Research question: We wish to investigate whether the age of the artists involved in a recent auction of new art is distributed according to the table below.

the age of the artists | 1 (less than 20) | 2 (21-30) | 3 (31-55) | 4 (more than 55)
Percentage             |               10 |        10 |        40 |               40
Please complete the following table calculating the values of A, B and C.

| 1 (less than 20) | 2 (21-30) | 3 (31-55) | 4 (more than 55) | Total
Observed Count | 31               | 29        | A=***     | 22               | 105
Expected Count | 10.5             | 10.5      | 42        | B=***            | 105
chi squared    | C=***            | 32.595    | 8.595     | 9.524            | 90.738
A [integer]:	Answer 6 Question 1
B [use at least 3dp

To complete the table and calculate the values of \( A \), \( B \), and \( C \), we’ll need to determine:

1. **A**: The observed count for age group 3 (ages 31-55).
2. **B**: The expected count for age group 4 (ages more than 55).
3. **C**: The chi-squared value for age group 1 (less than 20).

Let’s go through each calculation based on the given dataTo complete the table and calculate the values of \( A \), \( B \), and \( C \), we’ll need to determine:

1. **A**: The observed count for age group 3 (ages 31-55).
2. **B**: The expected count for age group 4 (ages more than 55).
3. **C**: The chi-squared value for age group 1 (less than 20).

Let’s go through each calculation based on the given 

### Step 1: Calculate \( A \)
The total observed count is 105, and the observed counts for other age groups sum up to:
\[
31 + 29 + 22 = 82
\]
Thus, we find \( A \) by subtracting 82 from the total:
\[
A = 105 - 82 = 23
\]

### Step 2: Calculate \( B \)
The expected count for each age group is derived from the total count (105) and the specified percentages:
- Age group 4 has a percentage of 40%.
\[
B = 0.40 \times 105 = 42.0
\]

### Step 3: Calculate \( C \)
The chi-squared formula for each cell is:
\[
\chi^2 = \frac{(O - E)^2}{E}
\]
where \( O \) is the observed count, and \( E \) is the expected count.

For age group 1 (less than 20):
- Observed count \( O = 31 \)
- Expected count \( E = 10.5 \)

Calculating \( C \):
\[
C = \frac{(31 - 10.5)^2}{10.5} = \frac{420.25}{10.5} = 40.024
\]

### Completed Table:

| Age Group               | 1 (less than 20) | 2 (21-30) | 3 (31-55) | 4 (more than 55) | Total |
|-------------------------|------------------|-----------|-----------|-------------------|-------|
| **Observed Count**      | 31               | 29        | 23        | 22               | 105   |
| **Expected Count**      | 10.5             | 10.5      | 42        | 42               | 105   |
| **Chi-squared**         | 40.024          | 32.595    | 8.595     | 9.524            | 90.738 |

### Answers:
- **A** = 23 (observed count for age group 3)
- **B** = 42.0 (expected count for age group 4, to three decimal places)
- **C** = 40.024 (chi-squared for age group 1)

Would you like additional details on any part of the process, or have questions?

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**Follow-up Questions:**

1. How would changing the observed values affect the chi-squared statistic?
2. What does a high chi-squared value indicate in this context?
3. Can the chi-squared test be used with non-integer observed counts?
4. What is the degree of freedom for this chi-squared test?
5. How would the results change if the expected percentages for each age group were different?

**Tip:** For any chi-squared test, ensure observed and expected frequencies add up to the same total for an accurate result.

Research question: We wish to investigate whether the age of the artists involved in a recent auction of new art is distributed according to the table below.

the age of the artists | 1 (less than 20) | 2 (21-30) | 3 (31-55) | 4 (more than 55)
Percentage             |               10 |        10 |        40 |               40
Please complete the following table calculating the values of A, B and C.

| 1 (less than 20) | 2 (21-30) | 3 (31-55) | 4 (more than 55) | Total
Observed Count | 31               | 29        | A=***     | 22               | 105
Expected Count | 10.5             | 10.5      | 42        | B=***            | 105
chi squared    | C=***            | 32.595    | 8.595     | 9.524            | 90.738

Explore a chi-squared test example that examines the age distribution of artists in an art auction. Calculate observed and expected frequencies, and determine chi-squared values for various age groups to evaluate statistical significance. This example illustrates key statistical concepts and is ideal for high school students learning advanced statistics.

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