Solved Problem: Chi-Squared Goodness-of-Fit Test Interpretation

📚 Quick Study Guide

• 📊 Purpose: The Chi-Squared Goodness-of-Fit test determines if sample data matches a population.
• 📝 Null Hypothesis ($H_0$): The observed data follows the specified distribution.
• 🧪 Alternative Hypothesis ($H_1$): The observed data does not follow the specified distribution.
• 🔢 Test Statistic: $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$, where $O_i$ is the observed frequency and $E_i$ is the expected frequency.
• 🎓 Degrees of Freedom (df): $df = k - 1 - c$, where $k$ is the number of categories and $c$ is the number of estimated parameters.
• 🌍 P-value: The probability of obtaining a test statistic as extreme as, or more extreme than, the one actually observed, assuming the null hypothesis is true.
• 💡 Decision Rule: If the p-value is less than the significance level ($\alpha$), reject the null hypothesis.

Practice Quiz

1. Which hypothesis does the Chi-Squared Goodness-of-Fit test primarily assess?
   1. The equality of two population means.
   2. Whether sample data fits a hypothesized distribution.
   3. The correlation between two variables.
   4. The independence of two categorical variables.
2. What is the formula for the Chi-Squared test statistic in a Goodness-of-Fit test?
   1. $\sum (O_i - E_i)$
   2. $\sum \frac{(O_i - E_i)}{E_i}$
   3. $\sum \frac{(O_i - E_i)^2}{E_i}$
   4. $\sum (O_i - E_i)^2$
3. What does $O_i$ represent in the Chi-Squared Goodness-of-Fit formula?
   1. Expected frequency of category i.
   2. Observed frequency of category i.
   3. The mean of category i.
   4. The standard deviation of category i.
4. How are degrees of freedom calculated in a Goodness-of-Fit test?
   1. Number of categories + 1.
   2. Number of categories - 1.
   3. Number of observations - 1.
   4. Number of categories - number of parameters estimated.
5. If the p-value is 0.03 and the significance level is 0.05, what decision should be made?
   1. Accept the null hypothesis.
   2. Reject the null hypothesis.
   3. Fail to reject the null hypothesis.
   4. Increase the sample size.
6. In a Chi-Squared Goodness-of-Fit test for a six-sided die, what is the expected frequency for each side if the die is fair and rolled 60 times?
   1. 6
   2. 10
   3. 12
   4. 60
7. What does a large Chi-Squared test statistic suggest?
   1. A good fit between observed and expected values.
   2. A poor fit between observed and expected values.
   3. That the sample size is too small.
   4. That the significance level is too high.