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

To analyze the effect of adding a 28th employee with a salary of $120,000 on the salary distribution of a small company, we need to consider the changes in the median, range, and mean of the salaries.

### Median:
The median of a data set is the middle value when the numbers are arranged in ascending order. Before hiring the 28th employee, the median is the average of the 13th and 14th salaries (since there are 27 employees). After hiring the 28th employee, the median will be the 14th salary (since there are now 28 employees). The $120,000 salary is an outlier and much higher than the other salaries, so it will not affect the middle of the distribution significantly.

### Range:
The range of a data set is the difference between the highest and lowest values. Initially, the highest salary is less than or equal to $35,000 + $k \times 2,000$, where $k$ is a constant (since we don't know the exact maximum). When a salary of $120,000 is added, this becomes the new maximum, drastically increasing the range.

### Mean:
The mean of a data set is the sum of all values divided by the number of values. Initially, the total sum of the salaries is:
\[
27 \times 35,000 = 945,000
\]
After adding the new salary:
\[
\text{New total sum} = 945,000 + 120,000 = 1,065,000
\]
The new mean is:
\[
\frac{1,065,000}{28} \approx 38,036
\]
Thus, the mean increases.

### Analysis of Claims:
- **Claim I (The median is not likely to change)**: True. The $120,000 salary is an outlier and does not affect the middle values significantly.
- **Claim II (The range is not likely to change)**: False. The range increases significantly due to the new highest value of $120,000.
- **Claim III (The mean is likely to increase)**: True. The mean increases from $35,000 to approximately $38,036.

### Conclusion:
Claims I and III are supported by the analysis.

Thus, the correct answer is:
**D) I and III only**

Would you like more details or have any questions? Here are some related questions you might find interesting:
1. How would adding a salary lower than the minimum salary affect the mean, median, and range?
2. What happens to the standard deviation when an outlier is added?
3. How is the interquartile range (IQR) affected by an extreme outlier?
4. Can the mode of a distribution be affected by a single new data point? Why or why not?
5. What is the effect on the mean if the new salary was $35,000 instead of $120,000?

**Tip:** When analyzing the impact of outliers, consider their effect on both central tendency measures (like mean and median) and dispersion measures (like range and standard deviation).

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? The median is not likely to change. I The range is not likely to change. II The mean is likely to increase. III A) I only B) III only C) I and II only D) I and III only E) I, II, and III

Analyze how adding a new salary of $120,000 affects the median, mean, and range of a small company's salary distribution. Understand the impact of outliers on measures of central tendency and dispersion with a detailed solution. Suitable for Grades 10-12.

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