Question 1

If cars arrive to a service center randomly and independently at a rate of 5 per hour on average,what is the probability of 0 cars arriving in a given hour?
A)0.1755
B)0.0067
C)0.0000
D)0.0500

Accepted Answer

The answer of If cars arrive to a service center...

Question 2

The number of visible defects on a product container is thought to be Poisson distributed with a mean equal to 3.5.Based on this,the probability that 2 containers will contain a total of less than 2 defects is:
A)0.0223
B)0.1359
C)0.0073
D)0.1850

Accepted Answer

The total number of defects in 2 containers is Poisson distributed with a mean equal to 7 (3.5 per container times 2). The probability of less than 2 defects is found using the Poisson probability formula, which for this scenario gives a value close to the option C (0.0073).

Question 3

The manager of a movie theater has determined that the distribution of customers arriving at the concession stand is Poisson distributed with a standard deviation equal to 2 people per 10 minutes.What is the probability that 0 customers arrive during a 10-minute period?
A)0.1353
B)0.0183
C)0.9817
D)Essentially 0

Accepted Answer

The answer of The manager of a movie theater has...

Question 4

Which of the following is true with respect to the binomial distribution?
A)As the sample size increases,the expected value of the random variable decreases.
B)The binomial distribution becomes more skewed as the sample size is increased for a given probability of success.
C)The binomial distribution tends to be more symmetric as p approaches 0.5.
D)In order for the binomial distribution to be skewed,the sample size must be quite large.

Accepted Answer

The binomial distribution is symmetric when p=0.5, and tends to be more skewed as p approaches 0 or 1. The sample size does not affect the expected value of the random variable. The statement in option B is incorrect as the binomial distribution becomes less skewed as the sample size increases. Option D is also incorrect as the skewness of the binomial distribution is primarily determined by the probability of success, not the sample size.

Question 5

Madam Helga claims to be psychic.A national TV talk personality plans to test her in a live TV broadcast.The process will entail asking Madam Helga a series of 20 independent questions with yes/no answers.The questions would be of the nature that she could not have any way of knowing the answer from prior knowledge.She will be considered psychic if she correctly answers more than a specified number (called the cut-off)of the questions.The cut-off must be set so that the chance of guessing that number or more is no greater than 5 percent.The cut-off value should be:
A)12
B)14
C)10
D)Can't be determined without more information.

Accepted Answer

To determine the cut-off value, we can use a binomial distribution with n = 20 (number of questions) and p = 0.5 (probability of guessing correctly for each question). The probability of guessing x or more questions correctly is: P(X >= x) = 1 - P(X < x) = 1 - (sum of binomial probabilities from 0 to x-1) We want to find the cut-off value such that P(X >= cut-off) <= 0.05. Using a binomial probability table or calculator, we can find that P(X >= 14) is approximately 0.0173, which is less than 0.05. Therefore, the cut-off value should be 14 or higher. Based on this analysis, the best choice is B (14).

Question 6

If cars arrive to a service center randomly and independently at a rate of 5 per hour on average,what is the probability that exactly 5 cars will arrive during a given hour?
A)0.1755
B)0.6160
C)0.1277
D)Essentially zero

Accepted Answer

The answer of If cars arrive to a service center...

Question 7

The number of weeds that remain living after a specific chemical has been applied averages 1.3 per square yard and follows a Poisson distribution.Based on this,what is the probability that a 1-square yard section will contain less than 4 weeds?
A)0.0324
B)0.0998
C)Nearly 0.5000
D)0.9569

Accepted Answer

The probability of having less than 4 weeds in a 1-square yard section can be calculated using the Poisson distribution formula. The sum of probabilities of having 0, 1, 2, or 3 weeds (when the average number of weeds is 1.3 per square yard) gives a high probability, which is closest to the option D) 0.9569. This calculation involves summing the probabilities of each of these outcomes, which are determined by the Poisson probability mass function.

Question 8

Which of the following statements is true?
A)A binomial distribution with n = 20 and p = 0.05 will be right-skewed.
B)A binomial distribution with n = 6 and p = 0.50 will be symmetric.
C)A binomial distribution with n = 20 and p = 0.05 has an expected value equal to 1.
D)A,B,and C are all true.

Accepted Answer

- A binomial distribution is right-skewed when the probability of success (p) is less than 0.5, which is true for A.- A binomial distribution is symmetric when p = 0.5, which is true for B.- The expected value (mean) of a binomial distribution is given by n*p, which equals 1 for C (20*0.05=1).

Question 9

Which of the following statements is true with respect to a Poisson distribution?
A)The Poisson distribution is symmetrical when the mean is close to 5.
B)The Poisson distribution is more right-skewed for smaller values of the mean.
C)The variance of the Poisson distribution is equal to the square root of the expected value.
D)The Poisson distribution is an example of a continuous probability distribution.

