Large Company Must Hire A New President

Large Company Must Hire A New President The Boa

Chapter 5 Exercise 4a Large Company Must Hire A New President The Boa

CHAPTER 5 EXERCISE 4 A large company must hire a new president. The Board of Directors prepares a list of five candidates, all of whom are equally qualified. Two of these candidates are members of a minority group. To avoid bias in the selection of the candidate, the company decides to select the president by lottery.

a. What is the probability one of the minority candidates is hired? (Round your answer to 1 decimal place.)

b. Which concept of probability did you use to make this estimate? CLASSICAL, EMPIRICAL, RANDOMNESS, UNIFORMITY, INFERENCE

CHAPTER 5 EXERCISE 14 The chair of the board of directors says, "There is a 50% chance this company will earn a profit, a 30% chance it will break even, and a 20% chance it will lose money next quarter."

a. Use an addition rule to find the probability the company will not lose money next quarter. (Round your answer to 2 decimal places.)

b. Use the complement rule to find the probability it will not lose money next quarter. (Round your answer to 2 decimal places.)

CHAPTER 5 EXERCISE 22 A National Park Service survey of visitors to the Rocky Mountain region revealed that 50% visit Yellowstone Park, 40% visit the Tetons, and 35% visit both.

a. What is the probability a vacationer will visit at least one of these attractions? (Round your answer to 2 decimal places.)

b. What is the probability .35 called? (Click to select) Exclusive, Joint, Complement

c. Are the events mutually exclusive? (Click to select) Sometimes, No, Yes

CHAPTER 5 EXERCISE 34 P( A 1) = .20, P( A 2) = .40, and P( A 3) = .40. P( B 1| A 1) = .25. P( B 1| A 2) = .05, and P( B 1| A 3) = .10.

Use Bayes' theorem to determine P ( A 3| B 1). (Round your answer to 4 decimal places.) P( A 3| B 1) _____________

CHAPTER 5 EXERCISE 40 Solve the following: a. b. 9P 3 c. 7C 2

CHAPTER 6 EXERCISE 4 Which of these variables are discrete and which are continuous random variables?

a. The number of new accounts established by a salesperson in a year. (Click to select) Continuous, Discrete

b. The time between customer arrivals to a bank ATM. (Click to select) Discrete, Continuous

c. The number of customers in Big Nick’s barber shop. (Click to select) Continuous, Discrete

d. The amount of fuel in your car’s gas tank. (Click to select) Continuous, Discrete

e. The number of minorities on a jury. (Click to select) Continuous, Discrete

f. The outside temperature today. (Click to select) Continuous, Discrete

CHAPTER 6 EXERCISE 14 The U.S. Postal Service reports 95% of first-class mail within the same city is delivered within 2 days of the time of mailing.

Six letters are randomly sent to different locations.

a. What is the probability that all six arrive within 2 days? (Round your answer to 4 decimal places.)

b. What is the probability that exactly five arrive within 2 days? (Round your answer to 4 decimal places.)

c. Find the mean number of letters that will arrive within 2 days. (Round your answer to 1 decimal place.)

d-1. Compute the variance of the number that will arrive within 2 days. (Round your answer to 3 decimal places.)

d-2. Compute the standard deviation of the number that will arrive within 2 days. (Round your answer to 4 decimal places.)

e. Compute the standard deviation of the number that will arrive within 2 days. (Round your answer to 4 decimal places.)

CHAPTER 6 EXERCISE 20 In a binomial distribution, n = 12 and π = .60.

a. Find the probability for x = 5? (Round your answer to 3 decimal places.)

b. Find the probability for x ≤ 5? (Round your answer to 3 decimal places.)

c. Find the probability for x ≥ 6? (Round your answer to 3 decimal places.)

CHAPTER 6 EXERCISE 26 A population consists of 15 items, 10 of which are acceptable. In a sample of four items, what is the probability that exactly three are acceptable? Assume the samples are drawn without replacement. (Round your answer to 4 decimal places.)

Probability

CHAPTER 7 EXERCISE 4 According to the Insurance Institute of America, a family of four spends between $400 and $3,800 per year on all types of insurance. Suppose the money spent is uniformly distributed between these amounts.

a. What is the mean amount spent on insurance? Mean $

b. What is the standard deviation of the amount spent? (Round your answer to 2 decimal places.)

Standard deviation $

c. If we select a family at random, what is the probability they spend less than $2,000 per year on insurance per year? (Round your answer to 4 decimal places.)

d. What is the probability a family spends more than $3,000 per year? (Round your answer to 4 decimal places.)

CHAPTER 7 EXERCISE 10 The mean of a normal probability distribution is 60; the standard deviation is 5. (Round your answers to 2 decimal places.)

a. About what percent of the observations lie between 55 and 65? Percentage of observations %

b. About what percent of the observations lie between 50 and 70? Percentage of observations %

c. About what percent of the observations lie between 45 and 75? Percentage of observations %

CHAPTER 7 EXERCISE 14 A normal population has a mean of 12.2 and a standard deviation of 2.5. a. Compute the z value associated with 14.3. (Round your answer to 2 decimal places.) Z

b. What proportion of the population is between 12.2 and 14.3? (Round your answer to 4 decimal places.)

c. What proportion of the population is less than 10.0? (Round your answer to 4 decimal places.)

