Question

According to CDC estimates, at least 2.8 million people in the United States are sickened each year with antibiotic- resistant infections, and at least 35,000 die as a result. Antibiotic resistance occurs when disease - causing microbes become resistant to antibiotic drug therapy. Because this resistance is typically genetic and transferred to the next generations of microbes, it is a very serious public heath problem. Of the infections considered most serious by the CDC, gonorrehea has an estimated 1.14 million new cases occurring annually, and approximately 50℅ of those cases are resistant to any antibiotic. A public health clinic in California sees eight patients with gonorrehea in the given week.

A). What is the distribution of X, the number off these eight cases that are resistant to any antibiotic?
B). What are the mean and standard deviation of X?
C). Find the probablity that exactly one of the cases in resistant to any antibiotic.
What is the probablity that at least one case is resistant to any antibiotic? (Hint: It is easier to find the probablity that exactly zero of the eight cases were resistant.)

Answer 1

The distribution of X follows a binomial distribution with n=8 and p=0.5. The mean of X is 4, and the standard deviation is approximately 1.41. The probability that exactly one case is resistant is about 0.03125, and the probability that at least one case is resistant is approximately 0.9961.

Step-by-step explanation:

When addressing the problem of estimating the distribution of patients resistant to antibiotics in a given sample size, we use statistical methods. For this situation, the binomial distribution is appropriate because we are dealing with a fixed number of trials, two possible outcomes (resistant or not resistant), independent trials, and a constant probability of success (resistance).

Distribution of X

The distribution of X, the number of these eight cases that are resistant to any antibiotic, follows a binomial distribution with parameters n = 8 (number of trials) and p = 0.5 (probability of resistance).

Mean and Standard Deviation of X

The mean (μ) of a binomial distribution is given by μ = np, hence for our case, μ = 8 * 0.5 = 4. The standard deviation (σ) is given by σ = sqrt(np(1-p)), which calculates to σ = sqrt(8 * 0.5 * (1 - 0.5)) = sqrt(2) ≈ 1.41.

Probability Calculations

To find the probability that exactly one of the cases is resistant to any antibiotic, we use the binomial probability formula: P(X=k) = (n choose k) * p^k * (1-p)^(n-k). For k = 1, P(X=1) = (8 choose 1) * 0.5^1 * 0.5^(8-1) = 8 * 0.5^8 ≈ 0.03125. To find the probability that at least one case is resistant, we calculate the probability that zero cases are resistant and subtract from 1: P(X≥0) = 1 - P(X=0) = 1 - (8 choose 0) * 0.5^0 * 0.5^8 ≈ 1 - 0.5^8 ≈ 0.9961.