Chi-Square Test

Definition

The chi-square test is a statistical hypothesis test that is used to determine if there is a significant difference between the observed and expected frequencies in one or more categories. It is a versatile test that can be applied in various contexts, including testing the variance of a single population, evaluating the goodness-of-fit of a distribution, and assessing the independence of two variables.

5 Must Know Facts For Your Next Test

1. The chi-square test statistic is calculated by summing the squared differences between the observed and expected frequencies, divided by the expected frequencies.
2. The chi-square test is used to determine if the variance of a single population is equal to a hypothesized value.
3. The goodness-of-fit chi-square test is used to determine if a sample of data fits a specified probability distribution.
4. The test of independence chi-square test is used to determine if two categorical variables are independent of each other.
5. The comparison of chi-square tests is used to determine if the results of two or more chi-square tests are significantly different from each other.

Review Questions

• Explain how the chi-square test is used to test the variance of a single population.
  • The chi-square test for a single variance is used to determine if the variance of a population is equal to a hypothesized value. The test statistic is calculated by taking the sum of the squared differences between the observed and expected frequencies, divided by the expected frequencies. The resulting chi-square statistic is then compared to a critical value from the chi-square distribution, with the number of degrees of freedom equal to the number of observations minus one. If the test statistic is greater than the critical value, the null hypothesis that the population variance is equal to the hypothesized value is rejected, indicating that the observed variance is significantly different from the expected variance.
• Describe the purpose and process of the goodness-of-fit chi-square test.
  • The goodness-of-fit chi-square test is used to determine if a sample of data fits a specified probability distribution. The test statistic is calculated by taking the sum of the squared differences between the observed and expected frequencies, divided by the expected frequencies. The resulting chi-square statistic is then compared to a critical value from the chi-square distribution, with the number of degrees of freedom equal to the number of categories minus the number of estimated parameters. If the test statistic is greater than the critical value, the null hypothesis that the sample data fits the specified distribution is rejected, indicating that the observed frequencies are significantly different from the expected frequencies.
• Explain the purpose and interpretation of the test of independence chi-square test.
  • The test of independence chi-square test is used to determine if two categorical variables are independent of each other. The test statistic is calculated by taking the sum of the squared differences between the observed and expected frequencies, divided by the expected frequencies. The resulting chi-square statistic is then compared to a critical value from the chi-square distribution, with the number of degrees of freedom equal to the number of rows minus one, multiplied by the number of columns minus one. If the test statistic is greater than the critical value, the null hypothesis that the two variables are independent is rejected, indicating that the observed frequencies are significantly different from the expected frequencies and that the variables are not independent.

Related terms

Hypothesis Testing: The process of using statistical analysis to determine whether a hypothesis about a population parameter is likely to be true or false.

Degrees of Freedom: The number of values in the final calculation of a statistic that are free to vary.

p-value: The probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true.