What is the Goodness-of-Fit Test? Definition for University Statistics

📚 What is the Goodness-of-Fit Test?

The Goodness-of-Fit test is a statistical hypothesis test used to determine how well a sample data fits with a theoretical distribution. In simpler terms, it checks if your observed data matches what you'd expect to see based on a specific model or assumption. Here’s a quick rundown:

Quick Study Guide

• 💡 Purpose: To assess if observed data is consistent with a hypothesized distribution.
• 📊 Hypotheses:
  • $H_0$: The data follows the specified distribution.
  • $H_1$: The data does not follow the specified distribution.
• 🧪 Common Tests: Chi-Square Goodness-of-Fit test is the most common.
• 🔢 Test Statistic (Chi-Square): $$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$$
  • $O_i$ = Observed frequency
  • $E_i$ = Expected frequency
• 📈 Degrees of Freedom: $df = k - p - 1$, where:
  • $k$ = Number of categories
  • $p$ = Number of estimated parameters from the data
• ✅ Decision Rule: Reject $H_0$ if the calculated $\chi^2$ value is greater than the critical value from the Chi-Square distribution table at a chosen significance level ($\alpha$).

Practice Quiz

1. Question 1: What is the primary purpose of the Goodness-of-Fit test?
   1. A. To estimate population parameters.
   2. B. To determine how well sample data fits a theoretical distribution.
   3. C. To calculate the mean of a dataset.
   4. D. To compare two different datasets.
2. Question 2: Which of the following is the null hypothesis ($H_0$) in a Goodness-of-Fit test?
   1. A. The data does not follow any distribution.
   2. B. The data follows the specified distribution.
   3. C. The data is normally distributed.
   4. D. The data is uniformly distributed.
3. Question 3: What is the most commonly used test for assessing Goodness-of-Fit?
   1. A. T-test
   2. B. Z-test
   3. C. Chi-Square test
   4. D. ANOVA
4. Question 4: In the Chi-Square Goodness-of-Fit test, what does $O_i$ represent?
   1. A. Expected frequency
   2. B. Observed frequency
   3. C. Total frequency
   4. D. Relative frequency
5. Question 5: In the Chi-Square Goodness-of-Fit test, what does $E_i$ represent?
   1. A. Observed frequency
   2. B. Expected frequency
   3. C. Total frequency
   4. D. Relative frequency
6. Question 6: What is the formula for calculating the degrees of freedom ($df$) in a Chi-Square Goodness-of-Fit test, where $k$ is the number of categories and $p$ is the number of estimated parameters?
   1. A. $df = k + p - 1$
   2. B. $df = k - p + 1$
   3. C. $df = k - p - 1$
   4. D. $df = k + p + 1$
7. Question 7: If the calculated Chi-Square value is greater than the critical value at a chosen significance level, what decision should be made?
   1. A. Accept the null hypothesis.
   2. B. Fail to reject the null hypothesis.
   3. C. Reject the null hypothesis.
   4. D. Increase the significance level.