What are the differences between the Chi-Square Test for Independence and the Chi-Square Test for Homogeneity?
Independence: One sample, tests for association between two categorical variables. | Homogeneity: Multiple samples, tests if the distribution of a categorical variable is the same across populations.
What are the differences between Observed Frequency and Expected Frequency?
Observed Frequency: Actual counts from the sample data. | Expected Frequency: Counts expected if the null hypothesis is true (no association).
What are the differences between rejecting and failing to reject the null hypothesis?
Rejecting H0: There is statistically significant evidence for the alternative hypothesis. | Failing to Reject H0: There is not enough evidence to support the alternative hypothesis.
What is the formula for the Chi-Square test statistic?
\( \chi^2 = \sum \frac{(O - E)^2}{E} \)
What is the formula for Expected Frequency (E)?
\( E = \frac{(\text{row total}) \times (\text{column total})}{\text{grand total}} \)
What is the formula for Degrees of Freedom (df)?
\( df = (\text{number of rows} - 1) \times (\text{number of columns} - 1) \)
Explain the concept of the Chi-Square Test for Independence.
It determines if there is a relationship between two categorical variables within a single population. It compares observed data to expected data assuming no relationship.
Explain the concept of the Chi-Square Test for Homogeneity.
It determines if the distribution of a categorical variable is the same across multiple populations. It compares the distribution of data across these populations.
Explain the 'Large Counts' condition for Chi-Square tests.
All expected counts must be at least 5. This ensures that the chi-square distribution is a good approximation for the test statistic's distribution.
Explain the meaning of a small p-value in a Chi-Square test.
A small p-value (typically less than 0.05) suggests that the observed data is unlikely if the null hypothesis were true, leading us to reject H0.
Explain how to draw a conclusion from a Chi-Square Test.
Compare the p-value to the significance level (alpha). If p-value ≤ alpha, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. State the conclusion in the context of the problem.
