Question 1

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 more skewed as the mean decreases. The Poisson distribution is a discrete probability distribution, meaning it relates to count data, and not continuous data. The variance of the Poisson distribution is equal to the expected value, not the square root of the expected value as stated in option C. Option A is also incorrect as the Poisson distribution is not symmetrical.

Question 2

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 3

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)0.4531
B)0.2001
C)0.6482
D)0.1951

Accepted Answer

The average number of weeds in a 3-square yard section is 3 * 1.3 = 3.9. Using the Poisson distribution with λ = 3.9, the probability of having less than 4 weeds is calculated by summing the probabilities of having 0, 1, 2, and 3 weeds, which results in approximately 0.4531.

Question 4

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

A binomial distribution is appropriate when there are only two possible outcomes (success or failure) for each trial, the trials are independent, the probability of success is constant for all trials, and the number of trials is fixed. In this case, the random variable is the number of people who say yes (success) out of the sample surveyed, and the conditions for a binomial distribution are met. Therefore, the most appropriate probability distribution is a binomial distribution.

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

The probability of guessing the correct answer for any given question is 0.5 (assuming a fair coin flip). To find the cut-off, we need to calculate the probability of guessing at least that many questions correctly by chance alone. We can use the binomial distribution for this, where n=20 (number of questions) and p=0.5 (probability of guessing correctly). We need to find the smallest value of k such that P(X >= k) <= 0.05, where X is the number of questions guessed correctly. Using a calculator or software, we find that k=14 satisfies this requirement, since P(X >= 14) = 0.021. Therefore, Madam Helga must correctly answer at least 14 questions to be considered psychic with 95% confidence.

Question 6

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 7

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

The Poisson distribution is commonly used to model the number of random events occurring in a fixed interval of time or space, such as the number of potholes per mile of road. The other distributions listed are not appropriate for this situation.

Question 8

Many people believe that they can tell the difference between Coke and Pepsi.Other people say that the two brands can't be distinguished.To test this,a random sample of 20 adults was selected to participate in a test.After being blindfolded,each person was given a small taste of either Coke or Pepsi and asked to indicate which brand soft drink it was.Suppose 14 people correctly identified the soft drink brand.Which of the following conclusions would be warranted under the circumstance?
A)Since the chance of getting 14 correct is 0.0370,which is quite small,the study shows that people are not able to identify brands effectively.
B)Since the probability of getting 14 or more correct is 0.0577,which is quite low,this means that people are not effective in identifying the soft drink brand.
C)Since the probability of getting 14 or more correct is 0.0577,which is quite low,the conclusion could be that people are effective at identifying soft drink brands.
D)The expected value for this binomial distribution is very close to 14 so this supports that people cannot tell the difference.

Accepted Answer

The probability of getting 14 or more correct out of a sample of 20 is 0.0577, which is quite low. This means that there is evidence to suggest that people are effective at identifying soft drink brands, as they scored higher than would be expected by chance. Option A is incorrect because getting 14 correct out of 20 is not that rare (the probability is 0.0370) and therefore cannot be used to conclude that people cannot identify the brands effectively. Option B is also incorrect because it implies that people cannot identify soft drink brands effectively, which is not supported by the evidence. Option D is incorrect because it suggests that the expected value for the binomial distribution is close to 14, but we do not have enough information to infer this.

Question 9

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 10

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 answerers.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 out of 20 questions is only 0.0207, which means that there is a small likelihood that Madam Helga is guessing. This suggests that she may have some special ability, as her correct answers cannot be attributed to chance alone. Therefore, option A is the best conclusion to reach.

Question 11

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 12

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 13

Many people believe that they can tell the difference between Coke and Pepsi.Other people say that the two brands can't be distinguished.To test this,a random sample of 20 adults was selected to participate in a test.After being blindfolded,each person was given a small taste of either Coke or Pepsi and asked to indicate which brand soft drink it was.If people really can't tell the difference,the probability that fewer than 6 people will guess correctly is:
A)0.0148.
B)approximately 0.02.
C)0.0307.
D)0.0514.

Accepted Answer

The answer of Many people believe that they can tell...

Question 14

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 15

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 formula for finding the standard deviation of a binomial distribution is:

SD = sqrt(np(1-p))

Substituting n = 50 and p = 0.10, we get:

SD = sqrt(50*0.10*(1-0.10)) = sqrt(4.5) ≈ 2.12

Therefore, the standard deviation of the number of defective parts is about 2.12. Option D is the correct answer.

Question 16

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 17

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

This statement is true. As p approaches 0.5, the two possible outcomes become equally likely, resulting in a more symmetric distribution. A and B are false statements, and D is incorrect as the skewness of the distribution depends on the value of p and not necessarily on the sample size.

Question 18

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 value or mean of a binomial distribution with parameters n and p is given by E(X) = n * p. In this case, E(X) = 50 * 0.10 = 5. Therefore, the expected number of defective parts is 5.

Question 19

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 20

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 $P(0; 2.5) = \frac{e^{-2.5} 2.5^0}{0!} = e^{-2.5} \approx 0.0821$.

Quiz 5: Discrete Probability Distributions

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

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?

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?

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:

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?

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

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:

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 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?

Which of the following statements is true?

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:

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?

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

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 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?

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?