How to Perform a One-Sample Chi-Square Test for Variance

📚 Quick Study Guide

• 📐 Purpose: The One-Sample Chi-Square Test for Variance determines if the variance of a population matches a specified value.
• 📊 Null Hypothesis ($H_0$): The population variance equals the hypothesized variance ($\sigma^2 = \sigma_0^2$).
• 📈 Alternative Hypothesis ($H_1$): The population variance is not equal to, less than, or greater than the hypothesized variance ($\sigma^2 \neq \sigma_0^2$, $\sigma^2 < \sigma_0^2$, or $\sigma^2 > \sigma_0^2$).
• 🧮 Test Statistic: Calculated as $\chi^2 = \frac{(n-1)s^2}{\sigma_0^2}$, where $n$ is the sample size, $s^2$ is the sample variance, and $\sigma_0^2$ is the hypothesized variance.
• 🎓 Degrees of Freedom: $df = n - 1$
• 📊 Decision Rule: Reject $H_0$ if the calculated $\chi^2$ value falls in the critical region based on the chosen significance level ($\alpha$) and degrees of freedom.
• 💡 Assumptions: The population from which the sample is drawn should be normally distributed.

Practice Quiz

1. What is the primary purpose of the One-Sample Chi-Square Test for Variance?
   1. A. To compare the means of two samples.
   2. B. To test if the variance of a population matches a specified value.
   3. C. To test if a sample is normally distributed.
   4. D. To compare the medians of two samples.
2. What is the null hypothesis ($H_0$) in a One-Sample Chi-Square Test for Variance?
   1. A. $\sigma^2 \neq \sigma_0^2$
   2. B. $\sigma^2 > \sigma_0^2$
   3. C. $\sigma^2 = \sigma_0^2$
   4. D. $\sigma^2 < \sigma_0^2$
3. What is the formula for the test statistic in a One-Sample Chi-Square Test for Variance?
   1. A. $\chi^2 = \frac{(n)s^2}{\sigma_0^2}$
   2. B. $\chi^2 = \frac{(n-1)s}{\sigma_0^2}$
   3. C. $\chi^2 = \frac{(n-1)s^2}{\sigma_0^2}$
   4. D. $\chi^2 = \frac{(n+1)s^2}{\sigma_0^2}$
4. What does 'n' represent in the formula for the Chi-Square test statistic?
   1. A. Population size
   2. B. Sample mean
   3. C. Sample size
   4. D. Population variance
5. What are the degrees of freedom (df) for a One-Sample Chi-Square Test for Variance?
   1. A. $n$
   2. B. $n + 1$
   3. C. $n - 1$
   4. D. $n - 2$
6. What is a critical assumption for the One-Sample Chi-Square Test for Variance?
   1. A. The population is uniformly distributed.
   2. B. The population is normally distributed.
   3. C. The population is exponentially distributed.
   4. D. The population is binomially distributed.
7. If the calculated $\chi^2$ value falls in the critical region, what decision do you make?
   1. A. Accept the null hypothesis.
   2. B. Fail to reject the null hypothesis.
   3. C. Reject the null hypothesis.
   4. D. Increase the sample size.