Chi-Square Goodness-of-Fit Test

Definition

The chi-square goodness-of-fit test is a statistical test used to determine if a sample of data fits a particular probability distribution. It is commonly used to evaluate whether the observed frequencies of categorical data match the expected frequencies under a hypothesized model.

5 Must Know Facts For Your Next Test

1. The chi-square goodness-of-fit test is used to evaluate the fit between observed and expected frequencies for categorical data.
2. The test statistic for the chi-square goodness-of-fit test is calculated by summing the squared differences between observed and expected frequencies, divided by the expected frequencies.
3. If the p-value from the chi-square goodness-of-fit test is less than the chosen significance level (e.g., 0.05), the null hypothesis is rejected, indicating that the observed data do not fit the expected distribution.
4. The chi-square goodness-of-fit test assumes that the expected frequencies are known or can be calculated from the hypothesized distribution.
5. The test is sensitive to sample size, and larger samples tend to have more power to detect deviations from the hypothesized distribution.

Review Questions

• Explain the purpose of the chi-square goodness-of-fit test and the null hypothesis it examines.
  • The chi-square goodness-of-fit test is used to determine whether a sample of data follows a particular probability distribution. The null hypothesis for this test states that the observed frequencies of the categorical data match the expected frequencies under the hypothesized distribution. By comparing the observed and expected frequencies, the test can assess whether the sample data is consistent with the proposed model or distribution.
• Describe the relationship between the chi-square test statistic, degrees of freedom, and the p-value in the context of the goodness-of-fit test.
  • The chi-square test statistic is calculated by summing the squared differences between observed and expected frequencies, divided by the expected frequencies. The degrees of freedom for the test are equal to the number of categories minus 1, as one degree of freedom is lost when estimating the expected frequencies. The p-value represents the probability of obtaining a test statistic as extreme or more extreme than the observed value, given that the null hypothesis is true. If the p-value is less than the chosen significance level, the null hypothesis is rejected, indicating that the observed data do not fit the expected distribution.
• Discuss how the sample size and the assumptions of the chi-square goodness-of-fit test can impact the interpretation and validity of the results.
  • The chi-square goodness-of-fit test is sensitive to sample size, with larger samples tending to have more power to detect deviations from the hypothesized distribution. Additionally, the test assumes that the expected frequencies are known or can be calculated from the proposed distribution. If these assumptions are violated, the validity and interpretation of the test results may be compromised. Researchers must carefully consider the sample size and ensure that the necessary assumptions are met when conducting and interpreting the chi-square goodness-of-fit test.

Related terms

Null Hypothesis: The null hypothesis in a chi-square goodness-of-fit test states that the observed frequencies follow the expected distribution.

Degrees of Freedom: The degrees of freedom in a chi-square goodness-of-fit test are equal to the number of categories minus 1, as one degree of freedom is lost when estimating the expected frequencies.

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