To address this question, it sounds like you're analyzing a categorical association between student age groups and lecture-watching mode (livestream vs. in-person). This setup suggests using a chi-square test of independence, which evaluates if there's a significant association between two categorical variables. Here's a general breakdown for each part of the question:

Step-by-Step Explanation:

1. Degrees of Freedom: For a chi-square test of independence, the degrees of freedom (df) are calculated using: $\text{df} = (r - 1)(c - 1)$ where $r$ is the number of age groups (rows) and $c$ is the number of lecture-watching methods (columns). If we assume two age groups and two modes (livestream and in-person), this would yield: $\text{df} = (2 - 1)(2 - 1) = 1$ However, substitute in the actual number of groups to confirm. If it’s 2x2, the answer will be df = 1.

2. Test Statistic: The chi-square test statistic is calculated based on the observed and expected counts. The formula is: $\chi^2 = \sum \frac{(O - E)^2}{E}$ where $O$ is the observed frequency, and $E$ is the expected frequency. Using the observed counts and calculating the expected counts based on marginal totals can yield the chi-square statistic value to two decimal places.

3. p-value: The p-value is obtained from the chi-square distribution with the calculated degrees of freedom. After finding the chi-square test statistic, consult a chi-square table or calculator to obtain the p-value to three decimal places.

4. Decision of the Test: If the p-value is less than the significance level (commonly 0.05), reject the null hypothesis, concluding a statistically significant association between age and lecture mode. Otherwise, retain the null hypothesis if $p > 0.05$.

---

Please share any specific values for observed frequencies or further information if available, so I can help calculate exact answers for the test statistic, p-value, and decision.

Would you like me to go through this with specific values, or do you need further clarification on any steps?

---

Relative Questions

1. What are the assumptions required for a chi-square test of independence?
2. How would results differ if you used an ANOVA or t-test instead of a chi-square test for this setup?
3. What is the impact of sample size on the reliability of the chi-square test?
4. Can the chi-square test determine causation between age and lecture-watching mode?
5. How could a post-hoc test be used if multiple age groups were compared?

Tip

When conducting a chi-square test, ensure all expected counts are at least 5 to satisfy test validity requirements.