 hours (or 3 minutes).

The probability that there will be a gap of at least 2 minutes between two successive bike riders is the same as the probability that the time between two successive bike riders is greater than or equal to 2 minutes, which is:

P(time between two successive bike riders >= 2 minutes)
= P(time between two successive bike riders >= 2/60 hours)
= e^(-40*2/60)
= e^(-4/3)
≈ 0.2635

Therefore, the closest answer choice is C) 0.2635.

Question 15

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 16

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

First, we need to calculate the mean and standard deviation of the binomial distribution: 
mean = n * p = 780 * 0.38 = 296.4
standard deviation = sqrt(n * p * (1 - p)) = sqrt(780 * 0.38 * 0.62) = 14.8

Next, we use a continuity correction: 
P(X < 300) = P(X < 300.5) 
= P(Z < (300.5 - 296.4) / 14.8)  where Z is a standard normal variable
= P(Z < 0.28)

Using a standard normal table or calculator, we find that P(Z < 0.28) is approximately 0.590. Therefore, the answer is (C) 0.590.

Question 17

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

When using the normal distribution to approximate probabilities for a binomial random variable, a correction for continuity should be made by adding or subtracting 0.5 to the value of x. To find the probability of more than 12 successes, we should find P(x>12.5).

Question 18

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 exponential distribution has a parameter λ = 1/30 per second. The probability that more than 60 seconds elapse is given by P(X > 60) = e^(-λ*60) = e^(-60/30) = e^(-2) ≈ 0.135.

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

The probability density function of the exponential distribution is given by:

f(x) = (1/μ)e^(-x/μ)

where μ is the mean.

We are given μ = 10 minutes.

To find the probability that a component is assembled in 3 to 7 minutes, we need to find the area under the probability density function between 3 and 7 minutes. This can be done using integration:

P(3 ≤ X ≤ 7) = ∫3^7 (1/10)e^(-x/10) dx

Using a calculator or software, this evaluates to:

P(3 ≤ X ≤ 7) ≈ 0.2442

Therefore, the best choice is C. We can use the cumulative distribution function (CDF) of the exponential distribution to find the probability that a component is assembled in 3 to 7 minutes. Let X be the time required to assemble a component, then X follows an exponential distribution with parameter λ = 1/10 (since the mean time is 10 minutes).
Using the CDF of exponential distribution, we have:
P(3 ≤ X ≤ 7) = F(7) - F(3)
= (1 - e^(-7/10)) - (1 - e^(-3/10))
= e^(-3/10) - e^(-7/10)
≈ 0.2442
Therefore, the answer is C.

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 standard deviation of the number that she would expect to prefer chicken, when determining the probability that 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: sqrt(n * p * (1-p)), where n is the number of trials (780) and p is the probability of success (0.38). This results in a standard deviation of approximately 13.6.

Quiz 6: Continuous Distributions

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 exponential distribution is an example of ___.

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 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.

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?

For the normal approximation to be a good approximation for binomial distribution, n*p must be greater than ___________.

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.

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 _______

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 _____________.

For an exponential distribution with a lambda (λ) equal to 4, the standard deviation is 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 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.The probability that at least 2 minutes will elapse between bike riders is closest to _____________.

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 ___.

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 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.

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.