The distribution of the​ tuitions, fees, and room and board charges of a random sample of public​ 4-year degree-granting postsecondary institutions is shown in the pie chart. Make a frequency distribution for the data. Then use the table to estimate the sample mean and the sample standard deviation of the data set. Use 

​$26249.50 as the midpoint for 

$25,000 

or​ more."

To compute the **frequency distribution**, **sample mean**, and **sample standard deviation** based on the given tuition data (from a pie chart), you need to follow these steps:

### 1. **Frequency Distribution Table**:
We will first organize the tuition fees into class intervals, identify the frequency for each interval, and then calculate the midpoint for each class. You have also provided the midpoint of $25,000 or more as $26,249.50.

#### Typical Structure of the Frequency Distribution Table:

| Class Interval         | Frequency (f) | Midpoint (x)  | $f \times x$  | 
|------------------------|---------------|---------------|----------------|
| \$5,000 - \$9,999       | *f1*          | *m1*          | *f1 \times m1* |
| \$10,000 - \$14,999     | *f2*          | *m2*          | *f2 \times m2* |
| \$15,000 - \$19,999     | *f3*          | *m3*          | *f3 \times m3* |
| \$20,000 - \$24,999     | *f4*          | *m4*          | *f4 \times m4* |
| \$25,000 or more        | *f5*          | \$26,249.50   | *f5 \times 26,249.50* |

If you can provide me with the actual frequencies or percentages from the pie chart, I can help you complete this table.

### 2. **Calculate Sample Mean**:

The formula for the sample mean is:

\[
\bar{x} = \frac{\sum (f \times x)}{\sum f}
\]

Where:
- $ \bar{x} $ is the sample mean,
- $ f $ is the frequency of each class,
- $ x $ is the midpoint of each class,
- $ \sum (f \times x) $ is the sum of the product of frequencies and midpoints, and
- $ \sum f $ is the total number of observations (total frequency).

### 3. **Calculate Sample Standard Deviation**:

The formula for the sample standard deviation $s$ is:

\[
s = \sqrt{\frac{\sum (f \times (x - \bar{x})^2)}{n - 1}}
\]

Where:
- $ s $ is the sample standard deviation,
- $ f $ is the frequency of each class,
- $ x $ is the midpoint of each class,
- $ \bar{x} $ is the sample mean,
- $ n $ is the total frequency (total number of observations).

---

### Next Steps:
If you provide the frequencies or the actual percentages from the pie chart, I can calculate the sample mean and sample standard deviation for you.

Would you like to continue and input the necessary data from the pie chart?

---

#### Related Questions:
1. What is the total number of institutions in the sample?
2. How were the intervals for tuition charges determined in the sample?
3. How does the sample data compare to national averages for public institutions?
4. What is the significance of using the midpoint for calculations in this case?
5. How do outliers (extreme values) affect the standard deviation in tuition data?

#### Tip:
Always use the midpoints when calculating the mean and standard deviation for grouped data, as this ensures an accurate approximation of the dataset.

The distribution of the tuitions, fees, and room and board charges of a random sample of public 4-year degree-granting postsecondary institutions is shown in the pie chart. Make a frequency distribution for the data. Then use the table to estimate the sample mean and the sample standard deviation of the data set. Use $26,249.50 as the midpoint for $25,000 or more.

Learn how to compute the frequency distribution, sample mean, and sample standard deviation for the tuitions, fees, and room and board charges of public 4-year institutions. This problem covers key statistical concepts using midpoints and frequency distribution. Suitable for college-level statistics.

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