What are degrees of freedom in chi-square goodness-of-fit tests?

📚 Quick Study Guide

• 📊 Degrees of freedom (df) represent the number of independent pieces of information available to estimate a parameter.
• 🧮 In a chi-square goodness-of-fit test, the formula for degrees of freedom is: $df = k - 1 - c$, where:
  • $k$ = the number of categories or groups.
  • $c$ = the number of estimated parameters in the distribution.
• 🧪 If no parameters are estimated from the sample data (only category counts), then $df = k - 1$.
• 📝 The degrees of freedom influence the shape of the chi-square distribution, which is used to determine the p-value.
• 💡 A higher degrees of freedom generally leads to a chi-square distribution that is less skewed.

Practice Quiz

1. What does 'degrees of freedom' represent in a chi-square goodness-of-fit test?

   1. The total number of observations.
   2. The number of categories in the data.
   3. The number of independent pieces of information available.
   4. The significance level of the test.

2. What is the formula for degrees of freedom (df) in a chi-square goodness-of-fit test when no parameters are estimated?

   1. $df = n - 1$, where n is the total number of observations
   2. $df = k - 1$, where k is the number of categories
   3. $df = k - 2$, where k is the number of categories
   4. $df = n - k$, where n is the number of observations and k is the number of categories

3. In a chi-square goodness-of-fit test with 5 categories, where no parameters are estimated, what is the degrees of freedom?

   1. 6
   2. 5
   3. 4
   4. 3

4. If the degrees of freedom in a chi-square test is 3, and the number of categories is 4, how many parameters were estimated?

   1. 0
   2. 1
   3. 2
   4. 3

5. Which of the following is true about the chi-square distribution and degrees of freedom?

   1. Higher degrees of freedom always increase skewness.
   2. Degrees of freedom have no impact on the shape of the distribution.
   3. Lower degrees of freedom generally lead to a less skewed distribution.
   4. Higher degrees of freedom generally lead to a less skewed distribution.

6. A researcher is performing a chi-square goodness-of-fit test with 6 categories. They also estimate one parameter from the sample. What are the degrees of freedom?

   1. 6
   2. 5
   3. 4
   4. 3

7. A genetics researcher is testing if observed genotype frequencies match expected Mendelian ratios. There are three possible genotypes (AA, Aa, aa). No parameters are estimated. What is the degrees of freedom for the chi-square goodness-of-fit test?

   1. 1
   2. 2
   3. 3
   4. 4