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Chapter 5 © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. BUSINESS ANALYTICS: DATA ANALYSIS AND DECISION MAKING Normal, Binomial, Poisson, and Exponential Distributions Introduction Several specific distributions commonly occur in a variety of business situations: Normal distribution—a continuous distribution characterized by a symmetric bell-shaped curve Binomial distribution—a discrete distribution that is relevant when we sample from a population with only two types of members or when we perform a series of independent, identical experiments with only two possible outcomes Poisson distribution—a discrete distribution that describes the number of events in any period of time Exponential distributions—a continuous distribution that describes the times between events © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The Normal Distribution The single most important distribution in statistics is the normal distribution. It is a continuous distribution and is the basis of the familiar symmetric bell-shaped curve. Any particular normal distribution is specified by its mean and standard deviation. By changing the mean, the normal curve shifts to the right or left. By changing the standard deviation, the curve becomes more or less spread out. There are really many normal distributions, not just a single one. The normal distribution is a two-parameter family, where the two parameters are the mean and standard deviation. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Continuous Distributions and Density Functions (slide 1 of 2) For continuous distributions, instead of a list of possible values, there is a continuum of possible values, such as all values between 0 and 100 or all values greater than 0. Instead of assigning probabilities to each individual value in the continuum, the total probability of 1 is spread over this continuum. The key to this spreading is called a density function, which acts like a histogram. The higher the value of the density function, the more likely this region of the continuum is. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Continuous Distributions and Density Functions (slide 2 of 2) A density function, usually denoted by f(x), specifies the probability distribution of a continuous random variable X. The higher f(x) is, the more likely x is. The total area between the graph of f(x) and the horizontal axis, which represents the total probability, is equal to 1. f(x) is nonnegative for all possible values of X. Probabilities are found from a density function as areas under the curve. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The Normal Density The normal distribution is a continuous distribution with possible values ranging over the entire number line—from “minus infinity” to “plus infinity.” Only a relatively small range has much chance of occurring. The normal density function is actually quite complex, in spite of its “nice” bell-shaped appearance. The formula for the normal density function, where μ and σ are the mean and standard deviation, is: © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Standardizing: Z-Values The standard normal distribution has mean 0 and standard deviation 1, so it is denoted by N(0,1). It is also referred to as the Z distribution. To standardize a variable, subtract its mean and then divide the difference by the standard deviation: A Z-value is the number of standard deviations to the right or left of the mean. If Z is positive, the original value is to the right of the mean. If Z is negative, the original value is the left of the mean. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5.1: Standardizing.xlsx Objective: To use Excel® to standardize annual returns of various mutual funds. Solution: Data set includes the annual returns of 30 mutual funds. Calculate the mean and standard deviation of each annual return and then use the standardizing formula to calculate the corresponding Z-value. OR calculate the Z-values directly, using Excel’s STANDARDIZE function. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Normal Tables and Z-Values A common use for Z-values and the standard normal distribution is in calculating probabilities and percentiles by the traditional method. This method is based on a table of the standard normal distribution found in many statistics textbooks. An example of such a table is given below. The body of the table contains probabilities. The left and top margins contain possible values. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Normal Calculations in Excel Two types of calculations are typically made with normal distributions: finding probabilities and finding percentiles. The functions used for normal probability calculations are NORMDIST and NORMSDIST. The main difference between these is that the one with the “S” (for standardized) applies only to N(0, 1) calculations, whereas NORMDIST applies to any normal distribution. Percentile calculations that take a probability and return a value are often called inverse calculations. The Excel functions for these are named NORMINV and NORMSINV. Again, the “S” in the second of these indicates that it applies to the standard normal distribution. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5.2: Normal Calculations.xlsx (slide 1 of 2) Objective: To calculate probabilities and percentiles for standard normal and general normal distributions in Excel. Solution: For “less than” probabilities, use NORMDIST or NORMSDIST directly. For “greater than” probabilities, subtract the NORMDIST or NORMSDIST function from 1. For “between” probabilities, subtract the two NORMDIST or NORMSDIST functions. For percentile calculations, use the NORMINV or NORMSINV function with the specified probability as the first argument. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5.2: Normal Calculations.xlsx (slide 2 of 2) © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Empirical Rules Revisited Three empirical rules apply to many data sets: About 68% of the data fall within one standard deviation of the mean. About 95% fall within two standard deviations of the mean. Almost all fall within three standard deviations of the mean. For these rules to hold with real data, the distribution of the data must be at least approximately symmetric and bell-shaped. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Weighted Sums of Normal Random Variables One very attractive property of the normal distribution is that if you create a weighted sum of normally distributed random variables, the weighted sum is also normally distributed. This is true even if the random variables are not independent. If X1 through Xn are n independent and normally distributed random variables with common mean μ and common standard deviation σ, then the sum X1 + … + Xn is normally distributed with mean nμ, variance nσ2, and standard deviation √nσ. If a1 through an are any constants, then the weighted sum a1X1 + … + anXn is normally distributed with mean a1μ1 + … + anμn and variance a21 σ21 + … + a2n σ2n. