Understanding decision making in tests of independence (statistics)

📚 Quick Study Guide

• 📊 A test of independence determines whether there is an association between two categorical variables.
• 📝 The null hypothesis ($H_0$) states that the two variables are independent.
• 🧪 The alternative hypothesis ($H_1$) states that the two variables are dependent.
• 🔢 The test statistic is calculated using the formula: $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$, where $O_i$ is the observed frequency and $E_i$ is the expected frequency.
• 📈 The expected frequency is calculated as: $E_{ij} = \frac{(\text{row total}) \times (\text{column total})}{\text{grand total}}$.
• ⚙️ Degrees of freedom (df) are calculated as: $df = (r - 1)(c - 1)$, where $r$ is the number of rows and $c$ is the number of columns in the contingency table.
• 💡 Compare the calculated $\chi^2$ value with the critical value from the chi-square distribution table at a chosen significance level (e.g., $\alpha = 0.05$).
• ✅ If the calculated $\chi^2$ value is greater than the critical value, reject the null hypothesis.

Practice Quiz

1. Question 1: What is the null hypothesis in a test of independence?
   1. A) The two variables are dependent.
   2. B) The two variables are independent.
   3. C) There is no relationship between the variables.
   4. D) The variables are correlated.
2. Question 2: How are degrees of freedom calculated in a test of independence?
   1. A) $df = r + c - 1$
   2. B) $df = (r - 1)(c - 1)$
   3. C) $df = n - 1$
   4. D) $df = r \times c$
3. Question 3: What does a large $\chi^2$ value suggest in a test of independence?
   1. A) Strong evidence against the null hypothesis.
   2. B) Strong evidence in favor of the null hypothesis.
   3. C) No relationship between the variables.
   4. D) The sample size is too small.
4. Question 4: If the p-value is less than the significance level ($\alpha$), what decision should be made?
   1. A) Accept the null hypothesis.
   2. B) Reject the null hypothesis.
   3. C) Increase the sample size.
   4. D) Perform a different test.
5. Question 5: In a contingency table, what does $E_{ij}$ represent?
   1. A) The observed frequency.
   2. B) The expected frequency.
   3. C) The row total.
   4. D) The column total.
6. Question 6: Which of the following is an assumption for the Chi-Square test of independence?
   1. A) Data must be normally distributed.
   2. B) Expected cell counts should be at least 5.
   3. C) Data must be continuous.
   4. D) Variances of the populations must be equal.
7. Question 7: What type of data is used in a test of independence?
   1. A) Continuous data
   2. B) Numerical data
   3. C) Categorical data
   4. D) Ordinal data