Performing a Chi-Square Goodness-of-Fit Test: Manual Calculation Tutorial

📚 Quick Study Guide

• 🔍 The Chi-Square Goodness-of-Fit test determines if sample data matches a population distribution.
• 📝 Null Hypothesis ($H_0$): The observed data fits the expected distribution.
• 🧪 Alternative Hypothesis ($H_1$): The observed data does not fit the expected distribution.
• 🔢 The Chi-Square 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.
• 📊 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 determine significance.
• 💡 If the calculated $\chi^2$ value is greater than the critical value, reject the null hypothesis.

Practice Quiz

1. Question 1: What is the primary purpose of the Chi-Square Goodness-of-Fit test?
   1. A) To compare means of two samples.
   2. B) To determine if observed data fits an expected distribution.
   3. C) To measure the correlation between two variables.
   4. D) To estimate population parameters.
2. Question 2: Which of the following is the correct formula for the Chi-Square test statistic?
   1. A) $\chi^2 = \sum (O_i - E_i)$
   2. B) $\chi^2 = \sum \frac{(O_i - E_i)}{E_i}$
   3. C) $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$
   4. D) $\chi^2 = \sum \frac{(E_i - O_i)^2}{O_i}$
3. Question 3: What does $O_i$ represent in the Chi-Square formula?
   1. A) Expected frequency.
   2. B) Observed frequency.
   3. C) Critical value.
   4. D) Degrees of freedom.
4. Question 4: How are degrees of freedom calculated in a Goodness-of-Fit test?
   1. A) Number of categories + 1.
   2. B) Number of categories - 1.
   3. C) Number of categories - Number of estimated parameters.
   4. D) Number of categories - Number of estimated parameters - 1.
5. Question 5: If the calculated Chi-Square value is greater than the critical value, what decision should be made?
   1. A) Accept the null hypothesis.
   2. B) Reject the null hypothesis.
   3. C) Increase the sample size.
   4. D) Decrease the significance level.
6. Question 6: In a genetics experiment, you expect a 3:1 phenotypic ratio. What distribution is being tested?
   1. A) Normal distribution
   2. B) Binomial distribution
   3. C) Expected distribution
   4. D) Poisson distribution
7. Question 7: Which of the following is NOT a condition for the Chi-Square Goodness-of-Fit test to be valid?
   1. A) Data must be categorical.
   2. B) Expected frequencies must be sufficiently large (usually at least 5).
   3. C) Observations must be independent.
   4. D) Data must follow a normal distribution.