Question

A. Let x be a binomial random variable with n=20 and p=0.05. Calculate P(x≤2) using the cumulative binomial probabilities table in Appendix I of your book. Round your answer to three decimal places. Use the Poisson approximation to calculate P(x≤2). Round your answer to four decimal places.

B. Compare your results. Is the approximation accurate?

C. The number x of people entering the intensive care unit at a particular hospital on any one day has a Poisson probability distribution with mean equal to four persons per day. (Round your answers to six decimal places if necessary.)

D. What is the probability that the number of people entering the intensive care unit on a particular day is two?

E. What is the probability that the number of people entering the intensive care unit on a particular day is less than or equal to two?

Answer 1

To calculate P(x≤2) using the binomial distribution, we can use the cumulative binomial probabilities table to find the cumulative probabilities for x=0 and x=1, and sum them to get the desired probability. Using the Poisson approximation, we can calculate P(x≤2) by finding the Poisson probabilities for x=0 and x=1 when the mean is equal to np. The approximation is accurate as the difference between the probabilities is small. For a Poisson distribution with a mean of four, we can find the probability of a specific number of people entering the intensive care unit on a particular day, or calculate the probability of the number being less than or equal to a certain number.

Step-by-step explanation:

To calculate P(x≤2) using the cumulative binomial probabilities table, we need to find the cumulative probability for x=0 and x=1 for n=20 and p=0.05. From the table, we find that the cumulative probability for x=0 is approximately 0.358 and the cumulative probability for x=1 is approximately 0.762. Summing these probabilities, we get P(x≤2) ≈ 0.358 + 0.762 ≈ 1.120.

Using the Poisson approximation, we calculate P(x≤2) by finding the Poisson probability for x=0 and x=1 when the mean λ is equal to np. In this case, λ = 20 * 0.05 = 1. Using the Poisson probability formula, we find that P(x=0) = 0.3679 and P(x=1) = 0.3679. Summing these probabilities, we get P(x≤2) = 0.3679 + 0.3679 = 0.7358.

Comparing the results, we can see that the Poisson approximation is accurate as the difference between the probabilities is only 0.3842.

For part C, since the number of people entering the intensive care unit on a particular day follows a Poisson distribution with a mean of four persons per day, we can use the Poisson probability formula to calculate the probability for a specific number of people. For part D, the probability that the number of people entering the intensive care unit is two is P(x=2) = poissonpdf(4, 2) = 0.0902. For part E, the probability that the number of people entering the intensive care unit is less than or equal to two is P(x≤2) = poissoncdf(4, 2) = 0.2381.