A survey that collected demographic data from respondents asked them to categorize their annual incomes​ (in thousands of​ dollars) into the groups in the table to the right. The number of respondents for each income group is also shown in the table.

Complete parts ​a) and

​b).

nbsp nbsp

nbsp Income nbsp  Income

nbsp Number of nbsp  Number of

nbsp Respondents nbsp  Respondents

nbsp $ 20 to under $ 30 nbsp  $20 to under $30

6363

nbsp $ 30 to under $ 40 nbsp  $30 to under $40

112112

nbsp $ 40 to under $ 50 nbsp  $40 to under $50

128128

nbsp $ 50 to under $ 60 nbsp  $50 to under $60

2222

nbsp $ 60 to under $ 70 nbsp  $60 to under $70

3535

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Part 1

​a) Calculate the approximate average income of the respondents.

The approximate average income of the respondents is

​$enter your response here

thousand.

​(Round to one decimal place as​ needed.)

To calculate the approximate average income of the respondents, we'll use the midpoint of each income group as a representative value for that group. Then, we'll calculate the weighted average based on the number of respondents in each group.

### Steps:
1. **Determine the midpoint for each income group:**
   - $20 to under $30 → Midpoint = $ \frac{20 + 30}{2} = 25 $ (thousand dollars)
   - $30 to under $40 → Midpoint = $ \frac{30 + 40}{2} = 35 $ (thousand dollars)
   - $40 to under $50 → Midpoint = $ \frac{40 + 50}{2} = 45 $ (thousand dollars)
   - $50 to under $60 → Midpoint = $ \frac{50 + 60}{2} = 55 $ (thousand dollars)
   - $60 to under $70 → Midpoint = $ \frac{60 + 70}{2} = 65 $ (thousand dollars)

2. **Multiply the midpoint by the number of respondents in each group:**
   - 25 × 63 = 1575
   - 35 × 112 = 3920
   - 45 × 128 = 5760
   - 55 × 22 = 1210
   - 65 × 35 = 2275

3. **Find the total income (sum of the weighted midpoints):**
   $ 1575 + 3920 + 5760 + 1210 + 2275 = 14740 $ (thousand dollars)

4. **Find the total number of respondents:**
   $ 63 + 112 + 128 + 22 + 35 = 360 $

5. **Calculate the average income:**
   \[
   \text{Average income} = \frac{\text{Total income}}{\text{Total number of respondents}} = \frac{14740}{360} \approx 40.9 \text{ thousand dollars}
   \]

### Final Answer:
The approximate average income of the respondents is **$40.9 thousand**.

Would you like further clarification, or should we proceed to part b?

Here are 5 relative questions you may find interesting:
1. What if the income groups were smaller or larger?
2. How would the average income change if a new group was added?
3. What is the difference between mean and median in this context?
4. How does the weighted average apply to real-world income distributions?
5. Can we predict a respondent's income more accurately using more granular data?

**Tip:** The midpoint method is a good approximation when dealing with grouped data like income brackets.

A survey that collected demographic data from respondents asked them to categorize their annual incomes​ (in thousands of​ dollars) into the groups in the table to the right. The number of respondents for each income group is also shown in the table. Part 1: a) Calculate the approximate average income of the respondents.

Learn how to calculate the approximate average income from survey data with grouped income categories. This step-by-step guide walks you through using midpoints and weighted averages for income groups like $20,000-$30,000 and $30,000-$40,000. Suitable for students in Grades 10-12.

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