How to Perform a Two-Sample F-Test Step-by-Step Guide

📚 Quick Study Guide

🔍 The Two-Sample F-Test is used to compare the variances of two independent samples. 🧪 The null hypothesis ($H_0$) states that the variances are equal ($\sigma_1^2 = \sigma_2^2$). 📊 The alternative hypothesis ($H_1$) states that the variances are not equal ($\sigma_1^2 \neq \sigma_2^2$). 🔢 The F-statistic is calculated as: $F = \frac{s_1^2}{s_2^2}$, where $s_1^2$ and $s_2^2$ are the sample variances. 💡 Degrees of freedom are calculated as $df_1 = n_1 - 1$ and $df_2 = n_2 - 1$, where $n_1$ and $n_2$ are the sample sizes. 📈 Compare the calculated F-statistic to the F-distribution table to determine the p-value. A small p-value (typically < 0.05) indicates that the variances are significantly different. 📝 If $F > 1$, then $s_1^2$ is placed in the numerator, otherwise $s_2^2$ is placed in the numerator. This ensures a right-tailed test.

Practice Quiz

1. Which of the following is the primary purpose of the Two-Sample F-Test?
   1. A) To compare the means of two independent samples.
   2. B) To compare the variances of two independent samples.
   3. C) To determine correlation between two variables.
   4. D) To test for normality in a single sample.
2. What is the null hypothesis ($H_0$) in a Two-Sample F-Test?
   1. A) The means of the two populations are equal.
   2. B) The variances of the two populations are equal.
   3. C) There is no difference between the two samples.
   4. D) The standard deviations of the two populations are unequal.
3. The F-statistic is calculated as which of the following?
   1. A) $F = \frac{s_1}{s_2}$
   2. B) $F = \frac{s_1^2}{s_2^2}$
   3. C) $F = \frac{\bar{x_1}}{\bar{x_2}}$
   4. D) $F = \frac{s_1^2 + s_2^2}{2}$
4. What are the degrees of freedom for the F-statistic, given sample sizes $n_1 = 10$ and $n_2 = 15$?
   1. A) $df_1 = 10, df_2 = 15$
   2. B) $df_1 = 9, df_2 = 14$
   3. C) $df_1 = 11, df_2 = 16$
   4. D) $df_1 = 15, df_2 = 10$
5. A small p-value (e.g., < 0.05) in a Two-Sample F-Test indicates which of the following?
   1. A) The variances are equal.
   2. B) The variances are significantly different.
   3. C) The means are equal.
   4. D) The means are significantly different.
6. If the calculated F-statistic is 2.5 and the critical F-value is 2.0, what is the correct conclusion?
   1. A) Fail to reject the null hypothesis.
   2. B) Reject the null hypothesis.
   3. C) Accept the null hypothesis.
   4. D) The test is inconclusive.
7. What should you do if your F-test indicates unequal variances before conducting a t-test?
   1. A) Proceed with a pooled t-test.
   2. B) Proceed with a paired t-test.
   3. C) Use Welch's t-test (unpooled).
   4. D) Ignore the result and proceed with any t-test.