Research question 2: 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
QUESTIONS:

Report the number of the degrees of freedom [integer]:	Answer 5 Question 1
3
Report the value of the test statistic [use at least 2dp]:	Answer 6 Question 1
Report the 𝑝
-value [use at least 3dp]:	Answer 7 Question 1
What is the decision of the test?	Answer 8 Question 1

For this problem, we can use a chi-square goodness-of-fit test to determine if the age distribution of the artists matches the given percentages. Here’s a step-by-step solution:

### Step 1: Define Hypotheses
- **Null Hypothesis (H0):** The observed age distribution of artists is consistent with the expected distribution.
- **Alternative Hypothesis (H1):** The observed age distribution of artists is not consistent with the expected distribution.

### Step 2: Calculate Degrees of Freedom
Since we have 4 age categories, the degrees of freedom for a chi-square test are calculated as:
\[
\text{Degrees of Freedom} = \text{Number of Categories} - 1 = 4 - 1 = 3
\]
Thus, the degrees of freedom are **3**.

### Step 3: Determine Expected Frequencies
Let's assume we have the observed frequencies for each age group from the auction data (we'll call these values \(O_1, O_2, O_3,\) and \(O_4\)). We need to compare them to the expected frequencies based on the percentages in the table:
- Age < 20: 10%
- Age 21-30: 10%
- Age 31-55: 40%
- Age > 55: 40%

If the total number of artists is \(N\), the expected frequency \(E\) for each category would be calculated by multiplying \(N\) by the respective percentage.

### Step 4: Calculate Test Statistic
The chi-square test statistic is calculated as:
\[
\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}
\]
where \(O_i\) and \(E_i\) are the observed and expected frequencies for each category. Please provide the observed frequencies to continue with the exact calculations.

### Step 5: Find the p-value
Once we have the chi-square test statistic, we can determine the p-value by comparing it against the chi-square distribution with 3 degrees of freedom.

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### Next Steps and Your Questions

Would you like to proceed by providing the observed frequencies so we can complete the calculations for the test statistic, p-value, and decision of the test?

Here are some additional questions to explore:

1. What assumptions are required for the chi-square goodness-of-fit test?
2. How would the test change if the observed frequencies significantly deviated from expected values?
3. What are the implications if the test rejects the null hypothesis?
4. Can this test be applied to other data, such as auction prices instead of ages?
5. How does sample size impact the chi-square test results?

**Tip:** Remember that in chi-square tests, having larger observed frequencies typically improves test reliability.

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.

Age of the artists | 1 (less than 20) | 2 (21-30) | 3 (31-55) | 4 (more than 55)
Percentage | 10 | 10 | 40 | 40

Explore how to perform a chi-square goodness-of-fit test to determine if the age distribution of artists in a recent auction matches the expected percentages. Learn the steps for hypothesis testing, calculating degrees of freedom, and finding the chi-square test statistic. Suitable for college-level statistics students.

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