Lee, Mindy S. Marantz, Andrew J. Maxwell, Kelly K. McCann, Dwight L. Pace, Dahra D. Perkins, Laurie Radovsky, Mary S. Raleigh, Sonia A. Rapaport, Emma J. Reinhold, Mark L. Renneker, William A. Robinson, Aaron M. Roland, E. Scott Rosenbloom, Peter C. Rowe, Ilene S. Ruhoy, David S. Saperstein, David A. Schlosser, Jill R. Schofield, Janet E. Settle, Leonard B. By the way, the author of the two essays mentioned earlier was Thomas Bayes, who is a lot more famous for the second essay than the first essay, a failed attempt to use mathematics and logic to prove the existence of God.
In these situations, rules have been developed for counting the exact number of possible outcomes. The Section 4. In that later survey, respondents who answered that they had purchased a large TV were asked additional questions concerning whether the large TV was HDR-capable and whether the respondents had purchased a streaming media player in the past 12 months.
By analyzing the results of these surveys, you were able to uncover many pieces of valuable information that will help you plan a marketing strategy to enhance sales and better target those households likely to purchase multiple or more expensive products.
Thus the marketing strategy should target those households that have indicated an intent to purchase. SUMMARY This chapter develops the basic concepts of probability that serve as a foundation fo r other concepts that later chapters discuss.
Probability is a numeric value from 0 to I that represents the chance, likelihood, or possibility that a particular event will occur. In addition to simple probability, the chapter discusses conditional probabilities and independent events. The chapter contingency tables and decision trees summarize and present probability information. The chapter also introduces Bayes' theorem. Bellhouse, D. Hooper, W. Lowd, D. Paulos, J. New York: Hill and Wang, Silberman, S. Zell er, T. Suppose that the survey was based on respondents from each of the two generation groups.
Construct a contingency table. Give an example of a simple event and a joint event. What is the probability that a randomly selected respondent prefers hybrid investment advice?
What is the probability that a randomly selected respondent prefers hybrid investment advice and is a baby boomer? Are the events "generation group" and "prefers hybrid investment advice" independent? The report represents the results of an online survey conducted in with employees located across the United States. One area of focus was on employees' response to important metrics to consider when evaluating the effectiveness of employee engagement programs.
The findings are summarized in the following tables. Respondents, categorized by the type of website they builtbusiness or personal-were asked to indicate the primary purpose for building their website. If a website builder is selected at random, what is the probability that he or she a.
Suppose the randomly chosen employee does indicate that presenteeism is an important metric to consider when evaluating the effectiveness of employee engagement programs. What is the probability that the employee is a non-HR employee? Are "presenteeism is an important metric" and "employee" independent?
Is "absenteeism is an important metric" independent of "employee"? What is the probability that a randomly chosen employee a. Part of the survey is devoted to the topic of marketing analytics and understanding what factors prevent companies from using more marketing analytics. The following findings are based on responses from senior marketers within B2B firms and senior marketers within B2C firms. Comment on the results in a through f.
Follow up the " Using Statistics: 'The Choice Is Yours,' Revisited" on page 79 by constructing contingency tables of market cap and type, market cap and risk, market cap and rating, type and risk, type and rating, and risk and rating for the sample of retirement funds stored in Open EndRunGuide. Read the claims and examine the supporting data. In what ways is the claim mi sleading? Using the table found under the "Show Me the Winning Probabilities" subhead, calculate the proper probabilities for the group of investors.
Are there any probability calculations that would be appropriate for rating an investment service? Why or why not? CardioGood Fitness 1. For each CardioGood Fitness treadmill product line see CardioGood Fitness , construct two-way contingency tables of gender, education in years, relationship status, and selfrated fitness.
There will be a total of six tables for each treadmill product. For each table you construct, compute all conditional and marginal probabilities. Write a report detailing your findings to be presented to the management of CardioGood Fitness. Retirement Funds 2. Write a report summarizing your conclusions. For each of these contingency tables, compute all the conditional and marginal probabilities. Example Compute simple and joint probabilities for purchase behavior data in Table 4.
