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A large university has 500 single employees who are covered by its dental plan. Suppose thenumber of claims filed during the next year by such an employee is a Poisson rv with mean value$2.3 .$ Assuming that the number of claims filed by any such employee is independent of the number filed by any other employee, what is the approximate probability that the total numberof claims filed is at least 1200$?$

Intro Stats / AP Statistics

Chapter 4

Joint Probability Distributions and Their Applications

Section 11

Supplementary Exercises

Probability Topics

The Normal Distribution

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Okay. So Were given that 500 single employees are covered by this dental plan and the average amount of claims per person is independent and is equal to 2.3 which I have written down over here. And it's also poison Lee distributed. So for poison distribution, the standard deviation is equal to the square root of our main value. So this is equal to the square root of 2.3, which if we plug into a calculator is Around 1.5 two. And now that we have this, what we're trying to find is the probability that We have at least 1200 claims in one one year, Given that we have 500 single employees and the average amount of claims is 2.3. So what I'm going to do first is I'm gonna make I'm going to find the average amount of claims For 500 employees given that we have this 2.3 per employee. And way that we do that is we just take our average amount Just 2.3. And we multiply it by the amount of employees which is 500. And if we do this we'll see that this is equal to 1150. And now a good thing that we can do is what we're going to use the central limit theorem. Since our poison distribution given an and large enough, where in this case or end is 500, which is pretty large. We can use a standard normal distribution too. Actually look at our probabilities since we have enough um random samples are and and value is big enough. So I'm going to do that is we're going to set up a Z score where Z is equal to X. The minus our U. X. Divided by Our standard deviation which is 1.52 Multiplied by the square root of our end value, which in this case is 500. And so what I'm gonna do is plug in this 1150 For you over here and then plug in 1200 for X. So we get 1200 minus 11 50 divided by 1.5 to times square to 500, which is around 33 98 And then what I'm gonna do is find this value. So if we simplify this, we get 50 divided by 33.98. And if we simplify this further, we get This is equal to 1.47. And now we can just use the central limit theorem. So you can say that our P of X is greater than equal to 1200, It's equal to P of C is greater than or equal to 1.47. And we can also use the fact that anything, the probability of Z being greater than something is equal to one minus the probability of Z being less than that. So We can say this is equal to 1- the probability of Z being less than 1.47. And now we can just go look at our standard normal table, Which if we do, we'll see that the value for 1.47 is around .9-9 to so This is equal to 1 -19 to 9 to Which is equal to around .7, sorry, point oh seven. Which is then our probability that we have at least 1200 Claims in one year.

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