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5.3: Personnel Decisions.xlsx Objective: To determine test scores that can be used to accept or reject job applicants at ZTel. Solution: Scores of all applicants are approximately normally distributed with mean 525 and standard deviation 55. Calculate the percentage of applicants who are automatic accepts or rejects, given the current standards of 600 for automatic accept and 425 for automatic reject. Find new cutoff values that reject 10% and accept 15% of applicants. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5.4: Paper Machine Settings.xlsx Objective: To determine the machine settings that result in paper of acceptable quality at PaperStock Company. Solution: A given roll of paper must be rejected if its actual fiber content is less than 19.8 pounds or greater than 20.3 pounds. The variability in fiber content is 0.10 pound when the process is “good,” but increases to 0.15 pound when the machine goes “bad.” Calculate the probability that a given roll is rejected, for a setting of μ = 20, when the machine is “good” and when it is “bad.” © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5.5: Tax on Stock Returns.xlsx Objective: To determine the after-tax profit Howard Davis can be 90% certain of earning. Solution: Howard is in the 33% tax bracket, so his after-tax profit is 67% of his before-tax profit. He invests $10,000 in a certain stock, whose annual return is normally distributed with mean 5% and standard deviation 14%. Calculate the dollar amount such that Howard’s after-tax profit is 90% certain to be less than this amount; that is, calculate the 90th percentile of his after-tax profit. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5.6: Oven Demand Simulation.xlsx (slide 1 of 3) Objective: To construct and analyze a spreadsheet model for microwave oven demand over the next 12 years using Excel’s NORMINV function, and to show how models using the normal distribution can lead to nonsensical outcomes unless they are modified appropriately. Solution: Using historical data, the company assumes that demand in year 1 is normally distributed with mean 5000 and standard deviation 1500. It also assumes that demand in each subsequent year is normally distributed with mean equal to the actual demand from the previous year and standard deviation 1500. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5.6: Oven Demand Simulation.xlsx (slide 2 of 3) Using this model may lead to nonsensical results as shown below: © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5.6: Oven Demand Simulation.xlsx (slide 3 of 3) One way to modify the model is to let the standard deviation and mean move together. That is, if the mean is low, then the standard deviation will also be low. To be even safer, it is possible to truncate the demand distribution at some nonnegative value such as 250, as shown below. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The Binomial Distribution The binomial distribution is a discrete distribution that can occur in two situations: Consider a situation where there are n independent, identical trials, where the probability of a success on each trial is p and the probability of a failure is 1 – p. When sampling from a population with only two types of members (males and females, for example) When performing a sequence of identical experiments, each of which has only two possible outcomes Define X to be the random number of successes in the n trials. Then X has a binominal distribution with parameters n and p. In Excel, calculate binomial probabilities with the BINOMDIST function. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5.7: Binomial Calculations.xlsx Objective: To use Excel’s BINOMDIST and CRITBINOM functions for calculating binomial probabilities and percentiles in the context of flashlight batteries. Solution: Let X be the number of successes in 100 trials of flashlight batteries, where a success means that the battery is still functioning after eight hours. Find the probabilities of various events, using the BINOMDIST function, as shown in the spreadsheet below. Find the 95th percentile of the distribution of X, using the CRITBINOM function. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Mean and Standard Deviation of the Binomial Distribution It can be shown that the mean and standard deviation of a binomial distribution with parameters n and p are given by the following equations. The empirical rules discussed in Chapter 2 also apply, at least approximately, to the binomial distribution. There is about a 95% chance that the actual number of successes will be within two standard deviations of the mean. There is almost no chance that the number of successes will be more than three standard deviations from the mean. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The Binomial Distribution in the Context of Sampling If sampling is done without replacement, each member of the population can be sampled only once. That is, once a person is sampled, his or her name is struck from the list and cannot be sampled again. If sampling is done with replacement, then it is possible, although maybe not likely, to select a given member of the population more than once. Most real-world sampling is performed without replacement. The binomial model applies only to sampling with replacement. However, if no more than 10% of the population is sampled, the binomial model can be used safely even if sampling is performed without replacement. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The Normal Approximation to the Binomial If you graph the binomial probabilities, you will see an interesting phenomenon: the graph begins to look symmetric and bell-shaped when n is fairly large and p is not too close to 0 or 1. The normal distribution provides a very good approximation to the binomial under these conditions. One practical consequence of the normal approximation to the binomial is that the empirical rules apply very well to binomial distributions. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5.8: Beating the Market.xlsx Objective: To determine the probability of a mutual fund outperforming a standard market index at least 37 out of 52 weeks. Solution: The number of weeks where a given fund outperforms the market index is binomially distributed with n = 52 and p = 0.5. This probability is quite small (0.00159). Now let Y be the number of the 400 best mutual funds that beat the market at least 37 of 52 weeks. Y is also binomially distributed, with parameters n = 400 and p = 0.00159. The resulting probability is nearly 0.5. