Task summary: For this Assignment, reflect on the case presented. Think about what strategies you might use to calculate associated probabilities for Sleaze Hospital, and then address the series of questions for the completion of the Assignment. The size of the essay should be 1200 words, APA-style. Several samples are attached to know what the professor is looking for. Only use sources from the University Library for which you will get a username and password after the order is picked up. Please use a Zenmate VPN to change your address to USA.
Full order description: ? MAIN DETAILS:. If you randomly sample 200 of Sleaze’s invoices, what is the probability that you will find at least 15 invoices that overcharge the customer? What is the probability you won’t find any that undercharge the customer?. Find an integer, k, such that the probability is at least 0.99 that you will find at least k invoices that overcharge the customer. (Hint: Use trial and error with the BINOMDIST function to find k.). Suppose that when Sleaze overcharges Medicare, the distribution of the amount overcharged (expressed as a percentage of the correct billing amount) is normally distributed with mean 15% and standard deviation 4%.. What percentage of overbilled invoices are at least 10% more than the legal billing amount?. What percentage of all invoices are at least 10% more than the legal billing amount?. If your auditing company samples 200 randomly chosen invoices, what is the probability that it will find at least five where Medicare was overcharged by at least 10%?
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Week 2 Assignment
Dominique Roy
DDHA 8800
Walden University
11 March 2023
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The probability of an event is the number of possible outcomes divided by the total
number of possible effects. Probability assumes that all outcomes are equally likely. People take
advantage of the opportunity to predict a certain event (Mangiero, Qayyum, & Cante, 2021).
They use information from a previous scenario to predict the likelihood of future situations. If a
random variable like x has a binomial distribution, a general observation says that n is fixed. All
observations are generally equally independent, and each observation has only two outcomes,
true or false. The probability p is the probability of success. An Excel function calculates the
probability in the binomial distribution (Forsyth, 2018). This excerpt contains a management
report that provides a statistical assessment of the overbilling Medicare scenario at Sleaze
Hospital.
If you randomly sample 200 of Sleaze’s invoices, what is the probability that you will find
at least 15 invoices that overcharge the customer?
A possible claim is that each invoice produces two independent results, one of which is
overestimated. Since (x?15 | N=200, p=0.06), we need the function BINOM.DISTRIBUTION (1
, 200, 0.06, TRUE) to get the overbilling probability.
200
200 !
?0.06 x ?0.94 200? x =0.222
15 ( 200?x i ) !?x i !
P ( x ?15, 200, 0.06 )=?
i
i
With an estimated probability of 22.2%, at least 15 out of 200 audited invoices will be
found to be overpaid. These audit results confirm the financial history of Sleaze Hospital.
What is the probability of not finding any undercharged customers?
The chance that Sleaze Hospital has undercharged a customer is 0.01. We will use the
following method to get a zero bill for our 200 selected customers. The probability statement for
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this scenario is P (X?0 | N=200, p=0.01), where X represents the variable being tested. , N
represents the sample size to be audited and p is the probability of an undercounted invoice. This
is solved using the Microsoft Excel formula BINOMIAL DISTRIBUTION (0, 200, 0.01,
TRUE).
P ( x =0,200, 0.01 )=
200 !
?0.010?0.99200 =0.99200 =0.134
200 !?0 !
Based on the estimated probability of 13.4%, it illustrates that the possibility of getting
invoices where a client was under-billed is minimal.
Minimum number of invoices out of the 200 invoices such that the probability of
overcharging the customer exceeds 99%
Finding the value of effective trials with k is a probability function of the binomial
distribution P (X?K | N=200, p=0.06). The value of K can be determined by trial and error in the
BIOMAL DISTRIBUTION (K, 200, 0.06, TRUE) function.
# of trials
200
p of success
0.06
range
probability
5
0.017720855
8
0.146991061
10
0.340709137
11
0.45804338
12
0.57600174
13
0.684886381
14
0.777719638
15
0.851196174
16
0.905424202
17
0.942888372
18
0.967200227
19
0.982065034
20
0.990651832
20 total number of invoices were overbilled.
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The overbilled invoices are at least 10% or more then the legal billing amount.
The percentage calculation of overcharged customers will be done by using the Excel
function of “NORMDIST. (x, ?, ?, 1).
Total sample
Mean value
Standard
deviation
probability
Less than 0.10
At least 0.10
Invoices at least
10% more than legal
billing
% of all
customers charged more
than 10%
200
15.00
0.04
0.08
0.109154288
0.890845712
20
17.82
The table above shows that at least 89% of overcharged customers are charged at least
10% of what they should be paying. It is important to note that this determines who is
responsible for managing increased complexity and constant change, which continues with a
greater risk of R&D becoming a highly functional environment that cannot be avoided or
understood.
The probability that one randomly selected bill is overcharged by at least 10% more than
the legal billing amount.
In the Sleaze hospital scenario, the normal distribution represents the value of the
hospital’s overcharged bills. An overdrawn account appears with a probability of 0.06. This
account has ten things to test and has an average of 15% overcharge and a standard deviation of
%. 17.9% is the percentage of the required amount received from invoices.
P=P ( overcharged )?P ( overcharged by at least 10 % )=
0.89?20
=17.8 %
100
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If the auditing company samples 200 randomly chosen invoices, what is the probability that
it will find at least five where Medicare was overcharged by at least 10%?
To find at least five Medicares with a fee of at least 10%, where n = number of routes p =
probability calculation with an overestimated amount. X= B (n, p) “X=B (200, 0.06)” The above
function shows that each selected invoice manually generates two independent results, either not
overpaid or binomially distributed (Lin). and Pantano, 2022). The probability of receiving 5
invoices is greater than a random sample of 200 customers. It is concluded that the probability
that Medicare will find at least 5 bills that exceed a random sample of 200 customers is 1.
200
P ( x ?5, 200, 0.0417 )=?
5
200 !
?0.0417 x ?0.9583200?x =0.9226
( 200?x i ) 7 !?x i !
i
i
References
Albright, S. C., & Winston, W. L. (2014). Business analytics: Data analysis & decision
making. Cengage Learning. Online eBook: ISBN: 978-1305947542
Forsyth, D. (2018). Probability and statistics for computer science (pp. 36-42). Springer
International Publishing. https://link.springer.com/content/pdf/10.1007/978-3319- 64410-3.pdf
Mangiero, G. A., Qayyum, A., & Cante, C. J. (2021). Teaching statistics – A dynamic Excel
approach. Journal of Education for Business, 1https://doi.org/10.1080/08832323.2021.1974815
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Running Head: MEDICARE OVERBILLING ANALYSIS
Medicare Overbilling Analysis
Name
Institutional Affiliation
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MEDICARE OVERBILLING ANALYSIS
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Medicare Overbilling Analysis
The probability of an event is the measure of the degree of assurance that an event will
occur. Probability is expressed as any number between 0 and 1, where 0 defines impossibility
and 1 defines certainty of an event (Fulton, Mendez, Bastian, & Musal, 2012). If a random
variable X has a binomial distribution, then the number of all observations, n, is fixed, all
observations are mutually independent, and there are only two outcomes (success and failure) for
each observation. The probability p is the probability of success (Forbes, Evans, Hastings, &
Peacock, 2011). The excel function to calculate probabilities in a binomial distribution is
¿
. ( x , trials , p , cumulative )
(Microsoft, 2019). The normal distribution is a continuous
DIST
distribution of random independent variables and follows a bell shape (Forbes, Evans, Hastings,
& Peacock, 2011). The excel function to calculate the probabilities of a normal distribution is
¿ NORM . DIST ( x , ? , ? , cumulative ) , where x is the value for which we want to find the
probability,
? is the population mean, and ?
is the population standard deviation
(Microsoft, 2019). The binomial distribution is used in the calculation of probabilities in question
1 while the normal distribution concept is used in question 2.
Case Study
Your company is running a Medicare audit on Sleaze Hospital. Because Sleaze has a
history of overbilling, the focus of your audit is on checking whether the billing amounts are
correct. Assume that each invoice is for too high an amount with probability 0.06 and for too low
an amount with probability 0.01 (so that the probability of a correct billing is 0.93). Also, assume
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MEDICARE OVERBILLING ANALYSIS
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that the outcome for any invoice is probabilistically independent of the outcomes for other
invoices.
Question 1
a. If you randomly sample 200 of Sleaze’s invoices, what is the probability that you will find at
least 15 invoices that overcharge the customer?
Solution
Let X be a random variable representing the number of overcharged customers. Also consider
that each invoice generates only two independent outcomes: overcharged or not overcharged. X
follows a binomial distribution with parameters:
n- total number of trials (n=200)
p-denotes the probability that an invoice is overcharged (p=0.06).
X B (200,0.06)
The probability that at least 15 invoices are overcharged is given by
p ( x ?15 )=1? p ( x