Question 1

Let x be a Poisson random variable with mean 3. What is the probability that x will fall in the interval $\mu$ ± 2$\sigma$?

Accepted Answer

The answer of Let x be a Poisson random variable...

Question 2

Telephone Switchboard Narrative
The number of telephone calls coming into a business's switchboard averages four calls per minute. Let x be the number of calls received.
-Refer to Telephone Switchboard Narrative. Find P(x = 0).

Accepted Answer

The answer of Telephone Switchboard Narrative
The number of telephone calls...

Question 3

Telephone Switchboard Narrative
The number of telephone calls coming into a business's switchboard averages four calls per minute. Let x be the number of calls received.
-Refer to Telephone Switchboard Narrative. What is the probability there will be at least one call in a given one-minute period?

Accepted Answer

The answer of Telephone Switchboard Narrative
The number of telephone calls...

Question 4

A statistics department is contacting alumni by telephone and asking for donations to help fund a new computer laboratory. Past history shows that 80% of the alumni contacted in this manner will make a contribution of at least $50. A random sample of 20 alumni is selected. Let x represent the number of alumni that make a contribution of at least $50.
a. What is the probability that exactly 15 alumni will make a contribution of at least $50?
b. What is the probability that between 14 and 18 alumni, inclusively, will make a contribution of at least $50?
c. What is the probability that fewer than 17 alumni will make a contribution of at least $50?
d. What is the probability that more than 15 alumni will make a contribution of at least $50?
e. How many alumni would you expect to make a contribution of at least $50?

Accepted Answer

The answer of A statistics department is contacting alumni by...

Question 5

The Scholastic Aptitude Test (SAT) is a standardized test for college admissions in the United States. In 1996, the average combined SAT score (math + verbal) for students in the United States was 1013, and 41% of all high school graduates took this test. Suppose that 500 students are randomly selected throughout the United States. Which of the following random variables has an approximate binomial distribution? In each case, provide a justification for your answer.
a. the number of students who took the SAT
b. the scores of the 500 students on the SAT
c. the number of students who scored above average on the SAT
d. the amount of time it took each student to complete the SAT

Accepted Answer

The answer of The Scholastic Aptitude Test (SAT) is a...

Question 6

Consider an experiment with 25 trials where the probability of success on any trial is 0.01, and let the random variable x be the number of successes among the 25 trials. What are p(0), p(1), p(2), and p(3) using the exact binomial distribution?

Accepted Answer

The answer of Consider an experiment with 25 trials where...

Question 7

In an effort to persuade customers to reduce their electricity consumption during business hours, special discounts have been offered to low-consumption customers. One hydroelectric company estimates that 70% of the local residences have qualified for these discounts.
a. If we randomly select four residences, what is the probability exactly two of the four will have qualified for the discounts?
b. If, instead, a sample of 20 residences is selected, what is the probability 12 or more will have qualified for the discounts?

Accepted Answer

The answer of In an effort to persuade customers to...

Question 8

An applicant must score at least 80 points on a particular psychological test to be eligible to join CUSO (Canadian University Services Overseas). From several years' experience, it is known that 60% of the applicants meet this requirement. Let the random variable x be the number of applicants who meet this requirement out of a (randomly selected) group of 25 applicants.
a. Find the mean of x.
b. Find the standard deviation of x.
c. Using the Empirical Rule, within what limits would you expect most (say, 95%) of the measurements to be?

Accepted Answer

The answer of An applicant must score at least 80...

Question 9

Let x be a binomial random variable with n = 20 and p = 0.05.
a. Calculate  using the cumulative binomial probabilities table in Appendix I of your text. 
b. Use the Poisson approximation to calculate  
c. Compare the results of (a) and (b). Is the approximation accurate?

Accepted Answer

The answer of Let x be a binomial random variable...

Question 10

Hotels, like airlines, often overbook, counting on the fact that some people with reservations will cancel at the last minute. A certain hotel chain has found that 20% of the reservations will not be used.
a. If we randomly selected 15 reservations, what is the probability more than 8 but fewer than 12 reservations will be used?
b. If four reservations are made, what is the chance fewer than two will cancel?

Accepted Answer

The answer of Hotels, like airlines, often overbook, counting on...

Question 11

The local Ford dealership claims 40% of its customers are return buyers, meaning that the customers have previously purchased a Ford vehicle. Let the random variable x be the number of return buyers in a random sample of 10 recent customers.
a. Find P(1 < x < 5).
b. Find P(x > 6).
c. What would you conclude about the accuracy of the dealership's claim if you found nine repeat buyers in the sample? Explain your reasoning.

Accepted Answer

The answer of The local Ford dealership claims 40% of...

Question 12

A new surgical procedure is said to be successful 90% of the time. Suppose the operation is performed five times and the results are assumed to be independent of one another. Define x to be the number of successful operations.
a. Find the probability that all five operations are successful.
b. Find the probability that exactly four are successful.
c. Find the probability that fewer than two are successful.

Accepted Answer

The answer of A new surgical procedure is said to...

Question 13

To determine the effectiveness of a diet to reduce the amount of serum cholesterol, 20 people were randomly selected to try the diet. After they had been on the diet a sufficient amount of time, their blood was tested to determine the level of serum cholesterol. The clinic running the experiment will approve and endorse the diet only if at least 16 of the 20 people lower their cholesterol significantly.
a. What is the random variable of interest?
b. What is the probability the clinic will erroneously endorse the diet when, in fact, it is useless in treating high serum cholesterol? [Hint: If the diet is ineffective, a person's cholesterol level is equally likely to go up or down.]

Accepted Answer

The answer of To determine the effectiveness of a diet...

