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Antibiotic resistance. According to CDC estimates, more than 2 million people in the United States are sickened each year with antibiotic-resistant infections, with at least 23,000 dying as a result. Antibiotic resistance occurs when diseasecausing 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 health problem. Of the three infections considered most serious by the CDC, gonorrhea has an estimated 800,000 cases occurring annually, with approximately \(30 \%\) of those cases resistant to any antibiotic \({ }^{7}\) A public health clinic in California sees eight patients with gonorrhea in a given week. (a) What is the distribution of \(X\), the number of these eight cases that are resistant to any antibiotic? (b) What are the mean and standard deviation of \(X\) ? (c) Find the probability that exactly one of the cases is resistant to any antibiotic. What is the probability that at least one case is resistant to any antibiotic? (Hint: It is easier to first find the probability that exactly zero of the eight cases were resistant.)

Step by step solution

01

Identify the appropriate distribution

Since we are dealing with a fixed number of trials (8 cases in a week), each having two possible outcomes (resistant or not resistant), and assuming independence between cases with a constant probability of resistance, the Binomial distribution is suitable here. Let \( X \) represent the number of antibiotic-resistant gonorrhea cases among the 8 patients. Thus, \( X \sim \text{Binomial}(8, 0.3) \), where 0.3 is the probability of resistance.

02

Calculate the mean and standard deviation

The mean \( \mu \) of a Binomial distribution is given by \( n \times p \), and the standard deviation \( \sigma \) is given by \( \sqrt{n \times p \times (1-p)} \). Here, \( n = 8 \) and \( p = 0.3 \). Therefore, the mean is \( 8 \times 0.3 = 2.4 \) and the standard deviation is \( \sqrt{8 \times 0.3 \times 0.7} \approx 1.29 \).

03

Calculate the probability for exactly one resistant case

Using the Binomial probability formula, \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), calculate the probability for exactly one resistant case (\( k=1 \)). \( P(X=1) = \binom{8}{1} \times 0.3^1 \times 0.7^7 \approx 0.1094 \).

04

Find the probability for at least one resistant case

It is often easier to calculate \( P(X \geq 1) \) by first finding \( P(X = 0) \) and using the complement rule: \( P(X \geq 1) = 1 - P(X = 0) \). \( P(X = 0) = \binom{8}{0} \times 0.3^0 \times 0.7^8 \approx 0.0576 \). Thus, \( P(X \geq 1) = 1 - 0.0576 \approx 0.9424 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Antibiotic Resistance

Antibiotic resistance is a growing concern in our public health system. It occurs when bacteria or other microbes develop the ability to defeat the drugs designed to kill them. This means that infections caused by these microbes become much harder, and sometimes impossible, to treat. The issue is particularly concerning because it can spread quickly when resistant strains are passed from person to person. This problem is often genetic, as resistant bacteria can transfer their resistant characteristics to the next generations of bacteria. The Centers for Disease Control and Prevention (CDC) have identified several infections where antibiotic resistance is most serious. One of these is gonorrhea, which sees a significant number of cases each year. In the context of the example given, 30% of these infections resist antibiotics, a figure that highlights the public health challenge. Understanding the spread and impact of antibiotic resistance helps public health agencies develop strategies to combat it. This includes monitoring resistant cases and applying statistical methods to predict and curtail this spread.

Probability Calculation

Understanding probability is crucial when dealing with events that have multiple outcomes. In the case of antibiotic resistance in gonorrhea, we are interested in finding out how likely it is that a certain number of cases resist treatment in a sample of patients.We use the Binomial distribution to calculate such probabilities because each case of infection in a clinic either has resistance or doesn't. Given the probability of resistance is 0.3, and we have 8 cases, we can calculate the likelihood of specific outcomes.
- For example, to calculate the probability of exactly one case resistant to antibiotics, we use the formula for Binomial probabilities: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \( n \) is the total number of trials (cases), \( k \) the exact number of successes (resistant cases) we are interested in, and \( p \) the probability of success (resistance) for each trial. This method offers a systematic way to handle such probability questions, providing insight into both common and rare outcomes.

Statistical Analysis

Statistical analysis allows us to interpret data and deduce meaningful insights from complex situations. By applying statistical methods like the Binomial distribution, we can make informed predictions about health phenomena such as antibiotic resistance. In our example, we analyzed the distribution of resistant gonorrhea cases using statistical probabilities. This means not just finding the likelihood of exact resistance rates but also understanding broader patterns like the probability of having at least one resistant case among the 8 patients. This broader view is achieved using tools like: - **Complementary Probabilities**: Finding the likelihood of at least one event by subtracting the probability that none occur from 1. - **Probability Calculations**: As seen in earlier sections, these help calculate the chances of specific outcomes. Such statistical analyses inform decision-making, helping to allocate resources effectively and plan interventions based on data-driven predictions.

Mean and Standard Deviation

The concepts of mean and standard deviation are fundamental in the analysis of any dataset, including when dealing with binomial distributions.- **Mean** is the average expected outcome. In a binomial distribution, it is calculated as the number of trials multiplied by the probability of success, i.e., \( n \times p \). For our problem, this is calculated as \( 8 \times 0.3 = 2.4 \).
- **Standard Deviation** measures the spread of the distribution. It tells us how much variation or "dispersion" exists from the mean. In a binomial setup, it is determined by the square root of \( n \times p \times (1-p) \). This results in a value of approximately 1.29 in our case.
Understanding these metrics gives us insight into both the central tendency of our dataset and the variability. It helps predict how much actual outcomes might deviate from what is statistically expected, reflecting the consistency or spread of the data.