Chi-Square Test Statistic

Definition

The chi-square test statistic is a statistical measure used to determine the goodness-of-fit between an observed set of data and an expected set of data. It is a fundamental concept in hypothesis testing that helps assess whether the differences between observed and expected frequencies are statistically significant.

5 Must Know Facts For Your Next Test

1. The chi-square test statistic is calculated by summing the squared differences between observed and expected frequencies, divided by the expected frequencies.
2. The chi-square test statistic follows a chi-square distribution, with the number of degrees of freedom determined by the number of categories in the data.
3. A larger chi-square test statistic indicates a greater difference between the observed and expected frequencies, suggesting the data does not fit the expected distribution or proportions.
4. The p-value associated with the chi-square test statistic is used to determine the statistical significance of the observed differences, with a smaller p-value indicating stronger evidence against the null hypothesis.
5. The chi-square test for goodness-of-fit is used to determine if a sample of data fits a particular distribution, while the chi-square test for homogeneity is used to determine if two or more samples come from populations with the same distribution or proportions.

Review Questions

• Explain how the chi-square test statistic is calculated and what it represents.
  • The chi-square test statistic is calculated by summing the squared differences between the observed and expected frequencies, divided by the expected frequencies. This formula measures the discrepancy between the observed data and the expected data under a null hypothesis. A larger chi-square test statistic indicates a greater difference between the observed and expected frequencies, suggesting the data does not fit the expected distribution or proportions.
• Describe the relationship between the chi-square test statistic, degrees of freedom, and the p-value.
  • The chi-square test statistic follows a chi-square distribution, with the number of degrees of freedom determined by the number of categories in the data. The p-value associated with the chi-square test statistic is used to determine the statistical significance of the observed differences, with a smaller p-value indicating stronger evidence against the null hypothesis. The degrees of freedom, along with the calculated chi-square test statistic, are used to look up the corresponding p-value in a chi-square distribution table, which then informs the decision to reject or fail to reject the null hypothesis.
• Distinguish between the use of the chi-square test for goodness-of-fit and the chi-square test for homogeneity, and explain how the interpretation of the results differs between these two tests.
  • The chi-square test for goodness-of-fit is used to determine if a sample of data fits a particular distribution, while the chi-square test for homogeneity is used to determine if two or more samples come from populations with the same distribution or proportions. For the goodness-of-fit test, the null hypothesis is that the data follows the expected distribution, and a significant p-value would lead to the rejection of this null hypothesis. For the test of homogeneity, the null hypothesis is that the samples come from populations with the same distribution or proportions, and a significant p-value would indicate that the samples are not homogeneous and come from different populations.

Related terms

Goodness-of-Fit Test:

A statistical test used to determine if a sample of data fits a particular distribution or set of theoretical expectations.

Test for Homogeneity:

A statistical test used to determine if two or more samples come from populations with the same distribution or proportions.

Degrees of Freedom:

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