CHAPTER 7 EXERCISE 18 A normal population has a mean of 80.0 and a standard deviation of 14.0. a. Compute the probability of a value between 75.0 and 90.0. (Round intermediate calculations to 2 decimal places. Round final answer to 4 decimal places.)

b. Compute the probability of a value of 75.0 or less. (Round intermediate calculations to 2 decimal places. Round final answer to 4 decimal places.)

c. Compute the probability of a value between 55.0 and 70.0. (Round intermediate calculations to 2 decimal places. Round final answer to 4 decimal places.)

CHAPTER 7 EXERCISE 28 For the most recent year available, the mean annual cost to attend a private university in the United States was $26,889. Assume the distribution of annual costs follows the normal probability distribution and the standard deviation is $4,500. Ninety-five percent of all students at private universities pay less than what amount? (Round z value to 2 decimal places and your final answer to the nearest whole number.)

Sample Paper For Above instruction

Introduction

Probability theory plays a crucial role in decision-making processes within organizations, especially when selecting candidates or predicting outcomes of uncertain events. This paper explores various applications of probability concepts through exercises that illustrate real-world scenarios involving business decisions, environmental studies, and statistical analysis.

Probability in Candidate Selection

The first scenario involves a large company selecting a new president from five equally qualified candidates, two of whom are minorities. Since the selection is by lottery to minimize bias, the probability that a minority candidate is chosen can be calculated using the classical approach to probability—dividing favorable outcomes by total outcomes (Ott & Longnecker, 2015). With 2 minority candidates out of 5, the probability is 0.4 or 40%. This approach assumes each candidate has an equal chance, emphasizing the principle of fairness in probability calculation (Ross, 2014).

Analyzing Business Probabilities

The probability of the company earning a profit or breaking even is given as 50% and 30%, respectively, with the remaining 20% representing a loss. Using the addition rule, the probability that the company will not lose money is obtained by summing the probabilities of earning a profit or breaking even, which sums to 0.8 or 80%. Alternatively, using the complement rule, the probability of not losing money is 1 minus the probability of losing money (0.2), confirming the earlier result (DeGroot & Schervish, 2012).

Probability and Set Theory in Tourism

A survey indicates overlaps in visitor attractions—Yellowstone Park and the Tetons—with probabilities of visiting each and both being 50%, 40%, and 35%, respectively. The probability that a visitor goes to at least one of these parks can be calculated using the inclusion-exclusion principle, resulting in 0.55 or 55%. The probability of 0.35 is identified as the joint probability of visiting both parks, illustrating the concept of joint events. The events are not mutually exclusive because they can occur simultaneously, which is typical in overlapping sets (Kreyszig, 2011).

Using Bayes’ Theorem

In more complex probability scenarios, Bayes' theorem helps update probabilities based on new evidence. Given conditional probabilities, Bayes’ theorem enables calculation of the probability that a specific event (acceptability of items) is the case given an observation, such as the presence of a certain feature in the data. Calculating P(A3|B1) involves applying Bayes' theorem with the provided probabilities, which underscores the importance of conditional probability in statistical inference (Feller, 1968).

Discrete and Continuous Variables in Business

Distinguishing between discrete and continuous variables is essential for modeling real-world phenomena. For example, the number of new accounts is a discrete variable because it takes countable values, while the amount of fuel in a gas tank is continuous, as it can assume any value within a range. Recognizing variables’ types informs appropriate statistical analysis methods, like probability distributions suited for discrete or continuous data (Montgomery & Runger, 2014).

Binomial and Normal Distributions

Binomial distribution models the likelihood of a specified number of successes in a series of independent trials, such as the probability of exactly five letters arriving within two days out of six sent, given each has a 95% chance of arrival on time. Calculations involve binomial probability formulas, and calculating mean and variance helps predict outcomes (Wackerly, Mendenhall, & Scheaffer, 2014).

Similarly, normal distributions describe variables like insurance costs or test scores, where calculations of z-scores, probabilities, and percentiles rely on standard normal tables or software. For example, estimating the maximum amount paid by 95% of students involves using the inverse normal distribution, illustrating the application of central limit theorem principles (Devore, 2015).

Applications in Decision Making

These probability concepts demonstrate their utility in practical decision-making, from hiring processes and financial planning to quality control and environmental conservation. Correctly applying addition, multiplication, complement, and Bayes' rule facilitates informed decisions based on quantitative data, ultimately supporting strategic planning and risk management within organizations.

Conclusion

Understanding and applying probability principles are vital for analyzing uncertain events across various fields. The exercises reviewed illustrate key concepts such as probability calculation, set operations, conditional probability, and distribution analysis. Proficiency in these areas enables organizations and individuals to make better decisions amid uncertainty, emphasizing the importance of statistical literacy in the modern world.

References

• DeGroot, M. H., & Schervish, M. J. (2012). Probability and Statistics (4th ed.). Pearson.
• Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
• Kreyszig, E. (2011). Advanced Engineering Mathematics. John Wiley & Sons.
• Montgomery, D. C., & Runger, G. C. (2014). Applied Statistics and Probability for Engineers. Wiley.
• Ott, R. L., & Longnecker, M. (2015). An Introduction to Statistical Methods and Data Analysis. Cengage Learning.
• Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Wiley.
• Ross, S. M. (2014). Introduction to Probability Models. Academic Press.
• Wackerly, D. D., Mendenhall, W., & Scheaffer, R. L. (2014). Mathematical Statistics with Applications. Cengage Learning.
• Additional scholarly sources relevant to probability applications and decision theory.