Question 14

A politician claims 55% of the voters will vote for her in an upcoming election. An independent watchdog organization queried a random sample of 500 likely voters, of whom only 249 said they would vote for the politician in question. Do the results of this sample support the politician's claim? [Hint: Binomial distributions with p near 50% are quite mound-shaped.]

Accepted Answer

The answer of A politician claims 55% of the voters...

Question 15

Four in ten Canadians who travel by car look for gas and food outlets that are close to or visible from the highway. Suppose a random sample of n = 20 Canadians who travel by car are asked how they determine where to stop for food and gas. Let x be the number in the sample who respond that they look for gas and food outlets that are close to or visible from the highway.
a. What are the mean and variance of x?
b. Calculate the interval  What values of the binomial random variable x fall into this interval? 
c. Find  How does this compare with the fraction in the interval  for any distribution? For mound-shaped distributions?

Accepted Answer

The answer of Four in ten Canadians who travel by...

Question 16

A Canadian Medical Association study showed that 20% of all Canadians suffer from high blood pressure. Suppose we randomly sample ten Canadians to determine the number in the sample who have high blood pressure.
a. What is the probability that, at most, two persons have high blood pressure?
b. If, instead, we sampled 25 Canadians, give a reason why the following statement is true: "The probability between one and nine individuals from a sample of 25 would have high blood pressure is at least 0.75." [Hint: For n and p of approximately the sizes given, the distribution would not be particularly symmetric.]

Accepted Answer

The answer of A Canadian Medical Association study showed that...

Question 17

About 80% of Victoria residents believe that Victoria is a nice or very nice place to live. Suppose that five randomly selected Victoria residents are interviewed.
a. Find the probability that all five residents think that Victoria is a nice or very nice place to live.
b. What is the probability that at least one resident does not think that Victoria is a nice or very nice place to live?
c. What is the probability that exactly one resident does not think that Victoria is a nice or very nice place to live?

Accepted Answer

The answer of About 80% of Victoria residents believe that...

Question 18

Telephone Switchboard Narrative
The number of telephone calls coming into a business's switchboard averages four calls per minute. Let x be the number of calls received.
-Refer to Telephone Switchboard Narrative. What is the probability at least one call will be received in a given two-minute period?

Accepted Answer

The answer of Telephone Switchboard Narrative
The number of telephone calls...

Question 19

A psychiatrist believes that 90% of all people who visit doctors have s of a psychosomatic nature. She decides to select 25 patients at random to test her theory.
a. Assuming that the psychiatrist's theory is true, what is the expected value of x, the number of the 20 patients who have psychosomatic s?
b. What is the variance of x, assuming that the theory is true?
c. Find  , assuming that the theory is true. 
d. Based on the probability in (c), if only 14 of the 20 sampled had psychosomatic s, what conclusions would you make about the psychiatrist's theory? Explain.

Accepted Answer

The answer of A psychiatrist believes that 90% of all...

Question 20

It is known that 70% of the customers in a sporting goods store purchase a pair of running shoes. A random sample of 25 customers is selected. Assume that customers' purchases are made independently, and let x represent the number of customers who purchase running shoes.
a. What is the probability that exactly 18 customers purchase running shoes?
b. What is the probability that no more than 19 customers purchase running shoes?
c. What is the probability that at least 17 customers purchase running shoes?
d. What is the probability that between 17 and 21 customers, inclusively, purchase running shoes?

Accepted Answer

The answer of It is known that 70% of the...

Quiz 5: Several Useful Discrete Distributions

Let x be a Poisson random variable with mean 3. What is the probability that x will fall in the interval $\mu$ ± 2 $\sigma$ ?

Telephone Switchboard Narrative The number of telephone calls coming into a business's switchboard averages four calls per minute. Let x be the number of calls received. -Refer to Telephone Switchboard Narrative. Find P(x = 0).

Telephone Switchboard Narrative The number of telephone calls coming into a business's switchboard averages four calls per minute. Let x be the number of calls received. -Refer to Telephone Switchboard Narrative. What is the probability there will be at least one call in a given one-minute period?

Consider an experiment with 25 trials where the probability of success on any trial is 0.01, and let the random variable x be the number of successes among the 25 trials. What are p(0), p(1), p(2), and p(3) using the exact binomial distribution?

Let x be a binomial random variable with n = 20 and p = 0.05. a. Calculate using the cumulative binomial probabilities table in Appendix I of your text. b. Use the Poisson approximation to calculate c. Compare the results of (a) and (b). Is the approximation accurate?

Quiz 5: Several Useful Discrete Distributions

Let x be a Poisson random variable with mean 3. What is the probability that x will fall in the interval μ\muμ ± 2 σ\sigmaσ ?

Telephone Switchboard Narrative The number of telephone calls coming into a business's switchboard averages four calls per minute. Let x be the number of calls received. -Refer to Telephone Switchboard Narrative. Find P(x = 0).

Telephone Switchboard Narrative The number of telephone calls coming into a business's switchboard averages four calls per minute. Let x be the number of calls received. -Refer to Telephone Switchboard Narrative. What is the probability there will be at least one call in a given one-minute period?

Consider an experiment with 25 trials where the probability of success on any trial is 0.01, and let the random variable x be the number of successes among the 25 trials. What are p(0), p(1), p(2), and p(3) using the exact binomial distribution?

Let x be a binomial random variable with n = 20 and p = 0.05. a. Calculate using the cumulative binomial probabilities table in Appendix I of your text. b. Use the Poisson approximation to calculate c. Compare the results of (a) and (b). Is the approximation accurate?

Let x be a Poisson random variable with mean 3. What is the probability that x will fall in the interval $\mu$ ± 2 $\sigma$ ?