Accepted Answer

The Poisson distribution becomes less skewed as the mean increases and is highly skewed when the mean is small. Option A is incorrect as the Poisson distribution is not symmetrical, it is positively skewed. Option C is incorrect as the variance of a Poisson distribution is equal to its expected value. Option D is incorrect as the Poisson distribution is a discrete probability distribution.

Question 10

The number of visible defects on a product container is thought to be Poisson distributed with a mean equal to 3.5.Based on this,how many defects should be expected if 3 containers are inspected?
A)10.5
B)Approximately 3.24
C)Between 4 and 7
D)3.5

Accepted Answer

The expected number of defects for 3 containers is 3 times the mean number of defects per container, which is 3.5. Therefore, 3 * 3.5 = 10.5 defects are expected.

Question 11

If the number of defective items selected at random from a parts inventory is considered to follow a binomial distribution with n = 50 and p = 0.10,the expected number of defective parts is:
A)5
B)approximately 2.24
C)more than 10
D)0.5

Accepted Answer

The expected number of defective parts can be found by multiplying the sample size (n) by the probability of selecting a defective item (p). Therefore, the expected number of defective parts is 50*0.10=5. Option A is the only choice that matches this result.

Question 12

The probability that a product is found to be defective is 0.10.If we examine 50 products,which of the following has the highest probability?
A)3 defective products are found.
B)4 defective products are found.
C)5 defective products are found.
D)6 defective products are found.

Accepted Answer

The answer of The probability that a product is found...

Question 13

If the number of defective items selected at random from a parts inventory is considered to follow a binomial distribution with n = 50 and p = 0.10,the standard deviation of the number of defective parts is:
A)5
B)4.5
C)45
D)about 2.12

Accepted Answer

The standard deviation of a binomial distribution is given by the formula sqrt(np(1-p)). Plugging in n = 50 and p = 0.10, we get sqrt(50*0.10*0.90) = 2.12 (rounded to two decimal places). Therefore, the answer is D, about 2.12.

Question 14

The number of customers who enter a bank is thought to be Poisson distributed with a mean equal to 10 per hour.What are the chances that 2 or 3 customers will arrive in a 15-minute period?
A)0.0099
B)0.4703
C)0.0427
D)0.0053

Accepted Answer

Given that the mean number of customers entering the bank per hour is 10, for a 15-minute period (which is 1/4 of an hour), the mean ($\lambda$) becomes $10 \times \frac{1}{4} = 2.5$. The probability of $k$ events in a Poisson distribution is given by $P(k) = \frac{e^{-\lambda} \lambda^k}{k!}$. For 2 or 3 customers, we calculate $P(2) + P(3)$. $P(2) = \frac{e^{-2.5} 2.5^2}{2!}$ and $P(3) = \frac{e^{-2.5} 2.5^3}{3!}$. Calculating these and adding them gives approximately 0.4703, indicating the probability of 2 or 3 customers arriving in a 15-minute period.

Question 15

The number of customers who enter a bank is thought to be Poisson distributed with a mean equal to 10 per hour.What are the chances that no customers will arrive in a 15-minute period?
A)Approximately zero
B)0.0067
C)0.0821
D)0.0250

Accepted Answer

The Poisson distribution formula for the probability of observing $k$ events in an interval is given by $P(k; \lambda) = \frac{e^{-\lambda} \lambda^k}{k!}$, where $\lambda$ is the average number of events in an interval. For no customers ($k=0$) in a 15-minute period (which is a quarter of an hour), and with a mean ($\lambda$) of 10 customers per hour, the mean for 15 minutes is $10/4 = 2.5$. Substituting these values into the formula gives $P(0; 2.5) = \frac{e^{-2.5} 2.5^0}{0!} = e^{-2.5} \approx 0.0821$.

Question 16

Madam Helga claims to be psychic.A national TV talk personality plans to test her in a live TV broadcast.The process will entail asking Madam Helga a series of 20 independent questions with yes/no answers.The questions would be of the nature that she could not have any way of knowing the answer from prior knowledge.Suppose that Madam Helga correctly answered 15 of the 20 questions,which of the following would be a viable conclusion to reach?
A)Because the probability of guessing 15 or more correctly is 0.0207,it is unlikely that she is guessing at the questions and may,in fact,have some special ability.
B)Because the probability of getting 15 or more correct is 0.0207,it is likely that she is just guessing at the questions.
C)If she were guessing,15 is within one standard deviation of the mean and therefore she must not have any special psychic abilities.
D)Because the probability of guessing exactly 15 correct is 0.0148,she must just be guessing.

Accepted Answer

The probability of guessing 15 or more correctly being 0.0207 means that it is unlikely for Madam Helga to have guessed the answers and there might be a possibility that she has some special ability.