Key Technique Use Excel arithmetic formulas. The worksheet shown below already contains the probabilities for the online section example. For other problems, change those probabilities in the cell range BS:C6. In the new template, similar to the worksheet shown below, fill in the Sample Space area with the data.
Example Apply Bayes' theorem to the television marketing example that the Bayesian Analysis online topic discusses. The worksheet shown below already contains the Table 4. As you change the event names in cells, B5, B6, CS, and C6, the column A row labels for simple and joint probabilities and the addition rule change as well.
Excel can automate Poisson probability calculations, which can be tedious. Figure 5. DIST function to compute the column B probabilities. FAlSE 0. OIST A What is the probability that in a given minute, there are no new visitors to the website? That there is at least one new visitor to the website? The answer to the questions can be found in the boldface cells in the "Poisson Probabilities Table. If the production process is in control, the mean number of chocolate chips per cookie is 6.
What is the probability that in any particular cookie being inspected a. What is the probability that a. The mean number of customer service complaints in January is 3. Source: Data extracted from U. What is the probability that in any given day a. How many cookies in a batch of should the manager expect to discard if company policy requires that all chocolate chip cookies sold have at least four chocolate chips?
Department of Transportation maintains statistics for mishandled bags. In January , Delta mishandled 0. What is the probability that in the next month, Delta will have a. In January , American Airlines mishandled 0. What is the probability that in the next month, American Airlines will have a. An analysis of complaints registered recently indicates that the mean number of vehicle lease complaints registered by consumers is 3.
Sorce: Data extracted from bit. What is the probability that in a given day a. Give an operational definition for problem. Why is the operational definition important in interpreting the initial quality score? Power and Associates calculates and publishes various statistics concerning car quality.
The dependability score measures problems experienced during the past 12 months by owners of vehicles For these models of cars, Ford had 1. Past history indicates that a mean of 0. What properties must be true about the situation described here in order to use the Poisson distribution to calculate probabilities concerning the number of phone calls received in a one-minute period?
Source: Data extracted from M. I B Let X be equal to the number of problems with a Ford. What assumptions must be made in order for X to be distributed as a Poisson random variable? Are these assumptions reasonable? Making the assumptions as in a , if you purchased a Ford in the model year, what is the probability that in the past 12 months, the car had b.
If you purchased a Toyota in the model year, what is the probability that in the past 12 months the car had a. Compare your answers in a and b to those for the Ford in Problem 5. Assuming that this situation matches the properties discussed in a , what is the probability that during a one-minute period a.
What is the maximum number of phone calls that will be received in a one-minute period The Section 5. The company's accounting information system automatically reviews order forms from online customers for possible mistakes.
Any questionable invoices are tagged and included in a daily exceptions report. Knowing that the probability that an order will be tagged is 0.
There was a Other calculations determined that, on average, one would expect 0. Because the binomial distribution can be applied for any known probability and sample size, Ricknel staffers will be able to make inferences about the online ordering process and, more importantly, evaluate any changes or proposed changes to that process. This chapter discusses two important discrete probability distributions: the binomial and Poisson distributions.
McKean, and A. Introduction to Mathematical Statistics, 7th ed. New York: Pearson Education, Levine, D. Ramsey, and R. McGinty, J. Chapter Review Problems Head said he would use the money to pay off his mortgage and provide for his children, and he had no plans to quit his job.
GM-Canada purchased event insurance from a company specializing in promotions at sporting events such as a half-court basketball shot or a hole-in-one giveaway at the local charity golf outing.
The event insurance company estimates the probability of a contestant winning the contest and, for a modest charge, insures the event. The promoters pay the insurance premium but take on no added risk as the insurance company will make the large payout in the unlikely event that a contestant wins. To see how it works, suppose that the insurance company estimates that the probability a contestant would win a million-dollar shootout is 0.
Calculate the expected value of the profit made by the insurance company. Many call this kind of situation a wi n- win opportunity for the insurance company and the promoter. Do you agree? Based on this information, and assuming a binomial distribution, what do you think is the probability that the stock market will rise a.
For this situation, what assumption of the binomial distribution might not be valid? What is the probability that the Dow Jones Industrial Average wi ll increase in 11 or more of the 14 U.
