Steps to Set Up a Chi-Square Goodness-of-Fit Test: Hypotheses & Goal

📚 Quick Study Guide

• 📊 The Chi-Square Goodness-of-Fit test assesses if observed data matches expected values.
• 🤔 Null Hypothesis (H₀): The observed data follows the specified distribution.
• 🎯 Alternative Hypothesis (H₁): The observed data does not follow the specified distribution.
• 📐 The test statistic is calculated as: $ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} $, where $O_i$ is the observed frequency and $E_i$ is the expected frequency for each category.
• 🔑 The goal is to determine if the differences between observed and expected frequencies are statistically significant.
• ⚙️ Degrees of freedom (df) = Number of categories - Number of estimated parameters - 1.
• ⚖️ Compare the calculated $ \chi^2 $ value to a critical value from the Chi-Square distribution table to make a decision.

Practice Quiz

1. What is the primary goal of a Chi-Square Goodness-of-Fit test?

   1. To estimate population parameters.
   2. To determine if observed data fits an expected distribution.
   3. To compare the means of two groups.
   4. To measure the correlation between two variables.

2. Which of the following is the correct null hypothesis (H₀) for a Chi-Square Goodness-of-Fit test?

   1. The observed data does not follow the specified distribution.
   2. The observed data is significantly different from the expected data.
   3. The observed data follows the specified distribution.
   4. There is no relationship between the observed and expected data.

3. What does $O_i$ represent in the Chi-Square test statistic formula $ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} $?

   1. Expected frequency.
   2. Observed frequency.
   3. Critical value.
   4. Degrees of freedom.

4. What does $E_i$ represent in the Chi-Square test statistic formula $ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} $?

   1. Observed frequency.
   2. Test statistic.
   3. Expected frequency.
   4. P-value.

5. What is the alternative hypothesis (H₁) in a Chi-Square Goodness-of-Fit test?

   1. The observed data follows the specified distribution.
   2. The observed data is equal to the expected data.
   3. The observed data does not follow the specified distribution.
   4. There is no difference between the observed and expected data.

6. How are degrees of freedom (df) calculated in a Chi-Square Goodness-of-Fit test?

   1. Number of categories + 1.
   2. Number of categories.
   3. Number of categories - Number of estimated parameters - 1.
   4. Number of observations - 1.

7. What do you compare the calculated Chi-Square statistic to in order to make a decision?

   1. The mean.
   2. The standard deviation.
   3. A critical value from the Chi-Square distribution table.
   4. The p-value of a t-test.