*Note: Please make sure to properly format your answers. All dollar figures in the answers need to include the dollar sign and any amount over 1,000 should include the comma ($2,354.67). All percentage values in the answers need to include a percentage sign (%). For all items without specific rounding instructions, round your answers to two decimal places, show both decimal places (5.06).*

Before the last school year began, it was estimated that the average discretionary personal expenses each school year for a student attending a 4-year public college were $2,110. This past summer Ashley decided to poll seven of her friends attending a 4-year public college because she thought that estimate was low. She made a list of their actual school-year expenses:
$2,800, $1,990, $2,005, $2,400, $1,860, $2,200, $2,000
a. What is the mean of her friends' personal expenses? Round your answer to the nearest cent.  
b. Is the average higher or lower than the estimate?
c. What would Ashley's actual personal expenses for that school year have to be so that her amount and her friends' amounts together would have an average of $2,110? 

Let's solve each part step by step.

### Part (a): Calculating the mean of her friends' personal expenses

The data provided is:  
$2,800, $1,990, $2,005, $2,400, $1,860, $2,200, $2,000

The formula to calculate the mean is:

\[
\text{Mean} = \frac{\sum \text{expenses}}{\text{number of friends}}
\]

Let me calculate this first.The mean of her friends' personal expenses is **$2,179.29** (rounded to the nearest cent).

### Part (b): Is the average higher or lower than the estimate?

The estimated average was **$2,110**, and the mean we calculated is **$2,179.29**, which is higher than the estimate.

### Part (c): What would Ashley's personal expenses need to be for the overall average to be $2,110?

Let Ashley's personal expenses be $x$. Including Ashley, there will be 8 people in total.

The new mean, including Ashley's expenses, should be $2,110. Using the formula for mean:

\[
\frac{\sum (\text{expenses of 7 friends}) + x}{8} = 2110
\]

We can solve this equation for $x$. I'll compute that next.Ashley’s personal expenses for that school year would need to be **$1,625** in order for the overall average, including her and her friends' expenses, to be $2,110.

### Summary of Answers:
- **(a)** The mean of her friends' personal expenses is **$2,179.29**.
- **(b)** The average is **higher** than the estimate of $2,110.
- **(c)** Ashley's personal expenses would need to be **$1,625** to achieve an average of $2,110 for the group.

Would you like more details on any part, or assistance with another question?

Here are 5 related questions to consider:
1. How would the median expenses compare to the mean in this case?
2. What would happen if one expense was drastically higher, like $5,000?
3. How does the variance or standard deviation of these expenses measure the spread?
4. What is the impact of excluding Ashley from the average if her expenses are an outlier?
5. How would changing one friend's expense to the original estimate of $2,110 impact the average?

**Tip:** When calculating averages, always ensure you're working with the correct total number of data points.

Explore how to calculate the mean of personal expenses for college students and compare it to an estimated value. Includes step-by-step solutions and analysis. Suitable for students in Grades 9-12.

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