According to Medical Billing Advocates of America, three out of four times, the medical bills that they review contain errors. If a sample of 10 medical bills is selected, what is the probability that a. What are the mean and standard deviation of the probabi lity distribution? Compare the results of a through c to those of Problem 5. If a sample of I 0 Google searches is selected, what is the proba- If a sample of 10 businesses is selected, what is the probability that: a.
At least 8 use two or more social media channels for business? At most 6 use two or more social media channels for business? If you selected 10 businesses in a certain geographical area and only three use two or more social media channels for business, what conclusion might you reach about businesses in this geographical area?
Assuming bility that a. What assumptions did you have to make to answer a through c? Source: Consumer Response Annual Report, bit. Use the binomial model to answer the following questions: a.
What is the expected value, or mean, of the binomial distribution? What is the standard deviation of the binomial distribution? What is the probability that 10 of the 20 consumers cited "account management" as the type of complaint? What is the probability that no more than 5 of the consumers cited "account management" as the type of complaint?
What is the probability that 5 or more of the consumers cited "account management" as the type of complaint? What is the probability that none of the 20 consumers cited "deposit and withdrawal" as the type of complaint?
What is the probability that no more than 2 of the consumers cited "deposit and withdrawal" as the type of complaint? What is the probability that 3 or more of the consumers cited "deposit and withdrawal" as the type of complaint? One widely publicized stock market indicator in the United States that is an example of spurious correlation is the relationship between the winner of the National Football League Super Bowl and the performance of the Dow Jones Industrial Average in that year.
Of course, any correlation between these is spurious as one thing has absolute ly nothing to do with the other! Since the first Super Bowl was held in through , the indicator has been correct 38 out of 52 times.
What is the probability that the indicator would be correct 38 or more times in 50 years? What does this tell you about the usefulness of this indicator? Assume that the number of questionable insurance claims referred to investigators by Miami-Dade, Broward, and Palm Beach counties is distributed as a Poisson random variable with a mean of 7 per day. What assumptions need to be made so that the number of questionable insurance claims referred to investigators by MiamiDade, Broward, and Pal m Beach counties is distributed as a Poisson random variable?
Making the assumptions given in a , what is the probability that b. AMS marketing has been conducting an aggressive direct-marketing campaign that includes postal and electronic mailings and telephone solicitations. Feedback from these efforts indicates that including premium channels in this combined service is a very important factor for both current and prospective subscribers.
After several brainstorming sessions, the marketing department has decided to add premium cable channels as a no-cost benefit of subscribing to the 3-For-All service. The research director, Mona Fields, is planning to conduct a survey among prospective customers to determine how many premium channels need to be added to the 3-For-All service in order to generate a subscription to the service. Based on past campaigns and on industry-wide data, she estimates the following: Number of Free Premium Channels Probability of Subscriptions 0 0.
If a sample of 50 prospective customers is selected and no free premium channels are included in the 3-For-All service offer, given past results, what is the probability that a. Compare the results of a through c to those of Problem I. Suppose that in the actual survey of 50 prospective customers, 6 customers subscribe to the 3-For-All service offer.
What does this tell you about the previous estimate of the proportion of customers who would subscribe to the 3-For-All service offer? What do the results in e tell you about the effect of offering free premium channels on the likelihood of obtaining subscriptions to the 3-For-All service? Suppose that additional surveys of 50 prospective customers were conducted in which the number of free premium channels was varied. The results were as follows: Number of Free Premium Channels Number of Subscriptions 3 4 5 5 6 6 7 How many free premium channels should the research director recommend for inclusion in the 3-For-All service?
Digital Case Apply your knowledge about expected value in this continuing Digital Case from Chapters 3 and 4. Open BullsAndBears. Then answer the following: 1. Are there any "catches" about the claims the brochure makes for the rate ofreturn of Happy Bull and Worried Bear funds? Suppose that in the actual survey of 50 prospective customers, 4 customers subscribe to the 3-For-All service offer. What subjective data influence the rate-of-return analyses of these funds? Could EndRun be accused of making false and misleading statements?
Instead of offering no premium free channels as in Problem I , suppose that two free premium channels are included in the 3-For-All service offer. Given past results, what is the probability that 3.
