A Manager Records The Number Of Hours, { X $}$, Each Employee Works On His Or Her Shift And Develops The Probability Distribution Below. Fifty People Work For The Manager. How Many People Work 4 Hours Per

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Introduction

In a typical work environment, managers often face the challenge of managing employee work hours. To better understand this issue, a manager records the number of hours each employee works on their shift and develops a probability distribution. In this article, we will explore this probability distribution and use it to determine how many people work 4 hours per shift.

The Probability Distribution

The probability distribution developed by the manager is as follows:

Hours Worked Probability
2 0.15
3 0.25
4 0.30
5 0.20
6 0.10

Understanding the Probability Distribution

The probability distribution shows the likelihood of each employee working a certain number of hours per shift. For example, the probability of an employee working 2 hours is 0.15, which means that 15% of employees work 2 hours per shift. Similarly, the probability of an employee working 4 hours is 0.30, which means that 30% of employees work 4 hours per shift.

Calculating the Expected Value

To calculate the expected value of the probability distribution, we multiply each outcome by its probability and sum the results.

Expected Value = (2 x 0.15) + (3 x 0.25) + (4 x 0.30) + (5 x 0.20) + (6 x 0.10) Expected Value = 0.30 + 0.75 + 1.20 + 1.00 + 0.60 Expected Value = 3.85

Calculating the Variance

To calculate the variance of the probability distribution, we first calculate the squared differences between each outcome and the expected value.

Squared Differences = [(2 - 3.85)^2] + [(3 - 3.85)^2] + [(4 - 3.85)^2] + [(5 - 3.85)^2] + [(6 - 3.85)^2] Squared Differences = [(-1.85)^2] + [(-0.85)^2] + [0.15^2] + [1.15^2] + [2.15^2] Squared Differences = 3.4225 + 0.7225 + 0.0225 + 1.3225 + 4.6225 Squared Differences = 10.090

Variance = (1/5) x 10.090 Variance = 2.018

Determining the Number of People Working 4 Hours

To determine the number of people working 4 hours per shift, we can use the probability distribution. Since 30% of employees work 4 hours per shift, we can calculate the number of people working 4 hours as follows:

Number of People Working 4 Hours = (0.30) x 50 Number of People Working 4 Hours = 15

Therefore, 15 people work 4 hours per shift.

Conclusion

In this article, we explored a manager's probability distribution of employee work hours. We calculated the expected value and variance of the probability distribution and used it to determine the number of people working 4 hours per shift. The results show that 15 people work 4 hours per shift, which can be useful information for the manager to make informed decisions about employee work hours.

References

Mathematical Formulas

  • Expected Value = ∑ (x_i * p_i)
  • Variance = ∑ (x_i^2 * p_i) - (∑ (x_i * p_i))^2
  • Probability Distribution = P(x) = {p_1, p_2, ..., p_n}

Code

import numpy as np

probabilities = np.array([0.15, 0.25, 0.30, 0.20, 0.10])



outcomes = np.array([2, 3, 4, 5, 6])



expected_value = np.sum(outcomes * probabilities)



variance = np.sum(outcomes2 * probabilities) - expected_value2



num_people_working_4_hours = (0.30) * 50


print("Expected Value:", expected_value)
print("Variance:", variance)
print("Number of People Working 4 Hours:", num_people_working_4_hours)


**A Manager's Probability Distribution: Q&A**
=============================================

**Introduction**
---------------

In our previous article, we explored a manager's probability distribution of employee work hours. We calculated the expected value and variance of the probability distribution and used it to determine the number of people working 4 hours per shift. In this article, we will answer some frequently asked questions about the probability distribution and its applications.

**Q&A**
------

### Q: What is the probability distribution of employee work hours?

A: The probability distribution of employee work hours is a table that shows the likelihood of each employee working a certain number of hours per shift. In this case, the probability distribution is as follows:

| Hours Worked | Probability |
| --- | --- |
| 2 | 0.15 |
| 3 | 0.25 |
| 4 | 0.30 |
| 5 | 0.20 |
| 6 | 0.10 |

### Q: What is the expected value of the probability distribution?

A: The expected value of the probability distribution is the average number of hours worked per shift. It is calculated by multiplying each outcome by its probability and summing the results.

