Steps to make a decision in a chi-square test of independence

📚 Quick Study Guide

🔍 The Chi-Square Test of Independence determines if there is a statistically significant association between two categorical variables. 🧪 Null Hypothesis ($H_0$): The two variables are independent. 📈 Alternative Hypothesis ($H_1$): The two variables are dependent. 🔢 Calculate the expected frequencies ($E$) for each cell in the contingency table: $E = \frac{(\text{Row Total}) \times (\text{Column Total})}{\text{Grand Total}}$ 📊 Calculate the Chi-Square test statistic: $\chi^2 = \sum \frac{(O - E)^2}{E}$, where $O$ is the observed frequency and $E$ is the expected frequency. 📐 Determine the degrees of freedom: $df = (r - 1)(c - 1)$, where $r$ is the number of rows and $c$ is the number of columns in the contingency table. 📊 Compare the calculated $\chi^2$ value to the critical value from the Chi-Square distribution table, or calculate the p-value. ✅ If the $\chi^2$ value is greater than the critical value (or the p-value is less than the significance level $\alpha$), reject the null hypothesis.

Practice Quiz

1. What is the primary purpose of the Chi-Square test of independence?
   1. A. To determine the mean of a population.
   2. B. To assess the association between two categorical variables.
   3. C. To compare the variances of two populations.
   4. D. To predict future values based on past data.
2. What is the null hypothesis ($H_0$) in a Chi-Square test of independence?
   1. A. The two variables are dependent.
   2. B. The two variables are related.
   3. C. The two variables are independent.
   4. D. There is no relationship between the variables.
3. How are the expected frequencies ($E$) calculated in a Chi-Square test of independence?
   1. A. $E = \frac{(\text{Row Total}) + (\text{Column Total})}{\text{Grand Total}}$
   2. B. $E = (\text{Row Total}) \times (\text{Column Total}) \times (\text{Grand Total})$
   3. C. $E = \frac{(\text{Row Total}) \times (\text{Column Total})}{\text{Grand Total}}$
   4. D. $E = \frac{\text{Grand Total}}{(\text{Row Total}) \times (\text{Column Total})}$
4. What does 'O' represent in the Chi-Square test statistic formula $\chi^2 = \sum \frac{(O - E)^2}{E}$?
   1. A. Expected frequency
   2. B. Observed frequency
   3. C. Degrees of freedom
   4. D. Significance level
5. How are the degrees of freedom ($df$) calculated in a Chi-Square test of independence?
   1. A. $df = r + c - 1$
   2. B. $df = (r - 1) + (c - 1)$
   3. C. $df = (r - 1)(c - 1)$
   4. D. $df = r \times c$
6. If the calculated Chi-Square value is greater than the critical value, what decision should be made?
   1. A. Accept the null hypothesis.
   2. B. Reject the null hypothesis.
   3. C. Fail to reject the null hypothesis.
   4. D. Increase the significance level.
7. What is the significance level ($\alpha$) used for in a Chi-Square test?
   1. A. To determine the sample size.
   2. B. To define the critical value.
   3. C. To determine the probability of making a Type I error.
   4. D. All of the above.