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LDR 5301, Methods of Analysis for Business Operations 1

Course Learning Outcomes for Unit IV Upon completion of this unit, students should be able to:

3. Contrast the major differences between the normal distribution and the exponential and Poisson distributions. 3.1 Explain exponential distribution. 3.2 Explain when exponential distribution is useful for analyzing data. 3.3 Interpret the results of one type of distribution.

Course/Unit

Learning Outcomes Learning Activity

3.1

Unit Lesson Chapter 2, pp. 43–46 Article: “Understanding Hypothesis Testing Using Probability Distributions” Unit IV Essay

3.2

Unit Lesson Chapter 2, pp. 43–46 Article: “Understanding Hypothesis Testing Using Probability Distributions” Unit IV Essay

3.3

Unit Lesson Chapter 2, pp. 43–46 Article: “Understanding Hypothesis Testing Using Probability Distributions” Unit IV Essay

Required Unit Resources Chapter 2: Probability Concepts and Applications, pp. 43–46 In order to access the following resource, click the link below. LeBlond, D. (2009, Winter). Understanding hypothesis testing using probability distributions. Journal of

Validation Technology, 15(1), 45–61. https://search-proquest- com.libraryresources.columbiasouthern.edu/scholarly-journals/understanding-hypothesis-testing- using/docview/205481967/se-2?accountid=33337

Unit Lesson

Introduction In this unit, we will explore exponential distribution. We will look at how the distribution is used, what the components of the formula are, and how to evaluate distribution results: What does this data tell me? How am I able to use this data to solve problems or provide better service to my customers?

UNIT IV STUDY GUIDE Probability Distributions: Part 2

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UNIT x STUDY GUIDE Title

Exponential Distribution The exponential distribution is known as the negative distribution (Render et al., 2018, p. 46). The formula for this is as follows:

Where: X = random variable (service times) µ = average number of units the service facility can handle e = 2.718 (the base of the natural logarithms) As Render et al. (2018) noted, this type of distribution is most likely used when time is used to measure the reliability of a product or service to a customer by computing the probability within the event. Think about the customer first. There is the initial engagement, the problem or service issue is addressed, time is spent on helping the customer or performing the task, then the task or service is completed. The authors provide an excellent example in the textbook with Arnold’s Mufflers. Make sure to review the example that begins on page 47 of the textbook.

The exponential distribution graph provides an image of what an exponential distribution looks like. Note how it has a downward slope from the top of f(X), left, to the lower right (X). Step through the Arnold’s Muffler example. Note how the example provides all the data for the user to implement into the formula.

Example and Reflection on Data What is the important takeaway here, given we accomplished the number crunching with Arnold’s Mufflers? What do the numbers tell Arnold and us? How can we use this in our business to make better decisions or improve service for our customers? Here is the answer: We can see, in Arnold’s situation, the data displays that there is a probability that 78% of the time Arnold’s mechanic can install a new muffler in 30 minutes or less time (Render et al., 2018). Now the other side of the

equation or probability is that 22% of the time, it will take the mechanic longer to install the muffler (Render et al., 2018); this could be for many reasons—interruptions, rusted bolts, or service equipment that does not work. So, looking at this as Arnold (the owner), he can now begin to build a quality schedule for this type of maintenance. A smart thing for Arnold to do as well, given these probabilities and distributions, is to build in some slack time with scheduling. For example, he can schedule appointments at 9:00, 9:45, 10:30, and so on, giving his mechanics a 15-minute cushion. In the Arnold example, the key points to look at are service time (X), average number that can be served per time period (µ), and the constraint of time to finish (t).

