The average cost of a stay in a U.S. hospital in 2021 for a Medicare patient was $15,400. Assume that these costs are exponentially distributed. a. Determine the probability that a randomly selected patients stay in a U.S. hospital will cost more than $17,500. b. Calculate the probability that a randomly selected patients stay in a U.S. hospital will cost less than $12,300

Given that the hospital costs for a Medicare patient are exponentially distributed, we can model the problem using the probability density function (PDF) of an exponential distribution. The PDF of an exponential distribution is given by:

\[
f(x; \lambda) = \lambda e^{-\lambda x}
\]

Where:
- $ \lambda $ is the rate parameter (the inverse of the mean),
- $ x $ represents the random variable (in this case, the cost of the hospital stay).

### Step 1: Find the rate parameter $ \lambda $
The mean cost of a hospital stay is given as $15,400. For an exponential distribution, the mean $ \mu $ is related to the rate parameter $ \lambda $ by:

\[
\mu = \frac{1}{\lambda}
\]

So:

\[
\lambda = \frac{1}{\mu} = \frac{1}{15,400}
\]

### Step 2: Solving Part (a)
We need to determine the probability that a randomly selected patient’s stay will cost more than $17,500. The cumulative distribution function (CDF) of the exponential distribution is:

\[
F(x) = 1 - e^{-\lambda x}
\]

To find the probability that the cost exceeds $17,500, we use the complement of the CDF:

\[
P(X > 17,500) = 1 - F(17,500) = e^{-\lambda \cdot 17,500}
\]

### Step 3: Solving Part (b)
We need to determine the probability that a randomly selected patient’s stay will cost less than $12,300. This is simply the value of the CDF at $12,300$:

\[
P(X < 12,300) = F(12,300) = 1 - e^{-\lambda \cdot 12,300}
\]

Now, let me calculate these probabilities.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can guide you on how to calculate the probabilities yourself, or you can try again later when the tool is available.

Would you like to go through the steps manually, or do you have any other questions? Here are some related questions to deepen your understanding:

1. How is the rate parameter $ \lambda $ calculated in an exponential distribution?
2. What is the difference between the probability density function (PDF) and the cumulative distribution function (CDF)?
3. How would the probabilities change if the average cost of a hospital stay increased?
4. Can you explain how exponential distributions are used in real-life scenarios like hospital stays?
5. How do different distributions (e.g., normal or uniform) differ from the exponential distribution?

**Tip**: When dealing with exponential distributions, remember that they are often used to model time until an event happens, such as waiting times or costs with memoryless properties.

The average cost of a stay in a U.S. hospital in 2021 for a Medicare patient was $15,400. Assume that these costs are exponentially distributed. a. Determine the probability that a randomly selected patient's stay in a U.S. hospital will cost more than $17,500. b. Calculate the probability that a randomly selected patient's stay in a U.S. hospital will cost less than $12,300.

Explore how to calculate the probability of hospital stays costing more than $17,500 or less than $12,300 using exponential distribution. Learn about key concepts such as probability density functions (PDFs), cumulative distribution functions (CDFs), and the rate parameter λ. This problem is perfect for students in introductory probability or statistics courses.

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