Please answer all of the following completely.
1. Simple random sampling uses a sample of size n from a population of size N to obtain data that can be used to make inferences about the characteristics of a population. Suppose that, from a population of 45 bank accounts, we want to take a random sample of six accounts in order to learn about the population. How many different random samples of four accounts are possible?
2. The Powerball lottery is played twice each week in 31 states, the District of Columbia, and the Virgin Islands. To play Powerball, a participant must purchase a $2 ticket, select five numbers from the digits 1 through 49, and then select a Powerball number from the digits 1 through 37. To determine the winning numbers for each game, lottery officials draw five white balls out a drum of 49 white balls numbered 1 through 49 and one red ball out of a drum of 37 red balls numbered 1 through 37. To win the Powerball jackpot, a participant’s numbers must match the numbers on the five white balls in any order and must also match the number on the red Powerball. The numbers 5–16–22–23–29 with a Powerball number of 6 provided the record jackpot of $580 million ( Powerball website, November 29, 2012).
a) How many Powerball lottery outcomes are possible? (Hint: Consider this a two-step experiment. Select the five white ball numbers and then select the one red Powerball number.)
b) What is the probability that a $2 lottery ticket wins the Powerball lottery?
3. Consider the experiment of rolling a pair of dice. Suppose that we are interested in the sum of the face values showing on the dice.
a)How many sample points are possible? (Hint: Use the counting rule for multiple-step experiments.
b) List the sample points.
c) What is the probability of obtaining a value of 7?
d) What is the probability of obtaining a value of 9 or greater?
e) Because each roll has six possible even values (2, 4, 6, 8, 10, and 12) and only five possible odd values (3, 5, 7, 9, and 11), the dice should show even values more often than odd values. Do you agree with this statement? Explain.
f) What method did you use to assign the probabilities requested?
4. Fortune magazine publishes an annual list of the 500 largest companies in the United States. The corporate headquarters for the 500 companies are located in 38 different states The following table shows the eight states with the largest number of Fortune 500 companies (Money/CNN website, May 12, 2012).
State | Number of Companies |
California | 53 |
Illinois | 32 |
New Jersey | 21 |
New York | 50 |
Ohio | 28 |
Pennsylvania | 23 |
Texas | 52 |
Virginia | 24 |
Suppose one of the 500 companies is selected at random for a follow-up questionnaire.
a) What is the probability that the company selected has its corporate headquarters in California?
b) What is the probability that the company selected has its corporate headquarters in California, New York, or Texas?
c) What is the probability that the company selected has its corporate headquarters in one of the eight states listed above?
5. Data on U.S. work-related fatalities by cause follow (The World Almanac, 2012).
Cause of Fatality | Number of Fatalities |
Transportation incidents | 1795 |
Assaults and violent acts | 837 |
Contact with objects and equipment | 741 |
Falls | 645 |
Exposure to harmful substances or environments | 404 |
Fires and explosions | 113 |
Assume that a fatality will be randomly chosen from this population.
a) What is the probability the fatality resulted from a fall?
b) What is the probability the fatality resulted from a transportation incident?
c) What cause of fatality is least likely to occur? What is the probability the fatality resulted from this cause?
6. Suppose that we have a sample space with five equally likely experimental outcomes: E1, E2, E3, E4, and E5. Let A = {E1, E2}; B = { E3, E4}; C = { E2, E4, E5}
b) Find P(A U B). Are A and B mutually exclusive?
c) Find Ac, Cc, P(Ac), and P(Cc).
d) Find A U Bc and P(A U Bc).
e) Find P(B U C).
7. High school seniors with strong academic records apply to the nation’s most selective colleges in greater numbers each year. Because the number of slots remains relatively stable, some colleges reject more early applicants. Suppose that for a recent admissions class, an Ivy League college received 2851 applications for early admission. Of this group, it admitted 1033 students early, rejected 854 outright, and deferred 964 to the regular admission pool for further consideration. In the past, this school has admitted 18% of the deferred early admission applicants during the regular admission process. Counting the students admitted early and the students admitted during the regular admission process, the total class size was 2375. Let E, R, and D represent the events that a student who applies for early admission is admitted early, rejected outright, or deferred to the regular admissions pool.
a) Use the data to estimate P(E), P(R), and P(D).
b) Are events E and D mutually exclusive? Find P(E∩D).
c) For the 2375 students who were admitted, what is the probability that a randomly selected student was accepted during early admission?
d) Suppose a student applies for early admission. What is the probability that the student will be admitted for early admission or be deferred and later admitted during the regular admission process?