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5.9: Supermarket Spending.xlsx Objective: To use the normal and binomial distributions to calculate the typical number of customers who spend at least $100 per day and the probability that at least 30% of all 500 daily customers spend at least $100. Solution: Historical data indicate that the amount spent per customer is normally distributed with mean $85 and standard deviation $30. If 500 customers shop in a given day, calculate the mean and standard deviation of the number who spend at least $100. Then calculate the probability that at least 30% of the 500 customers spend at least $100. This is the probability that a binomially distributed random variable, with n = 500 and p = 0.309, is at least 150. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5.10: Airline Overbooking.xlsx (slide 1 of 2) Objective: To assess the benefits and drawbacks of airline overbooking. Solution: Assume that the no-show rate is 10%—that is, each ticketed passenger shows up with probability 0.90. For a flight with 200 seats, calculate the probability that more than 205 passengers show up; that more than 200 passengers show up; that at least 195 seats are filled; and that at least 190 seats are filled. Use the BINOMDIST function and a data table to determine the probabilities. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5.10: Airline Overbooking.xlsx (slide 2 of 2) To see how sensitive these probabilities are to the number of tickets issued, create a one-way data table, as shown at the bottom of the spreadsheet below. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5.11: Election Returns.xlsx Objective: To use a binomial model to determine whether early returns reflect the eventual winner of an election between two candidates. Solution: Suppose that a small percentage of the votes have been counted and the Republican is currently ahead 540 to 460. On what basis can the networks declare the Republican the winner, if there are millions of voters? Use a binomial model to see how unlikely the event “at least 540 out of 1000” is, assuming that the Democrat will be the eventual winner. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5.12: Basketball Simulation.xlsx Objective: To formulate a nonbinomial model of basketball shooting, and to use it to find the probability of a “450 shooter” making at least 13 out of 25 shots. Solution: Assume the shooter makes 45% of his shots in the long run. Use simulation to create a model that implies that the shooter gets better the more shots he makes and worse the more he misses. Consider his nth shot. If he has made his last k shots, assume the probability of making shot n is 0.45 + kd1. If he has missed his last k shots, assume the probability of making shot n is 0.45 − kd2. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The Poisson and Exponential Distributions In most statistical applications, the Poisson and exponential distributions play a much less important role than the normal and binomial distributions. However, in many applied management science models, the Poisson and exponential distributions are key distributions. For example, much of the study of probabilistic inventory models, queuing models, and reliability models relies heavily on these two distributions. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The Poisson Distribution (slide 1 of 3) The Poisson distribution is a discrete distribution. It usually applies to the number of events occurring within a specified period of time or space. Its possible values are all of the nonnegative integers: 0, 1, 2, and so on—there is no upper limit. Even though there is an infinite number of possible values, this causes no real problems because the probabilities of all sufficiently large values are essentially 0. The Poisson distribution is characterized by a single parameter, usually labeled λ (Greek lambda), which must be positive. It is both the mean and the variance of the Poisson distribution. It is often called a rate—arrivals per hour, for example. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The Poisson Distribution (slide 2 of 3) All Poisson distributions have the same basic shape as in the figure below. That is, they first increase and then decrease. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The Poisson Distribution (slide 3 of 3) Typical examples of the Poisson distribution: A bank manager is studying the arrival pattern to the bank. The events are customer arrivals, the number of arrivals in an hour is Poisson distributed, and λ represents the expected number of arrivals per hour. A retailer is interested in the number of customers who order a particular product in a week. Then the events are customer orders for the product, the number of customer orders in a week is Poisson distributed, and λ is the expected number of orders per week. In Excel, calculate Poisson probabilities with the POISSON function. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5.13: Poisson Demand Distribution.xlsx (slide 1 of 2) Objective: To model the probability distribution of monthly demand for plasma screen TVs with a particular Poisson distribution. Solution: Because the histogram of demands from previous months resembles a Poisson distribution, try modeling the monthly demand with a Poisson distribution. The historical average demand per month is about 17, so let the mean demand per month λ = 17. Now test the Poisson model by calculating the probabilities of various events. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5.13: Poisson Demand Distribution.xlsx (slide 2 of 2) © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The Exponential Distribution (slide 1 of 2) The most common probability distribution used to model the times between customer arrivals, often called interarrival times, is the exponential distribution. In general, the continuous random variable X has an exponential distribution with parameter λ (with λ > 0) if the density function of X has the form: The mean and standard deviation of this distribution are both equal to the reciprocal of the parameter λ. For any exponential distribution, the probability to the left of a given value x > 0 can be calculated with Excel’s EXPONDIST function. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The Exponential Distribution (slide 2 of 2) The exponential density function has the shape shown below. Because this density function decreases continuously from left to right, its most likely value is x = 0. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.