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 solve this question, let's complete the table step-by-step.

### Step 1: Calculating \( A \) (Observed Count for the age group 31-55)

We are given that the total observed count is 105. We already have observed counts for the other three age groups:
- \( 31 \) for age group "less than 20"
- \( 29 \) for age group "21-30"
- \( 22 \) for age group "more than 55"

Using the total count, we can calculate \( A \) as follows:
\[
A = 105 - (31 + 29 + 22) = 105 - 82 = 23
\]

So, **\( A = 23 \)**.

### Step 2: Calculating \( B \) (Expected Count for the age group more than 55)

The expected count for each age group is based on the percentages provided:
- **Less than 20**: 10% of 105
- **21-30**: 10% of 105
- **31-55**: 40% of 105
- **More than 55**: 40% of 105

For age group "more than 55":
\[
B = 0.40 \times 105 = 42
\]

So, **\( B = 42 \)**.

### Step 3: Calculating \( C \) (chi-squared value for age group less than 20)

The chi-squared statistic for each group is calculated by the formula:
\[
\chi^2 = \frac{(\text{Observed Count} - \text{Expected Count})^2}{\text{Expected Count}}
\]

For age group "less than 20":
\[
C = \frac{(31 - 10.5)^2}{10.5} = \frac{(20.5)^2}{10.5} = \frac{420.25}{10.5} = 40.0238
\]

So, **\( C \approx 40.024 \)** (rounded to three decimal places).

### Completed Table

Now we can fill in the missing values in the table:

|                    | 1 (less than 20) | 2 (21-30) | 3 (31-55) | 4 (more than 55

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 analysis to determine if artist ages from a recent auction match expected distribution. Calculate missing values, including observed and expected counts and chi-squared statistics. Suitable for high school statistics and data analysis.

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