8. Assume that we have two events, A and B, that are mutually exclusive. Assume further that we know P(A) = 0.30 and P(B) = 0.40.
a) What is P(A ∩ B)?
b) What is P(A | B)?
c) A student in statistics argues that the concepts of mutually exclusive events and independent events are really the same, and that if events are mutually exclusive they must be independent. Do you agree with this statement? Use the probability information in this problem to justify your answer.
d) What general conclusion would you make about mutually exclusive and independent events given the results of this problem?
9. A joint survey by Parade magazine and Yahoo! found that 59% of American workers say that if they could do it all over again, they would choose a different career (USA Today, September 24, 2012). The survey also found that 33% of American workers say they plan to retire early and 67% say they plan to wait and retire at age 65 or older. Assume that the following joint probability table applies.
Retire Early | ||||
Yes | No | |||
Career | Same | .20 | .21 | .41 |
Different | .13 | .46 | .59 | |
.33 | .67 |
a) What is the probability a worker would select the same career?
b) What is the probability a worker who would select the same career plans to retire early?
c) What is the probability a worker who would select a different career plans to retire early?
d) What do the conditional probabilities in parts (b) and (c) suggest about the reasons workers say they would select the same career?
10. The prior probabilities for events A1 and A2 are P(A1) = 0.70 and P(A2) = 0.30. It is also known that P(A1 ∩ A2) = 0. Suppose and P(B| A1) = 0.20 and P(B| A2) = 0.05.
a) Are A1 and A2 mutually exclusive? Explain.
b) Compute P(A1 ∩ B) and P(A2 ∩ B).
c) Compute P(B).
d) Apply Bayes’ theorem to compute P(A1 | B) and P(A2 | B).
11. A consulting firm submitted a bid for a large research project. The firm’s management initially felt they had a 50–50 chance of getting the project. However, the agency to which the bid was submitted subsequently requested additional information on the bid. Past experience indicates that for 75% of the successful bids and 40% of the unsuccessful bids the agency requested additional information.
a) What is the prior probability of the bid being successful (that is, prior to the request for additional information)?
b) What is the conditional probability of a request for additional information given that the bid will ultimately be successful?
c) Compute the posterior probability that the bid will be successful given a request for additional information.
12. Consider the experiment of a worker assembling a product.
a) Define a random variable that represents the time in minutes required to assemble the product.
b) What values may the random variable assume?
c) Is the random variable discrete or continuous?
13) To perform a certain type of blood analysis, lab technicians must perform two procedures. The first procedure requires either one or two separate steps, and the second procedure requires one, two, or three steps.
a) List the experimental outcomes associated with performing the blood analysis.
b) If the random variable of interest is the total number of steps required to do the complete analysis (both procedures), show what value the random variable will assume for each of the experimental outcomes.
14. The probability distribution for the random variable x follows.
x | f(x) |
20 | .20 |
25 | .15 |
30 | .25 |
35 | .40 |
a) Is this probability distribution valid? Explain.
b) What is the probability that x = 30?
c) What is the probability that x is less than or equal to 25?
d) What is the probability that x is greater than 30?
*15. Employee retention is a major concern for many companies. A survey of Americans asked how long they have worked for their current employer (Bureau of Labor Statistics website, December 2015). Consider the following example of sample data of 2000 college graduates who graduated five years ago.
Time with Current Employer (years) | Number |
1 | 506 |
2 | 390 |
3 | 310 |
4 | 218 |
5 | 576 |
Let x be the random variable indicating the number of years the respondent has worked for her/his current employer.
a) Use the data to develop an empirical discrete probability distribution for x.
b) Show that your probability distribution satisfies the conditions for a valid discrete probability distribution.
c) What is the probability that a respondent has been at her/his current place of employment for more than 3 years?