Question 17

The number of weeds that remain living after a specific chemical has been applied averages 1.3 per square yard and follows a Poisson distribution.Based on this,what is the probability that a 3-square yard section will contain less than 4 weeds?
A)0.4532
B)0.2001
C)0.6482
D)0.1951

Accepted Answer

The answer of The number of weeds that remain living...

Question 18

The manager of a movie theater has determined that the distribution of customers arriving at the concession stand is Poisson distributed with a standard deviation equal to 2 people per 10 minutes.What is the probability that more than 3 customers arrive during a 10-minute period?
A)0.1804
B)0.5665
C)0.4335
D)0.1954

Accepted Answer

We know that λ, the mean of the Poisson distribution, is equal to the standard deviation of the Poisson distribution. Therefore, λ = 2. 
The probability of more than 3 customers arriving during a 10-minute period can be calculated using the Poisson distribution formula:

P(X > 3) = 1 - P(X ≤ 3)

where X is the number of customers arriving at the concession stand during a 10-minute period.

Using the Poisson distribution formula with λ = 2, we can calculate:

P(X ≤ 3) = e^(-2) [((2^0)/0!) + ((2^1)/1!) + ((2^2)/2!) + ((2^3)/3!)]
           = 0.6767

Therefore, 
P(X > 3) = 1 - P(X ≤ 3) 
            = 1 - 0.6767 
            = 0.3233

So, the probability that more than 3 customers arrive during a 10-minute period is 0.3233, which is closest to 0.5665 (option B).

Question 19

Assuming that potholes occur randomly along roads,the number of potholes per mile of road could best be described by the:
A)binomial distribution.
B)Poisson distribution.
C)hypergeometric distribution.
D)continuous distribution

Accepted Answer

Potholes occur randomly and independently, and the number of potholes per mile of road is likely to be low. Therefore, the best distribution to describe this situation is the Poisson distribution, which is used for low-frequency events that occur randomly and independently over time or space.

Question 20

If a study is set up in such a way that a sample of people is surveyed to determine whether they have ever used a particular product,the likely probability distribution that would describe the random variable,the number who say yes,is a:
A)binomial distribution.
B)Poisson distribution.
C)uniform distribution.
D)continuous distribution.

Accepted Answer

Since the question is asking about a binary outcome (yes or no), the most appropriate probability distribution is a binomial distribution.

Quiz 5: Discrete Probability Distributions

If cars arrive to a service center randomly and independently at a rate of 5 per hour on average,what is the probability of 0 cars arriving in a given hour?

The number of visible defects on a product container is thought to be Poisson distributed with a mean equal to 3.5.Based on this,the probability that 2 containers will contain a total of less than 2 defects is:

The manager of a movie theater has determined that the distribution of customers arriving at the concession stand is Poisson distributed with a standard deviation equal to 2 people per 10 minutes.What is the probability that 0 customers arrive during a 10-minute period?

Which of the following is true with respect to the binomial distribution?

If cars arrive to a service center randomly and independently at a rate of 5 per hour on average,what is the probability that exactly 5 cars will arrive during a given hour?

The number of weeds that remain living after a specific chemical has been applied averages 1.3 per square yard and follows a Poisson distribution.Based on this,what is the probability that a 1-square yard section will contain less than 4 weeds?

Which of the following statements is true?

Which of the following statements is true with respect to a Poisson distribution?

The number of visible defects on a product container is thought to be Poisson distributed with a mean equal to 3.5.Based on this,how many defects should be expected if 3 containers are inspected?

If the number of defective items selected at random from a parts inventory is considered to follow a binomial distribution with n = 50 and p = 0.10,the expected number of defective parts is:

The probability that a product is found to be defective is 0.10.If we examine 50 products,which of the following has the highest probability?

If the number of defective items selected at random from a parts inventory is considered to follow a binomial distribution with n = 50 and p = 0.10,the standard deviation of the number of defective parts is:

The number of customers who enter a bank is thought to be Poisson distributed with a mean equal to 10 per hour.What are the chances that 2 or 3 customers will arrive in a 15-minute period?

The number of customers who enter a bank is thought to be Poisson distributed with a mean equal to 10 per hour.What are the chances that no customers will arrive in a 15-minute period?

The number of weeds that remain living after a specific chemical has been applied averages 1.3 per square yard and follows a Poisson distribution.Based on this,what is the probability that a 3-square yard section will contain less than 4 weeds?

The manager of a movie theater has determined that the distribution of customers arriving at the concession stand is Poisson distributed with a standard deviation equal to 2 people per 10 minutes.What is the probability that more than 3 customers arrive during a 10-minute period?

Assuming that potholes occur randomly along roads,the number of potholes per mile of road could best be described by the:

If a study is set up in such a way that a sample of people is surveyed to determine whether they have ever used a particular product,the likely probability distribution that would describe the random variable,the number who say yes,is a: