Medicare Overbilling Analysis

Probability that will over-charge or under-charge the customer 

According to the information given; the highest invoice value = 0.06 

The lowest invoice value = 0.01 

Correct billing = 0.93 

The outcome of each is independent of the other. It therefore implies that, in case all the probabilities co-exist, the result will be the product of the marginals. 

The sample number is 200 and we have to get 15 invoices that will be on the higher side. 

Taking the number of invoices that overcharge to be X, and in this case, it will be an independent variable such that the customer is over-charged or not over-charged. 

We use the Binomial Distribution function; 

N1p 

N= number of trials, (200 invoices) 

P= Probability of over-charged invoices, 0.06 

X = 15 

The probability of getting over-charged invoices is p (x≥15) = 1-p (x<15) 

= 1-p (x≤14) 

Computing the above values on Excel we get 

BINODIST (14, 200, 0.06, TRUE) = 1 - 0.77772 

= 0.22228 

0.22 is the probability of getting at least 15 invoices that overcharge in the given sample. 

Calculating the probability of getting under-charged customers, we still use the Binomial Distribution 

Let y be the number of under-charged customers 

N= 200 

P = probability that each invoice will be under-charged (0.01) 

Q=0.99 

The above information can be summarized as p(y=0) 

Computing in excel, BINODIST using this statistical function (0,200,0.01, TRUE) 

The probability is 0.13398 

Finding the value of k, that will give a probability of at least 0.99 

K will be represented by; 

P (x≤k) ≥0.99 

The binomial probability will be derived using excel statistical function; 

BINOMDIST (k, n, p, cum) 

Trial and error methodology will be utilized for various values of k until the required value of probability (0.99) is obtained. 

Using the excel statistical BINOMDIST, 

K – number to be derived from trial and error 

N= 200 

P= 0.06 

cum- (value of 1) 

After computation in excel, the value of k that represents 0.99 probability is is 20. 

(c) % of overbilled invoices (10% more) 

Let x be the number of customers that are charged 10% more 

It can be represented in the equation as; 

P(x>0.10) 

Also, P (≥0.10) = 1 – p (x≤0.10) 

Using the statistical formulae, NORMDIST (X, µ, б,1) 

X – value of probability 

µ, -Normal population mean 

б- The standard deviation 

1-Logical value 

µ = 0.15, б=0.04 for x =0.10, we find the difference in probability as shown below 

Range Probability Formula 

< 1.10 0.105649774 = NORMDIST (01, B1, B2, 1) 

At least 0.10 0.894350225 = 1.B5 

From the above computation, it is clear that 89% of customers are charged at least 10 % more than they should pay. 

% of all over-billed invoices (10% or more) 

In the random sample of 200 clients, the percentage probability of overcharge is 0.06 

Let y be the number of all customers who are over-charged 

N=200, p =0.06 

Y. B(n1p) 

Y.B(200,0.06) 

P. (Y≤y) ≥ 0.99 

Using excel statistical formulae CRITBINOM (n1 p1 dp) and the result is 17.8% 

The same result can be arrived as below; 

From the previous calculation, 89% is the percentage of customers who will be charged more and represents 20 clients 

The probability percentage that all invoices will be charged at least 10% = 89/100*20 

= 17.8% 

(e) The probability of at least 5 overcharges (at least 10%) 

From the previous summation, 17.8% of invoices re charged 10% and more 

P = 0.178, n = 200 

The probability that at least five customers are overcharged by at least 10% can be represented as follows; 

Let y be the probability 

Y. B(n1p) 

Y.B (200, 0.178) 

The probability equation is as follows 

P (y ≥5) = 1-p(y<5) 

= 1-p(y≤µ) 

Using BINODIST (K, n, p, cum) 

Range Probability Formulae 

<µ 1.47572 BINOMDIST (µ, B1, B2, 1) 

At least 5 1 1.B5 

The probability in a 200 sample is 1 . 

References 

Gil, A. N. M. (2020). Asymptotic inversion of the binomial and negative binomial cumulative distribution functions. arXiv preprint arXiv:2001.03953. 

Biscarri, W., Zhao, S. D., R. J. (2018). A simple and fast method for computing the Poisson binomial distribution function. Computational Statistics & Data Analysis, 122, 92-100. 

Shonkwiler, J. S. (2016). Variance of the truncated negative binomial distribution. Journal of econometrics, 195(2), 209-210. 

Chai, C. W. (2016). Analytic and Numerical Studies on Generalized Co-evolving Agent-based Models (Doctoral dissertation, The Chinese University of Hong Kong (Hong Kong)).