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Part 1
In​ 2021, the average hourly compensation​ (salary plus​ benefits) for federal workers was ​$52.8852.88. The accompanying table contains a random sample of the hourly amounts paid to state and local government workers. Complete parts a through d.
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Part 1
a. If hourly pay is thought to be normally​ distributed, what proportion of federal government employees receive pay higher than the average for federal​ employees?
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To determine the proportion of federal government employees who receive pay higher than the average hourly compensation, we need to make a few assumptions based on the information provided:

### Step-by-Step Process:

1. **Average Hourly Compensation of Federal Workers**:
   The average hourly compensation for federal workers is given as $ \mu = 52.88 $.

2. **Normal Distribution Assumption**:
   We are assuming that the hourly pay of federal government employees is normally distributed. For any normal distribution, 50% of the data lies above the mean, and 50% lies below.

3. **Proportion of Employees Paid More Than the Mean**:
   Since we are asked to find the proportion of employees earning **more** than the average, and we are assuming a normal distribution, the answer is directly tied to the properties of the normal distribution.
   
   For any normal distribution:
   - The mean splits the distribution into two equal halves.
   - This means that **50%** of the employees earn **more** than the mean, and the remaining 50% earn less than the mean.

Thus, **50%** of federal government employees are paid more than the average hourly compensation of $ 52.88 $.

Would you like further details or have any questions?

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Here are 5 related questions to deepen understanding:

1. How would the proportion change if the distribution were skewed?
2. What is the standard deviation's role in determining the spread of wages around the mean?
3. How can we use z-scores to find proportions for non-average wages?
4. What if the data were bimodal instead of normally distributed?
5. How would the sample size affect the accuracy of these estimates?

**Tip**: In normal distribution, the mean is the midpoint, so 50% of values lie above it regardless of the standard deviation.

In 2021, the average hourly compensation (salary plus benefits) for federal workers was $52.88. The accompanying table contains a random sample of the hourly amounts paid to state and local government workers. Complete parts a through d. a. If hourly pay is thought to be normally distributed, what proportion of federal government employees receive pay higher than the average for federal employees?

Discover how to calculate the proportion of federal government employees earning more than the average pay of $52.88 per hour. This statistical problem uses concepts of normal distribution and the empirical rule, ideal for high school students in Grades 10-12.

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