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Health Care: Office Visits What is the age distribution of patients who make office visits to a doctor or nurse? The following table is based on information taken from the Medical Practice Characteristics section of the Statistical Abstract of the United States (116th Edition). \begin{tabular}{l|ccccc} \hline Age group, years & Under 15 & \(15-24\) & \(25-44\) & \(45-64\) & 65 and older \\ \hline Percent of office visitors & \(20 \%\) & \(10 \%\) & \(25 \%\) & \(20 \%\) & \(25 \%\) \\ \hline \end{tabular} Suppose you are a district manager of a health management organization (HMO) that is monitoring the office of a local doctor or nurse in general family practice. This morning the office you are monitoring has eight office visits on the schedule. What is the probability that (a) at least half the patients are under 15 years old? First, explain how this can be modeled as a binomial distribution with 8 trials, where success is visitor age is under 15 years old and the probability of success is \(20 \%\). (b) from 2 to 5 patients are 65 years old or older (include 2 and 5 )? (c) from 2 to 5 patients are 45 years old or older (include 2 and 5 )? Hint: Success is 45 or older. Use the table to compute the probability of success on a single trial. (d) all the patients are under 25 years of age? (e) all the patients are 15 years old or older?

Step by step solution

01

Understanding Binomial Distribution

We are analyzing scenarios using the Binomial Distribution, which is suitable because each patient's visit can be considered a trial with two outcomes: either belonging to a target age group (success) or not. The number of trials (n) is 8, corresponding to the 8 office visits scheduled. The probability of success for each visit depends on the age group under consideration.

02

Calculating Probability for Part (a)

To find the probability that at least half (4 or more) of the patients are under 15 years, where the probability of a visit being by someone under 15 is 20% (p = 0.2). Calculation involves finding \(P(X \geq 4)\) for a Binomial distribution with \(n = 8\) and \(p = 0.2\). Calculate each component:\[ P(X \geq 4) = 1 - P(X < 4) = 1 - \left(P(X=0) + P(X=1) + P(X=2) + P(X=3)\right) \]Use the Binomial formula \(P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\) to compute individually and sum.

03

Calculating Probability for Part (b)

Calculate the probability of 2 to 5 patients being 65 years or older, with \(p = 0.25\). Find \(P(2 \leq X \leq 5)\) where:\[ P(2 \leq X \leq 5) = P(X=2) + P(X=3) + P(X=4) + P(X=5) \]Use the Binomial formula with n = 8, p = 0.25.

04

Calculating Probability for Part (c)

Calculate the probability of 2 to 5 patients being 45 years or older, where probability of success is the sum of probabilities of being 45-64 and 65+ years, \(p = 0.20 + 0.25 = 0.45\). Compute \(P(2 \leq X \leq 5)\) for \(n = 8\) using:\[ P(2 \leq X \leq 5) = P(X=2) + P(X=3) + P(X=4) + P(X=5) \]

05

Calculating Probability for Part (d)

Calculate the probability that all patients are under 25 years, so 8 out of 8 visits are by patients under 25 (probability includes under 15 and 15-24). Thus, \(p = 0.20 + 0.10 = 0.30\). Use:\[ P(X=8) = \binom{8}{8} \times 0.30^8 \]

06

Calculating Probability for Part (e)

Evaluate the probability that all patients are 15 years or older, meaning 0 out of 8 patients are under 15 (\(p\) for under 15 is 0.20, so 1 minus this gives probability of 15 or older, \(p = 0.80\)). Find \(P(X=0)\) with:\[ P(X=0) = \binom{8}{0} \times 0.80^8 \]

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Key Concepts

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

Probability Calculations

Probability calculations can often feel like a puzzle. When scenarios involve a specific number of trials, each with two possible outcomes (success or failure), the Binomial Distribution comes handy. This is exactly the scenario we're facing in our office visit problem. Probability calculations help us comprehend scenarios where we gauge the likelihood of certain number of patients falling into particular age categories.

To solve these kinds of problems, we rely on the binomial formula:

• First, determine the number of trials, which is 8 office visits in this case.
• Then, compute the probability for a single trial of the specific outcome you're interested in.
• You utilize the formula: \[ P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \]where \( \binom{n}{k} \) represents the binomial coefficient, \( p \) is the probability of success, and \( n-k \) incorporates probabilities of all other possibilities.

The combination of these gives you the probability of k successes in n trials. In a real-world context, these calculations allow health practitioners and managers to make informed decisions based on the likelihood of patient categories.

Age Distribution

Understanding age distribution can greatly influence how healthcare services are provided. The Medical Practice Characteristics provide a percentage breakdown of visitors over different age categories. This knowledge aids health management organizations in resource allocation and service planning.

Age distribution describes how a population is spread out across different age groups. For office visits to a healthcare professional, the distribution tells us which age groups most frequently visit:

• Under 15: 20%
• 15-24: 10%
• 25-44: 25%
• 45-64: 20%
• 65 and older: 25%

Such distribution offers insights into demographics and can hint at which age group may require more attention or specific types of healthcare services. This kind of data informs decision-making for health policy development and allocation of medical resources. Understanding who your main clients are allows for tailored services that deliver quality care.

Health Statistics

Health statistics are crucial for improving the quality of healthcare services. They offer insights into who uses services, their health trends, and how diseases are spread across age groups. In the context of our exercise, these statistics inform probability calculations on the likelihood of patients fitting certain age categories during their visits.

When health statistics are broken down by age, they provide critical information about public health needs. By examining how office visits correlate with age groups:

• We can understand how different age categories affect the frequency of visits.
• Managerial strategies can be devised to improve service efficiency and patient satisfaction.
• Healthcare professionals can prepare for increased visits from specific age groups and tailor their services accordingly.

Dynamic usage of health statistics ensures that healthcare professionals are proactive rather than reactive, easing the administrative bottlenecks and providing better healthcare outcomes for all age demographics.