To determine the test statistic for the given data, we will perform a Chi-Square Goodness-of-Fit Test because we are comparing observed frequencies (actual number eliminated) to expected frequencies (expected number eliminated). Here's how the test statistic ($\chi^2$) is calculated:

Formula:

$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$ Where:

• $O_i$ = Observed frequency
• $E_i$ = Expected frequency

Observed and Expected Values:

Step-by-Step Calculation:

1. For each position, calculate $(O_i - E_i)^2$.
2. Divide the squared difference by $E_i$.
3. Sum all the values.

Let me calculate the value for you.

Data for calculation

data = { "Singing Position": [1, 2, 3, 4, 5, 6, "7-12"], "Observed (O)": [20, 12, 9, 8, 6, 5, 9], "Expected (E)": [12.9, 12.9, 9.9, 7.9, 6.4, 5.5, 13.5] }

Convert data into a DataFrame

df = pd.DataFrame(data)

Calculate (O - E)^2 / E for each position

df["(O - E)^2"] = (df["Observed (O)"] - df["Expected (E)"]) ** 2 df["(O - E)^2 / E"] = df["(O - E)^2"] / df["Expected (E)"]

Calculate the Chi-Square statistic

chi_square_statistic = df["(O - E)^2 / E"].sum()

Display results

import ace_tools as tools; tools.display_dataframe_to_user(name="The calculated test statistic ($\chi^2$) is approximately 5.62.

Would you like further details on interpreting this result or calculating the p-value? Here are some related questions to consider:

1. How do you determine the degrees of freedom for this test?
2. What is the critical value at a 0.05 significance level for this test?
3. How do you interpret the result in terms of the null hypothesis?
4. What are the assumptions of the Chi-Square Goodness-of-Fit Test?
5. How do you use the p-value to confirm or reject the null hypothesis?

Tip: Always compare the calculated test statistic with the critical value or use the p-value to make decisions about the null hypothesis.