What is the Chi-Square Test Statistic for Independence?

📚 Quick Study Guide

• 📊 The Chi-Square Test for Independence determines if there's a significant association between two categorical variables.
• 📝 The null hypothesis ($H_0$) states that the two variables are independent. The alternative hypothesis ($H_1$) states that they are dependent.
• 🔢 The Chi-Square test statistic is calculated as: $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$, where $O_i$ is the observed frequency and $E_i$ is the expected frequency in each category.
• 🧪 The expected frequency is calculated as: $E_{ij} = \frac{(\text{Row Total}) \times (\text{Column Total})}{\text{Grand Total}}$.
• 📈 The degrees of freedom (df) are calculated as: $\text{df} = (\text{Number of Rows} - 1) \times (\text{Number of Columns} - 1)$.
• 🧐 A larger Chi-Square value suggests a stronger association between the variables.
• ✅ Compare the calculated $\chi^2$ value to a critical value from the Chi-Square distribution table, or use a p-value to determine statistical significance.

Practice Quiz

1. What does the Chi-Square Test for Independence primarily assess?

   1. A. The mean difference between two groups
   2. B. The correlation between two continuous variables
   3. C. The association between two categorical variables
   4. D. The variance within a single group

2. What is the null hypothesis ($H_0$) in a Chi-Square Test for Independence?

   1. A. The variables are dependent.
   2. B. The variables are correlated.
   3. C. The variables are independent.
   4. D. There is no relationship between the variables.

3. Which formula is used to calculate the Chi-Square test statistic?

   1. A. $\sum (O_i - E_i)$
   2. B. $\sum \frac{(O_i - E_i)}{E_i}$
   3. C. $\sum \frac{(O_i - E_i)^2}{O_i}$
   4. D. $\sum \frac{(O_i - E_i)^2}{E_i}$

4. How is the expected frequency ($E_i$) calculated in a Chi-Square Test?

   1. A. $\frac{(\text{Row Total}) + (\text{Column Total})}{\text{Grand Total}}$
   2. B. $\frac{(\text{Row Total}) \times (\text{Column Total})}{\text{Grand Total}}$
   3. C. $\frac{(\text{Grand Total})}{(\text{Row Total}) \times (\text{Column Total})}$
   4. D. $(\text{Row Total}) \times (\text{Column Total}) \times {\text{Grand Total}}$

5. What do degrees of freedom (df) represent in the Chi-Square Test?

   1. A. The number of observations in the sample
   2. B. The number of variables being tested
   3. C. The number of categories in the variables
   4. D. The number of independent pieces of information available to estimate a parameter

6. What does a larger Chi-Square value typically indicate?

   1. A. Weaker association between variables
   2. B. Stronger association between variables
   3. C. No association between variables
   4. D. Perfect independence between variables

7. What is compared to the calculated Chi-Square value to determine statistical significance?

   1. A. The mean value
   2. B. The median value
   3. C. A critical value from the Chi-Square distribution or a p-value
   4. D. The standard deviation