key term - Test of Independence

Definition

A test of independence is a statistical hypothesis test used to determine whether two categorical variables are independent or related. It examines if the observed frequencies in a contingency table differ significantly from the expected frequencies under the assumption of independence.

5 Must Know Facts For Your Next Test

1. The test of independence is used to determine if two categorical variables are independent or related, based on the data in a contingency table.
2. The null hypothesis for the test of independence states that the two variables are independent, meaning the observed frequencies in the contingency table match the expected frequencies under the assumption of independence.
3. The test statistic for the test of independence is the chi-square statistic, which measures the discrepancy between the observed and expected frequencies.
4. The p-value from the test of independence represents the probability of observing the given test statistic (or a more extreme value) if the null hypothesis is true.
5. If the p-value is less than the chosen significance level (typically 0.05), the null hypothesis is rejected, indicating that the two variables are not independent and are related.

Review Questions

• Explain the purpose of the test of independence and how it is used to analyze the relationship between two categorical variables.
  • The test of independence is used to determine whether two categorical variables are independent or related. It examines if the observed frequencies in a contingency table, which displays the joint distribution of the two variables, differ significantly from the expected frequencies under the assumption of independence. The null hypothesis states that the two variables are independent, and the test statistic used is the chi-square statistic, which measures the discrepancy between the observed and expected frequencies. If the p-value from the test is less than the chosen significance level, the null hypothesis is rejected, indicating that the two variables are not independent and are related.
• Describe the role of the contingency table in the test of independence and how it is used to calculate the expected frequencies.
  • The contingency table is central to the test of independence, as it displays the frequency distribution of the two categorical variables being analyzed. The expected frequencies under the assumption of independence are calculated based on the marginal totals of the contingency table. Specifically, the expected frequency for each cell is the product of the corresponding row and column totals divided by the total number of observations. The chi-square statistic is then calculated by comparing the observed frequencies in the contingency table to these expected frequencies, providing a measure of the discrepancy between the two.
• Explain the interpretation of the p-value from the test of independence and how it is used to draw conclusions about the relationship between the variables.
  • The p-value from the test of independence represents the probability of observing the given test statistic (or a more extreme value) if the null hypothesis of independence is true. If the p-value is less than the chosen significance level, typically 0.05, the null hypothesis is rejected, indicating that the two variables are not independent and are related. In other words, the low p-value suggests that the observed frequencies in the contingency table are unlikely to have occurred by chance if the variables were truly independent. This allows the researcher to conclude that the two variables are associated and that there is a significant relationship between them.

Related terms

Contingency Table:

A table that displays the frequency distribution of two categorical variables, allowing for the analysis of the relationship between them.

Chi-Square Test:

A statistical test used to determine if there is a significant difference between the observed and expected frequencies in one or more categories.

Null Hypothesis:

The hypothesis that there is no relationship or difference between the variables being tested, which is assumed to be true until proven otherwise.