When the number of trials, n , is large, binomial probability tables may not be

Question

When the number of trials, n, is large, binomial probability tables may not be available. Furthermore, if a computer is not available, hand calculations will be tedious. As an alternative, the Poisson distribution can be used to approximate the binomial distribution when n is large and p is small. Here the mean of the Poisson distribution is taken to be = np. That is, when n is large and p is small, we can use the Poisson formula with = np to calculate binomial probabilities; we will obtain results close to those we would obtain by using the binomial formula. A common rule is to use this approximation when n / p 500.

     To illustrate this approximation, in the movie Coma, a young female intern at a Boston hospital was very upset when her friend, a young nurse, went into a coma during routine anesthesia at the hospital. Upon investigation, she found that 13 of the last 27,000 healthy patients at the hospital had gone into comas during routine anesthesias. When she confronted the hospital administrator with this fact and the fact that the national average was 4 out of 40,000 healthy patients going into comas during routine anesthesias, the administrator replied that 13 out of 27,000 was still quite small and thus not that unusual.

Note: It turned out that the hospital administrator was part of a conspiracy to sell body parts and was purposely putting healthy adults into comas during routine anesthesias. If the intern had taken a statistics course, she could have avoided a great deal of danger.)

(a) Use the Poisson distribution to approximate the probability that 13 or more of 27,000 healthy patients would slip into comas during routine anesthesias, if in fact the true average at the hospital was 4 in 40,000 . Hint: = np = 27,000 (4/40,000 ) = 2.7 . (Leave no cell blank. You must enter "0" for the answer to grade correctly. Do not round intermediate calculations. Round final answer to 5 decimal places.)

Probability                    

(b) Given the hospital's record and part a, what conclusion would you draw about the hospital's medical practices regarding anesthesia?

The hospital's rate of comas is (Click to select)higher than normalunusually high.

Explanation / Answer

a)As we know that the formula of Poission distibution is

P(X=x)=(e^(-u)) u^x/x! -----(A)

Now it has been asked to calculate the probability that 13 or more of 27,000 healthy patients would slip into comas during routine anesthesias

So we are concerned with x>=13

P(X>=13) is what we need to calculate

and the same can be represented as

P(X>=13)=1-P(X<13) = 1-[P(X=0)+P(X=1)+---P(X=12)] -----(B)

To calculate the same we need the help of R. Still you can always use equation B by putting the values of B as given in Equation A but it will require a lot of time and so the help of a software is necessary

So writting the following command in R will give you the following result

1-sum(dpois(0:12,2.7))

Ans= 5.399487e-06 = 0.000005399487 is the probability of  13 or more of 27,000 healthy patients would slip into comas during routine anesthesias

b) so from the above probability we can firmly say that 5 healthy patients out of a million patients at the hospitals would slip into comas during routine anesthesias which is a very small number while the hospital's quoted figure of 4/40,000 is quite higher and we can infer that the hospital administrator was part of a conspiracy to sell body parts and was purposely putting healthy adults into comas during routine anesthesias.

I hope this has helped your understanding. Please upvote the ans if it has really helped . Good Luck!!

Related Questions

Navigate

• Browse (All)
• Browse W
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.