Expected Value = (2 x 0.15) + (3 x 0.25) + (4 x 0.30) + (5 x 0.20) + (6 x 0.10)
Expected Value = 0.30 + 0.75 + 1.20 + 1.00 + 0.60
Expected Value = 3.85

### Q: What is the variance of the probability distribution?

A: The variance of the probability distribution is a measure of the spread of the data. It is calculated by taking the squared differences between each outcome and the expected value, and then summing the results.

Variance = (1/5) x [(2 - 3.85)^2 + (3 - 3.85)^2 + (4 - 3.85)^2 + (5 - 3.85)^2 + (6 - 3.85)^2]
Variance = (1/5) x [(-1.85)^2 + (-0.85)^2 + 0.15^2 + 1.15^2 + 2.15^2]
Variance = (1/5) x [3.4225 + 0.7225 + 0.0225 + 1.3225 + 4.6225]
Variance = (1/5) x 10.090
Variance = 2.018

### Q: How many people work 4 hours per shift?

A: To determine the number of people working 4 hours per shift, we can use the probability distribution. Since 30% of employees work 4 hours per shift, we can calculate the number of people working 4 hours as follows:

Number of People Working 4 Hours = (0.30) x 50
Number of People Working 4 Hours = 15

### Q: What are the applications of the probability distribution?

A: The probability distribution can be used in various applications, such as:

* **Workforce planning**: The probability distribution can be used to determine the number of employees needed to meet the demands of the business.
* **Employee scheduling**: The probability distribution can be used to create employee schedules that take into account the likelihood of each employee working a certain number of hours per shift.
* **Resource allocation**: The probability distribution can be used to allocate resources, such as equipment and materials, based on the likelihood of each employee working a certain number of hours per shift.

### Q: How can I create a probability distribution for my business?

A: To create a probability distribution for your business, you can follow these steps:

1. **Collect data**: Collect data on the number of hours worked by each employee per shift.
2. **Analyze the data**: Analyze the data to determine the likelihood of each employee working a certain number of hours per shift.
3. **Create a table**: Create a table that shows the likelihood of each employee working a certain number of hours per shift.
4. **Calculate the expected value and variance**: Calculate the expected value and variance of the probability distribution.

**Conclusion**
----------

In this article, we answered some frequently asked questions about the probability distribution of employee work hours. We discussed the expected value and variance of the probability distribution, and its applications in workforce planning, employee scheduling, and resource allocation. We also provided a step-by-step guide on how to create a probability distribution for your business.

**References**
--------------

* [1] Probability Distribution. (n.d.). Retrieved from <https://en.wikipedia.org/wiki/Probability_distribution>
* [2] Expected Value. (n.d.). Retrieved from <https://en.wikipedia.org/wiki/Expected_value>
* [3] Variance. (n.d.). Retrieved from <https://en.wikipedia.org/wiki/Variance>

**Mathematical Formulas**
-------------------------

* Expected Value = ∑ (x_i \* p_i)
* Variance = ∑ (x_i^2 \* p_i) - (∑ (x_i \* p_i))^2
* Probability Distribution = P(x) = {p_1, p_2, ..., p_n}

**Code**
------

```python
import numpy as np

# Define the probability distribution
probabilities = np.array([0.15, 0.25, 0.30, 0.20, 0.10])

# Define the outcomes
outcomes = np.array([2, 3, 4, 5, 6])

# Calculate the expected value
expected_value = np.sum(outcomes * probabilities)

# Calculate the variance
variance = np.sum(outcomes**2 * probabilities) - expected_value**2

# Calculate the number of people working 4 hours
num_people_working_4_hours = (0.30) * 50

print("Expected Value:", expected_value)
print("Variance:", variance)
print("Number of People Working 4 Hours:", num_people_working_4_hours)
</code></pre>