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To solve this problem, we need to use the properties of a normal distribution. Given that salaries are normally distributed, we can compute the required proportion using the z-score formula.

### Given Information:
- Mean salary, $\mu = 41,750$
- Standard deviation, $\sigma = 7,500$
- We are asked to find the proportion of employees earning **at most** $48,000.

### Step 1: Compute the z-score

The z-score tells us how many standard deviations a value is from the mean. The formula for the z-score is:

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

where:
- $x = 48,000$ (the salary threshold we are interested in),
- $\mu = 41,750$ (the mean),
- $\sigma = 7,500$ (the standard deviation).

Substituting the values:

\[
z = \frac{48,000 - 41,750}{7,500} = \frac{6,250}{7,500} \approx 0.8333
\]

### Step 2: Find the Proportion

Now that we have the z-score $z \approx 0.8333$, we use the standard normal distribution table (or a calculator) to find the proportion of the population that lies below this z-score.

Looking up the z-score of $0.8333$ in a standard normal table or using a calculator, we get:

\[
P(z < 0.8333) \approx 0.7977
\]

### Conclusion:

The proportion of employees earning at most $48,000 is approximately **0.7977**. So, about 79.77% of the employees earn less than or equal to $48,000.

Would you like any further details or clarification?

Here are 5 related questions for further exploration:
1. How would the result change if the standard deviation was smaller, say $5,000?
2. What percentage of employees earn more than $48,000?
3. How do we interpret the z-score in a real-world context?
4. What proportion of employees earn between $35,000 and $50,000?
5. How would we calculate the salary range for the middle 68% of employees?

**Tip:** The z-score is a powerful tool in statistics for comparing individual data points to the overall distribution.

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 $7,500. 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 $48,000. Round your answer to at least four decimal places.

Learn how to calculate the proportion of employees earning at most $48,000 using normal distribution with a mean salary of $41,750 and a standard deviation of $7,500. This solution applies statistical concepts such as the z-score and normal distribution tables. Suitable for students in Grades 11-12 and early college statistics courses.

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