Chi-Square Test

Definition

The chi-square test is a statistical hypothesis test used to determine if there is a significant difference between observed and expected frequencies or proportions in one or more categories. It is a versatile test that can be applied in various contexts, including contingency tables, discrete distributions, and tests of independence or variance.

5 Must Know Facts For Your Next Test

1. The chi-square test is used to determine if there is a significant association or relationship between two categorical variables.
2. The test statistic for the chi-square test is calculated as the sum of the squared differences between observed and expected frequencies, divided by the expected frequencies.
3. The chi-square test follows a chi-square distribution, with the number of degrees of freedom depending on the number of categories or variables being tested.
4. A small p-value (typically less than the chosen significance level, such as 0.05) indicates that the observed data is unlikely to have occurred by chance under the null hypothesis, and the null hypothesis is rejected.
5. The chi-square test can be used to compare two independent population proportions, test the independence of two categorical variables, and test the variance of a single population.

Review Questions

• Explain how the chi-square test is used in the context of contingency tables.
  • In the context of contingency tables, the chi-square test is used to determine if there is a significant association between two categorical variables. The test compares the observed frequencies in each cell of the contingency table to the expected frequencies under the null hypothesis of no association. The test statistic is calculated as the sum of the squared differences between observed and expected frequencies, divided by the expected frequencies. A small p-value indicates that the observed data is unlikely to have occurred by chance if the null hypothesis is true, and the null hypothesis of no association is rejected.
• Describe how the chi-square test can be used to analyze a discrete distribution, such as the playing card experiment.
  • In the context of a discrete distribution, such as the playing card experiment, the chi-square test can be used to determine if the observed frequencies of the different card suits or values differ significantly from the expected frequencies under the null hypothesis of a uniform distribution. The test statistic is calculated as the sum of the squared differences between observed and expected frequencies, divided by the expected frequencies. A small p-value indicates that the observed data is unlikely to have occurred by chance if the null hypothesis is true, and the null hypothesis of a uniform distribution is rejected.
• Discuss how the chi-square test is used to test the independence of two categorical variables.
  • The chi-square test of independence is used to determine if two categorical variables are independent or if there is a significant association between them. The test compares the observed frequencies in the cells of a contingency table to the expected frequencies under the null hypothesis of independence. The test statistic is calculated as the sum of the squared differences between observed and expected frequencies, divided by the expected frequencies. A small p-value indicates that the observed data is unlikely to have occurred by chance if the null hypothesis of independence is true, and the null hypothesis is rejected, concluding that the two variables are not independent.

Related terms

Null Hypothesis: The null hypothesis represents the assumption of no significant difference or relationship between the variables being tested.

Test Statistic: The test statistic is a numerical value calculated from the observed data that is used to determine the probability of the observed results under the null hypothesis.

P-value: The p-value is the probability of obtaining a test statistic as extreme or more extreme than the observed value, given that the null hypothesis is true.