Part I:
A hospital researcher is interested in the number of times the average post-op patient will ring the nurse during a  12
-hour shift. For a random sample of  50
patients, the following information was obtained.

x
0	1	2	3	4	5
P(x)
350
950
1650
1350
750
250

Determine the expected value  μ=
?

The expected value is:  μ=

Number

To determine the expected value \(\mu\) of the number of times the average post-op patient will ring the nurse during a 12-hour shift, we need to use the probability distribution provided.

The expected value \(\mu\) is calculated using the formula:

\[
\mu = \sum (x \cdot P(x))
\]

where \(x\) is the number of times a patient rings the nurse, and \(P(x)\) is the probability of each occurrence.

**Given Data:**

- \(x = 0, 1, 2, 3, 4, 5\)

- Frequencies: \(350, 950, 1650, 1350, 750, 250\)

**Total Sample Size:**

The total number of patients in the sample is:

\[
N = 350 + 950 + 1650 + 1350 + 750 + 250 = 5300
\]

**Calculate the Probability \(P(x)\):**

\[
P(x) = \frac{\text{Frequency of } x}{\text{Total Sample Size}}
\]

- \(P(0) = \frac{350}{5300}\)
- \(P(1) = \frac{950}{5300}\)
- \(P(2) = \frac{1650}{5300}\)
- \(P(3) = \frac{1350}{5300}\)
- \(P(4) = \frac{750}{5300}\)
- \(P(5) = \frac{250}{5300}\)

Now, let's calculate the expected value \(\mu\).

\[
\mu = \left(0 \cdot \frac{350}{5300}\right) + \left(1 \cdot \frac{950}{5300}\right) + \left(2 \cdot \frac{1650}{5300}\right) + \left(3 \cdot \frac{1350}{5300}\right) + \left(4 \cdot \frac{750}{5300}\right) + \left(5 \cdot \frac{250}{5300}\right)
\]

Now, let's compute this value:

\[
\mu = \sum (x \cdot P(x))
\]The expected value \(\mu\) of the number of times the average post-op patient will ring the nurse during a 12-hour shift is approximately \(2.37\).

### Additional Questions

1. How is the expected value useful in understanding the nurse call behavior in the hospital setting?
2. What would the expected value indicate if it were significantly higher or lower?
3. How would you compute the variance of the number of calls using the given probabilities?
4. What assumptions are made when calculating the expected value from this data?
5. How would changes in the sample size affect the reliability of the expected value?
6. Could the distribution of calls be considered a Poisson distribution? Why or why not?
7. How might the hospital use this expected value to improve patient care?
8. What additional data might help refine the expected value calculation?

### Tip

Always ensure that the total probability sums to 1, which serves as a quick check that the probabilities are correctly calculated.

Learn how to calculate the expected value of nurse call frequency during a 12-hour shift using probability distributions. This problem involves determining the average number of times patients ring the nurse based on given probabilities. Suitable for advanced level learners interested in probability and statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation