Chi-Square Goodness-of-Fit vs. Chi-Square Test of Independence: Key Differences

📚 Quick Study Guide

• 🔍 Chi-Square Goodness-of-Fit: Tests if an observed frequency distribution matches an expected distribution. It assesses how well sample data fits a hypothesized distribution.
• 🧪 Null Hypothesis (Goodness-of-Fit): The observed distribution fits the expected distribution.
• 📊 Test Statistic (Goodness-of-Fit): $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$, where $O_i$ is the observed frequency and $E_i$ is the expected frequency.
• 🔢 Degrees of Freedom (Goodness-of-Fit): $df = k - 1$, where $k$ is the number of categories.
• 🤝 Chi-Square Test of Independence: Tests if two categorical variables are independent. It examines whether the distribution of one variable differs based on the values of another variable.
• 🧬 Null Hypothesis (Independence): The two variables are independent.
• 📈 Test Statistic (Independence): $\chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}}$, where $O_{ij}$ is the observed frequency in cell (i, j) and $E_{ij} = \frac{(Row Total \times Column Total)}{Grand Total}$.
• 🌍 Degrees of Freedom (Independence): $df = (r - 1)(c - 1)$, where $r$ is the number of rows and $c$ is the number of columns in the contingency table.

Practice Quiz

1. Which test determines if observed sample data matches an expected distribution?
   1. Chi-Square Test of Independence
   2. Chi-Square Goodness-of-Fit
   3. T-test
   4. ANOVA
2. What is the null hypothesis for the Chi-Square Test of Independence?
   1. The variables are dependent.
   2. The variables are related.
   3. The variables are independent.
   4. There is no relationship between the variables.
3. The degrees of freedom for a Chi-Square Goodness-of-Fit test with 5 categories is:
   1. 5
   2. 6
   3. 4
   4. 25
4. The degrees of freedom for a Chi-Square Test of Independence in a 3x2 contingency table is:
   1. 6
   2. 5
   3. 2
   4. 4
5. What does a high Chi-Square statistic generally indicate?
   1. A good fit between observed and expected values.
   2. Strong evidence against the null hypothesis.
   3. The variables are independent.
   4. The sample size is too small.
6. Which test uses a contingency table?
   1. Chi-Square Goodness-of-Fit
   2. Chi-Square Test of Independence
   3. Both
   4. Neither
7. What is the primary difference between the two tests in terms of data type?
   1. Goodness-of-fit uses continuous data, while independence uses categorical.
   2. Independence uses continuous data, while goodness-of-fit uses categorical.
   3. Both use only categorical data.
   4. Both use only continuous data.