The expected-return analysis seems to show that the Worried Bear fund has a greater expected return than the Happy Bull fund.
Should a rational investor never invest in the Happy Bull fund? Example Compute the expected value, variance, and standard deviation for the number of interruptions per day data of Table 5. PHStat Use Binomial. In the procedure's dialog box shown below : 1. Enter 4 as the Sample Size. Enter 0. Enter 0 as the Outcomes From value and enter 4 as the Outcomes To value. Enter a Title, check Histogram, and click OK. Workbook Use the Discrete Variable workbook as a model.
The worksheet contains the column A and B entries needed to compute the expected value, variance, and standard deviation for the example. These formulas use the expected value that cell B4 in the COMPUTE worksheet of the same workbook computes first three rows shown below and are equivalent to the fourth column calculations in Table 5.
Enter the probability distribution data into columns A and B and, if necessary, extend column C, by first selecting cell C7 and then copying that cell down as many rows as necessary. If the probabiljty distribution has fewer than six outcomes, select the rows that contain the extra, unwanted outcomes, right-click, and then click Delete in the shortcut menu. Workbook Use the Binomial workbook as a template and model. The worksheet already contains the entries needed for the example.
For other problems, change the sample size in cell B4 and the probability of an event of interest in cell B5. If necessary, extend the binomial probabilities table by first selecting cell rangeABl8 and then copying that cell range down as many rows as necessary. To construct a histogram of the probability distribution, use the Appendix Section B.
PHStat Use Poisson. In this procedure's dialog box shown below : 1. Check Histogram to construct a histogram of the Poisson probability distribution. Workbook Use the Poisson workbook as a template. The worksheet already contains the entries for the example.
For other problems, change the mean or expected number of events of interest in cell E4. To differentiate MyTVLab from the other companies that sell similar services, you decide to create a "Why Choose Us" web page to help educate new and prospective subscribers about all that MyTVLab offers.
As part of that page, you have produced a new video that samples the content MyTVLab streams as well as demonstrates the relative ease of setting up MyTVLab on many types of devices. You want this video to download with the page so that a visitor can jump to different segments immediately or view the video later, when offline.
You know from research Kishnan and Sitaraman and past observations, Internet visitors will not tolerate waiting too long for a web page to load. One wait time measure is load time, the time in seconds that passes from first pointing a browser to a web page until the web page is fully loaded and content such as video is ready to be viewed.
You have set a goal that the load time for the new sales page should rarely exceed 10 seconds too long for visitors to wait and, ideally, should rarely be less than 1 second a waste of company Internet resources. To measure this time, you point a web browser at the MyTVLab corporate test center to the new sales web page and record the load time.
In your first test, you record a time of 6. You repeat the test and record a time of 7. Though consistent to your goal, you realize that two load times do not constitute strong proof of anything, especially as your assistant has performed his own test and recorded a load time of 8. Could you use a method based on probability theory to ensure that most load times will be within the range you seek? Could you use these facts to assure yourself that the load time goal you have set for the new sales page is likely to be met?
In the MyTVLab scenario, you are examining the load time, a continuous numerical variable. You are no longer considering a table of discrete specific values, but a continuous range of values. For example, the phrase "load times are between 5 and 9 seconds" includes any value between 5 and 9 and not just the values 5, 6, 7, 8, and 9.
If you plotted the phrase on a graph, you would draw a continuous line from 5 to 9 and not just plot five discrete points. If that area under the curve follows the well-known pattern of certain continuous distributions, you can use the continuous probability distribution for that pattern to estimate the likelihood that a load time is within a range of values.
In the MyTVLab scenario, the past load times of a similar page describes a pattern that conforms to the pattern associated with the normal distribution, the subject of Section 6.
That would allow you, as the vice president for sales and marketing, to use the normal distribution with the statistics given to determine if your load time goal is likely to be met. Figure 6. The online Section 6. Distributions such as the right-skewed exponential distribution do not. In symmetrical distributions the mean equals the median, whereas in a right-skewed distribution the mean is greater than the median.