16. The following table provides a probability distribution for the random variable x.
x | f(x) |
3 | .25 |
6 | .50 |
9 | .25 |
a) Compute E(x), the expected value of x.
b) Compute σ2, the variance of x.
c) Compute σ, the standard deviation of x.
17. The American Housing Survey reported the following data on the number of times that owner-occupied and renter-occupied units had a water supply stoppage lasting 6 or more hours over a 3-month period.
Number of Units (1000s) | ||
Number of Times | Owner Occupied | Renter Occupied |
0 | 439 | 394 |
1 | 1100 | 760 |
2 | 249 | 221 |
3 | 98 | 92 |
4 times or more | 120 | 111 |
a) Define a random variable that owner-occupied units had a water supply stoppage lasting 6 or more hours in the past 3 months and develop a probability distribution for the random variable. (Let represent 4 or more times.)
b) Compute the expected value and variance for x.
c) Define a random variable y = the number of times that renter-occupied units had a water supply stoppage lasting 6 or more hours in the past 3 months and develop a probability distribution for the random variable. (Let y = 4 represent 4 or more times.)
d) Compute the expected value and variance for y.
e) What observations can you make from a comparison of the number of water supply stoppages reported by owner-occupied units versus renter-occupied units?
*18. Consider a binomial experiment with n = 20 and p = 0.65.
a) Compute f(12).
b) Compute f(16).
c) Compute P(x ≥ 16).
d) Compute P(x ≤ 15).
e) Compute E(x).
f) Compute Var(x) and .
*19. Market-share-analysis company Net Applications monitors and reports on Internet browser usage. According to Net Applications, in the summer of 2014, Google’s Chrome browser exceeded a 20% market share for the first time, with a 20.37% share of the browser market (Forbes website, December 15, 2014). For a randomly selected group of 20 Internet browser users, answer the following questions.
a) Compute the probability that exactly 8 of the 20 Internet browser users use Chrome as their Internet browser.
b) Compute the probability that at least 3 of the 20 Internet browser users use Chrome as their Internet browser.
c) For the sample of 20 Internet browser users, compute the expected number of Chrome users.
d) For the sample of 20 Internet browser users, compute the variance and standard deviation for the number of Chrome users.
*20. Consider a Poisson distribution with a mean of three occurrences per time period.
a) Write the appropriate Poisson probability function.
b) What is the expected number of occurrences in three time periods?
c) Write the appropriate Poisson probability function to determine the probability of x occurrences in three time periods.
d) Compute the probability of two occurrences in one time period.
e) Compute the probability of six occurrences in three time periods.
f) Compute the probability of five occurrences in two time periods.
*21. Airline passengers arrive randomly and independently at the passenger-screening facility at a major international airport. The mean arrival rate is 10 passengers per minute.
a) Compute the probability of no arrivals in a one-minute period.
b) Compute the probability that three or fewer passengers arrive in a one-minute period.
c) Compute the probability of no arrivals in a 15-second period.
d) Compute the probability of at least one arrival in a 15-second period.
*22. Suppose N = 12 and r = 4 . Compute the hypergeometric probabilities for the following values of n and x.
a) n = 4, x = 1.
b) n = 2, x = 2.
c) n = 2, x = 0.
d) n = 4, x = 2.
e) n = 4, x = 4.
*23. Suppose N = 15 and r = 4. What is the probability of x = 3 for n =10?
*24. The Zagat Restaurant Survey provides food, decor, and service ratings for some of the top restaurants across the United States. For 15 restaurants located in Boston, the average price of a dinner, including one drink and tip, was $48.60. You are leaving on a business trip to Boston and will eat dinner at three of these restaurants. Your company will reimburse you for a maximum of $50 per dinner. Business associates familiar with these restaurants have told you that the meal cost at one-third of these restaurants will exceed $50. Suppose that you randomly select three of these restaurants for dinner.
a) What is the probability that two of the meals will exceed the cost covered by your company?
b) What is the probability that all three of the meals will exceed the cost covered by your company?
c) What is the probability that none of the meals will exceed the cost covered by your company?
d) What is the probability that one of the meals will exceed the cost covered by your company?