University Statistics Chi-Square Test Questions and Answers

📚 Quick Study Guide

• 🧮 The Chi-Square test is used to determine if there is a statistically significant association between two categorical variables.
• 📊 There are two main types: the Chi-Square test for independence and the Chi-Square goodness-of-fit test.
• 🧪 The Chi-Square test for independence examines whether the distribution of categories in one variable differs based on the categories of the second variable.
• 📈 The Chi-Square goodness-of-fit test assesses whether the observed distribution of a categorical variable matches an expected distribution.
• 📝 The formula for the Chi-Square test statistic is: $ \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 for each category.
• 🔑 Degrees of freedom (df) are calculated differently for each test type: For independence, $ df = (number\ of\ rows - 1) * (number\ of\ columns - 1) $; For goodness-of-fit, $ df = number\ of\ categories - 1 $.
• 🧐 A small p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, suggesting a significant association (independence test) or a poor fit (goodness-of-fit test).

Practice Quiz

1. Which of the following is a primary use of the Chi-Square test?
   1. A. Determining the correlation between two continuous variables.
   2. B. Assessing the association between two categorical variables.
   3. C. Comparing the means of two groups.
   4. D. Predicting future values based on past data.
2. What does a Chi-Square test for independence assess?
   1. A. Whether two variables are linearly related.
   2. B. Whether the distributions of categories in two variables are related.
   3. C. Whether a single variable follows a normal distribution.
   4. D. Whether the means of two groups are equal.
3. In the Chi-Square formula, what does $O_i$ represent?
   1. A. Expected frequency.
   2. B. Observed frequency.
   3. C. Total frequency.
   4. D. Relative frequency.
4. For a Chi-Square test of independence in a 3x4 contingency table, what are the degrees of freedom?
   1. A. 6
   2. B. 7
   3. C. 8
   4. D. 12
5. What does a small p-value (e.g., p ≤ 0.05) in a Chi-Square test typically indicate?
   1. A. Strong evidence in favor of the null hypothesis.
   2. B. Strong evidence against the null hypothesis.
   3. C. No relationship between the variables.
   4. D. A perfect fit between observed and expected values.
6. Which test assesses if the observed distribution of a categorical variable matches an expected distribution?
   1. A. T-test
   2. B. ANOVA
   3. C. Chi-Square goodness-of-fit test
   4. D. Regression analysis
7. If the calculated Chi-Square statistic is 0, what can you conclude?
   1. A. There is a perfect match between observed and expected frequencies.
   2. B. There is a strong association between the variables.
   3. C. The sample size is too small.
   4. D. The null hypothesis is false.