Posted: July 25th, 2024
MAB_E4
In the past few units, you have learned about normal distributions, Poisson distributions, and now exponential distributions. For this assignment, you will compose an essay in which you contrast the major differences between the normal distribution from Unit III and the exponential and Poisson distributions. You should demonstrate what you have learned about exponential distribution by explaining what it is and describing the situations in which it is useful for analyzing data. In addition, ensure that your essay meets the following criteria.
· Provide a real-world example (not already used in the textbook or unit lessons) in a diagram of one of the three distributions, and explain it.
· Explain what the data tells you.
· Given your selection, discuss how the data and results can be implemented in business or your life.
Your essay must be at least three pages in length, and you must integrate at least two academic resources as references. You should also follow the traditional essay format, and ensure that you have good introduction, body, and conclusion sections. Adhere to APA Style when constructing this assignment, including in-text citations and references for all sources that are used. Please note that no abstract is needed.
LDR 5301, Methods of Analysis for Business Operations 1
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
3.1
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
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
https://search-proquest-com.libraryresources.columbiasouthern.edu/scholarly-journals/understanding-hypothesis-testing-using/docview/205481967/se-2?accountid=33337
LDR 5301, Methods of Analysis for Business Operations 2
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)
LDR 5301, Methods of Analysis for Business Operations 3
UNIT x STUDY GUIDE
Title
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.)
LDR 5301, Methods of Analysis for Business Operations 4
UNIT x STUDY GUIDE
Title
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
Exponential distribution as demonstrated by COVID-19
(Givingbacktosociety, 2020)
LDR 5301, Methods of Analysis for Business Operations 5
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
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.
Learning Activity
Required Unit Resources
Unit Lesson
Introduction
Exponential Distribution
Example and Reflection on Data
Examples
Conclusion
References
Learning Activities (Nongraded)
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