Each of the three distributions also has unique properties. The normal distribution is not only symmetrical, but bell-shaped, a shape that loosely suggests the profile of a bell. Being bell-shaped means that most values of the continuous variable will cluster around the mean. Although the values in a normal distribution can range from negative infinity to positive infinity, the shape of the normal distribution makes it very unlikely that extremely large or extremely small values will occur.
The uniform distribution, also known as the rectangular distribution, contains values that are equally distributed in the range between the smallest value and the largest value. In a uniform distribution, every value is equally likely. The exponential distribution contains values from zero to positive infinity and is rightskewed, making the mean greater than the median. Its shape makes it unlikely that extremely large values will occur. Besides visualizations such as those in Figure 6.
A probability density function for a specific continuous probability distribution , represented by the symbol JCX , defines the distribution of the values for a continuous variable and can be used as the basis for calculations that determine the likelihood or probability that a value will be within a certain range.
The most commonly used continuous probability distribution, the normal distribution, plays an important role in statistics and business. Because of its relationship to the Central Limit Theorem see Section 7. For business, many continuous variables used in decision making have distributions that closely resemble the normal distribution. The normal distribution can be used to estimate the probability that values occur within a specific range or interval.
This probability corresponds to an area under a curve that the normal distribution defines. Therefore, when using the normal distribution to estimate values of a continuous variable, the probability that the variable will be exactly a specified value is always zero. For the MyTVLab scenario, the load time for the new sales page would be an example of a continuous variable whose distribution approximates the normal distribution.
This approximation enables one to estimate probabilities such as the probability that the load time would be between 7 and 10 seconds, the probability that the load time would be between 8 and 9 seconds, or the probability that the load time would be between 7. Exhibit 6. The distributions of many business decision-making continuous variables share the first three properties, sufficient to allow the use of the normal distribution to estimate the probability for specific ranges or intervals of values.
Its mean and median are equal. Values cluster around the mean. Interquartile range is roughly 1. The distribution has an infinite range - oo oo. Six standard deviations Table 6. Due to minor irregularities in the machinery and the water pressure, the fill amounts will vary slightly from the desired target amount, which is a bit more than l. O liters to prevent underfilling of bottles and the subsequent consumer unhappiness that such underfilling would cause.
TABLE 6. Alternative Spirituality and Religion Review 1 2 : Oliver, Simon and Warrier, Maya eds London and New York: T. Journal of Contemporary Religion 25 1 : ISSN: electronic paper 6.
Evans, Mark ISBN hbk and pbk. Fieldwork in Religion 42 2 : Studying Local Churches: A Handbook. London: SCM Press. Fieldwork in Religion, 4. Review: Cowan, Douglas E.
Bromley Malden, MA: Blackwell. Fieldwork in Religion 3 1. Bharat, Sandy Christ across the Ganges: Hindu Responses to Jesus. Winchester, U. Faith and Freedom 61 1 , Spring and Summer: Both: , London, Arrow Books. Fieldwork in Religion, 2 1 : Review: Daniell, David.
The Bible in English. Faith and Freedom 56, part 2, no. ISSN X 6. Review: Walliss, John. Aldershot: Ashgate. Review: Chatterjee, Aroup. Mother Teresa: The Final Verdict. Kolkata, India: Meteor Books. Review: Mitchell, Donald W. Buddhism: Introducing the Buddhist Experience. New York: Oxford University Press. Buddhist Studies Review, 20 2 : ISSN 6. Oxford: Oxford University Press.
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Faith and Freedom 55 1 , Spring and Summer: Review: Barrett, David V. The New Believers. Review: Hodgkinson, Liz World Faiths Encounter, 29, July: Review: King, Sallie B. Ingram Richmond: Curzon Press. Buddhist Studies Review, 17 2 : Review: Berners-Lee, Tim London: Orion Business Books. Faith and Freedom 53 2 , Autumn and Winter: London: Chapman. Faith and Freedom 53 1 : Review: Kiely, Robert ed.
Buddhist Studies Review 15 2, November Review: Seager, Richard Hughes World Faiths Encounter 19, March Review: Knitter, Paul London: Orbis. World Faiths Encounter 19, March: London: Lutterworth.
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