(Render et al., 2018, p. 46)

Exponential distribution (Render et al., 2018)

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Now, look at how this problem was worked in the textbook using the exponential distribution formula given earlier. As the variables are inserted into the formula, we can see that the computations indicate the area under the curve. Note from the graph on page 47 that the times go from zero time to complete to 2 ½ hours to complete. Another factor here to consider the size of Arnold’s Muffler Shop. Does it have one bay or as many as five? Here is where multiple areas under the curve can be projected for the given task. For example, it might take 30 minutes to install a standard muffler on a standard American-made car. However, what if it is a tractor-trailer truck? What if it is a Ferrari? The second car may be more complex. Consider the necessary tools, the expertise, and the variable parts needed. Examples Here are some real-world examples that reflect an exponential distribution: The first example is an everyday household item: a battery, specifically the decrease of battery power when used in devices. Think about this problem. How long should AA or AAA batteries last? If we consider smoke alarms, the rule of thumb is to change them out every year when the time changes to daylight savings time. Why? It is a matter of safety. Below is a chart that displays the probability of failure of a battery over time (days of use).

DAYS IN USE PROBABILITY OF FAILURE 0 0 2 0.0198 10 0.095 32 0.27385 99 0.62842

Note: It is obvious the battery has a higher probability of failure the longer it is in use. This also means that the power generated within the battery is reduced. A second example is the sound of loud music from a party or concert as attendees leave the area. In this situation, the decibel level decreases as distance increases, so this follows the same curve as shown earlier in this lesson. Here is a final example of the exponential distribution with a different slope. This is an example of the spread of COVID-19, the disease caused by the novel coronavirus. The slope rises from left to right as the number of incidents increases, not decreases.

(Becht, 2014.)

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Conclusion

In this unit, we looked at the exponential distribution curve along with its formula and the impact certain events and data have on the shape of the curve. Some good examples were provided to you to make the concepts of the exponential distribution clearer. We looked at the data results of a battery being used on a device. We all know that batteries do not last forever (unless they are rechargeable, to some extent). A normal household battery will decline with power output over time. The decrease is less power and hence a distribution curve that is negative, moving downward from left to right. Why is this important? Think about your home’s smoke alarm. Would you feel safe in your home knowing that your smoke detector/carbon monoxide detector’s batteries are nine months old? Will it still work? Another example regarding running a business was that of Arnold’s Mufflers. Arnold can determine how to schedule repairs based on time to accomplish and schedule his empty bays accordingly. Therefore, there are business applications as well as public health applications. During the worldwide pandemic of COVID-19 (the novel coronavirus), health experts wanted to see a negative sloped distribution curve (left to right) meaning less infections, deaths, and hospitalizations through the use of social distancing, hand washing, and other measures. However, before these policies were in place and practiced, COVID-19 had an exponential positive powerful upside from bottom left to top right. The big takeaway here is you are now equipped to look at data from a different perspective. You can graph the data and get a better representation of what the numbers indicate; so, if you are tasked in your place of work with a data scheduling problem or data analysis problem, you can now investigate if the exponential distribution can help you solve it.

References Becht, K. (2014, April 29). Exponential distributions. CK-12 Foundation.

https://www.ck12.org/probability/exponential-distributions/rwa/exponential-distribution/ Givingbacktosociety. (2020, April 5). Coronavirus - daily and cumulative count [Graph].

https://commons.wikimedia.org/wiki/File:Coronavirus_-_daily_and_cumulative_count-1.png

Exponential distribution as demonstrated by COVID-19 (Givingbacktosociety, 2020)

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UNIT x STUDY GUIDE Title

Render, B., Stair, R. M., Jr., Hanna, M. E., & Hale, T. S. (2018). Quantitative analysis for management (13th ed.). Pearson. https://online.vitalsource.com/#/books/9780134518558

Learning Activities (Nongraded) Nongraded Learning Activities are provided to aid students in their course of study. You do not have to submit them. If you have questions, contact your instructor for further guidance and information. For an overview of the chapter equations, review the Key Equations on page 51 of the textbook. Then, complete problems 2–26 and 2–28 on page 57, and the Self-Test problems as a review, if needed. You can use the key in the back of the book in Appendix G to check your answers for the problems and Appendix H to check your answers for self-tests.