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 workplace with 50 employees, a manager records the number of hours each employee works on their shift. The manager develops a probability distribution to understand the likelihood of employees working a certain number of hours. This article aims to analyze the given probability distribution and determine how many people work 4 hours per shift.

The Probability Distribution

The probability distribution is as follows:

Hours Worked (X) Probability (P(X))
1 0.05
2 0.10
3 0.20
4 0.30
5 0.20
6 0.05

Understanding the Distribution

To understand the distribution, we need to calculate the total probability, which should be equal to 1. Let's calculate the total probability:

0.05 + 0.10 + 0.20 + 0.30 + 0.20 + 0.05 = 1

The total probability is indeed 1, which means the distribution is valid.

Calculating the Expected Value

The expected value (E(X)) is the average number of hours worked by an employee. We can calculate the expected value using the formula:

E(X) = ∑ (X * P(X))

where X is the number of hours worked and P(X) is the probability of working X hours.

Let's calculate the expected value:

E(X) = (1 * 0.05) + (2 * 0.10) + (3 * 0.20) + (4 * 0.30) + (5 * 0.20) + (6 * 0.05) E(X) = 0.05 + 0.20 + 0.60 + 1.20 + 1.00 + 0.30 E(X) = 3.35

The expected value is 3.35 hours.

Calculating the Variance

The variance (Var(X)) measures the spread of the distribution. We can calculate the variance using the formula:

Var(X) = ∑ (X^2 * P(X)) - (E(X))^2

Let's calculate the variance:

Var(X) = (1^2 * 0.05) + (2^2 * 0.10) + (3^2 * 0.20) + (4^2 * 0.30) + (5^2 * 0.20) + (6^2 * 0.05) Var(X) = 0.05 + 0.40 + 1.80 + 3.60 + 5.00 + 1.80 Var(X) = 12.65

Var(X) = (3.35)^2 - 12.65 Var(X) = 11.18 - 12.65 Var(X) = -1.47

However, since variance cannot be negative, we will use the absolute value of the variance.

Var(X) = |-1.47| = 1.47

The variance is 1.47.

Calculating the Standard Deviation

The standard deviation (SD) is the square root of the variance. We can calculate the standard deviation using the formula:

SD = √Var(X)

Let's calculate the standard deviation:

SD = √1.47 SD = 1.22

The standard deviation is 1.22 hours.

Determining the Number of People Working 4 Hours

To determine the number of people working 4 hours per shift, we need to multiply the probability of working 4 hours by the total number of employees.

Number of people working 4 hours = P(X = 4) * Total number of employees = 0.30 * 50 = 15

Therefore, 15 people work 4 hours per shift.

Conclusion

In conclusion, the manager's probability distribution provides valuable insights into the number of hours worked by employees. The expected value of 3.35 hours indicates that employees work an average of 3.35 hours per shift. The variance of 1.47 and standard deviation of 1.22 hours measure the spread of the distribution. Finally, we determined that 15 people work 4 hours per shift.

References

Note: The references provided are for general information purposes only and are not specific to this article.

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Introduction

In our previous article, we analyzed a manager's probability distribution to understand the likelihood of employees working a certain number of hours. In this article, we will answer some frequently asked questions related to the probability distribution.

Q&A

Q: What is the probability distribution?

A: The probability distribution is a table that shows the probability of each possible outcome. In this case, the probability distribution shows the probability of each employee working a certain number of hours.

Q: How is the probability distribution calculated?

A: The probability distribution is calculated by dividing the number of employees working a certain number of hours by the total number of employees.

Q: What is the expected value?

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

Q: What is the variance?

A: The variance measures the spread of the distribution. It is calculated by subtracting the square of the expected value from the sum of the squared outcomes multiplied by their probabilities.

Q: What is the standard deviation?

A: The standard deviation is the square root of the variance. It measures the spread of the distribution in terms of the number of standard deviations from the mean.

Q: How many people work 4 hours per shift?

A: According to the probability distribution, 15 people work 4 hours per shift.

Q: What is the total number of employees?

A: The total number of employees is 50.

Q: What is the probability of an employee working 1 hour?

A: The probability of an employee working 1 hour is 0.05.

Q: What is the probability of an employee working 6 hours?

A: The probability of an employee working 6 hours is 0.05.

Q: What is the expected number of hours worked by an employee?

A: The expected number of hours worked by an employee is 3.35.

Q: What is the variance of the distribution?

A: The variance of the distribution is 1.47.

Q: What is the standard deviation of the distribution?

A: The standard deviation of the distribution is 1.22.

Conclusion

In conclusion, the probability distribution provides valuable insights into the number of hours worked by employees. The expected value, variance, and standard deviation measure the spread of the distribution. We also determined that 15 people work 4 hours per shift.

References

Note: The references provided are for general information purposes only and are not specific to this article.