According to a recent survey, the salaries of entry-level positions at a large company have a mean of $41,750 and a standard deviation of $6500. Assuming that the salaries of these entry-level positions are normally distributed, find the proportion of employees in entry-level positions at the company who earn at most $53,000. Round your answer to at least four decimal places.

To solve this problem, we need to determine the proportion of employees who earn at most $53,000. This involves calculating the cumulative probability of a normal distribution.

Given:
- Mean ($\mu$) = \$41,750
- Standard deviation ($\sigma$) = \$6,500
- We want to find $P(X \leq 53,000)$, where $X$ is the salary.

### Step 1: Calculate the z-score
The z-score is calculated using the formula:

\[
z = \frac{X - \mu}{\sigma}
\]

Substituting the values:

\[
z = \frac{53,000 - 41,750}{6,500} = \frac{11,250}{6,500} \approx 1.7308
\]

### Step 2: Find the cumulative probability
We now need to find the cumulative probability corresponding to this z-score from the standard normal distribution table or using a calculator.

The cumulative probability for a z-score of 1.7308 is approximately 0.9582.

### Conclusion:
The proportion of employees in entry-level positions at the company who earn at most $53,000 is approximately **0.9582**.

Would you like more details, or do you have any questions?

Here are 8 related questions that might interest you:
1. How do you find the probability of earning more than a certain salary in this scenario?
2. What percentage of employees earn between $35,000 and $50,000?
3. How would the calculation change if the standard deviation were larger?
4. What is the probability of an employee earning exactly $53,000?
5. Can you calculate the salary that corresponds to the top 10% of employees?
6. What does it mean when a distribution is "normally distributed"?
7. How would you calculate the z-score for a salary of $40,000?
8. How would a change in the mean affect the proportion of employees earning below a certain amount?

**Tip:** Understanding the z-score is crucial when working with normal distributions, as it standardizes data points for comparison.

Learn how to calculate the proportion of entry-level employees earning at most $53,000 using normal distribution. This problem involves calculating cumulative probability, z-score, and understanding normal distribution properties. Suitable for advanced high school students and beyond.

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