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To solve this problem, let's analyze each of the three claims based on the information provided:

1. **Initial Information:**
   - The distribution of 27 salaries has a mean of $35,000 and a standard deviation of $2,000.
   - A new employee is hired with a salary of $120,000.

2. **Effect of the New Salary on the Mean:**
   - The mean is calculated by dividing the total sum of all salaries by the number of salaries.
   - Adding a significantly higher salary ($120,000) to the existing salaries will increase the total sum, thus increasing the mean. **So, claim III ("The mean is likely to increase") is true.**

3. **Effect of the New Salary on the Median:**
   - The median is the middle value in a list of numbers ordered from smallest to largest.
   - Initially, there are 27 salaries, so the median is the 14th salary.
   - After adding one more salary, making it 28 salaries, the median will be the average of the 14th and 15th salaries in the ordered list.
   - Unless the new salary of $120,000 changes the order of the original 14th and 15th salaries, the median is **not likely to change**. Therefore, claim I is **true**.

4. **Effect of the New Salary on the Range:**
   - The range is the difference between the highest and lowest salaries.
   - If $120,000 is higher than the previous maximum salary, the range will increase. However, since we do not have information about the previous highest salary, we cannot definitively say whether the range will change.
   - Therefore, claim II ("The range is not likely to change") is **not necessarily true** because it depends on the previous maximum salary.

Given this analysis, the correct answer is **D (I and III only)**.

Would you like a more detailed explanation or have any questions about this problem? Here are some related questions you might find useful:

1. What happens to the mean if the new salary was exactly the average of the existing salaries?
2. How would the median change if the new salary was the lowest among all the salaries?
3. How does standard deviation change with the addition of a new outlier salary?
4. What would the effect be on the range if the new salary was lower than the lowest existing salary?
5. How would the distribution of salaries change if multiple new employees with higher salaries were added?

**Tip:** Remember, the mean is sensitive to extreme values, whereas the median is more robust to outliers.

The distribution of 27 salaries at a small company has mean $35,000 and standard deviation $2,000. Suppose the company hires a 28th employee at a salary of $120,000. Which of the following claims about the new salary distribution is supported? I. The median is not likely to change. II. The range is not likely to change. III. The mean is likely to increase.

This problem explores how hiring a new employee at a higher salary affects the mean, median, and range in a company with 27 employees. With a new salary of $120,000, learn how to determine changes in statistical measures like the mean and whether the range or median is affected.

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