๐ Topic Summary
Goodness-of-fit tests are statistical tests used to determine how well a sample of data fits with a theoretical distribution. The Chi-Square test is a common method. It involves comparing observed frequencies with expected frequencies under the assumed distribution. A large difference between observed and expected frequencies suggests that the data does not fit the distribution well. The null hypothesis typically states that the data follows the specified distribution, and the alternative hypothesis states that it does not.
To perform a goodness-of-fit test, you calculate a test statistic (e.g., the Chi-Square statistic), which measures the discrepancy between the observed and expected values. This statistic is then compared to a critical value from a Chi-Square distribution (or other appropriate distribution) to determine if the null hypothesis should be rejected. The degrees of freedom for the test are typically the number of categories minus the number of parameters estimated from the data, minus 1.
๐งฎ Part A: Vocabulary
Match the terms with their definitions:
| Term |
Definition |
| 1. Observed Frequency |
A. The number of times a category is expected to occur under the null hypothesis. |
| 2. Expected Frequency |
B. A statistical test used to determine if observed sample data matches expected values. |
| 3. Chi-Square Test |
C. The probability of obtaining test results at least as extreme as the results actually observed, assuming that the null hypothesis is correct. |
| 4. P-value |
D. The number of times a category actually occurs in the sample data. |
| 5. Degrees of Freedom |
E. The number of independent pieces of information used to calculate a statistic. |
Answers: 1-D, 2-A, 3-B, 4-C, 5-E
๐ Part B: Fill in the Blanks
The goodness-of-fit test is used to assess whether a sample data set is consistent with a _________ distribution. The _________ hypothesis assumes that there is no significant difference between the observed and expected values. A small _________ value indicates strong evidence against the null hypothesis, suggesting that the data does not fit the specified distribution well. The Chi-Square statistic measures the _________ between observed and expected frequencies. The _________ are used to determine the critical value for the test.
Answers: theoretical, null, p-, discrepancy, degrees of freedom
๐ค Part C: Critical Thinking
Explain in your own words why it's important to check the assumptions of a goodness-of-fit test before interpreting the results. What could happen if the assumptions are violated?