Significance Level and P-Value in Two-Sample F-Tests

📚 Quick Study Guide

• 📊 Significance Level (α): The probability of rejecting the null hypothesis when it is actually true (Type I error). Common values are 0.05 (5%) and 0.01 (1%).
• 🔎 P-value: The probability of obtaining test results at least as extreme as the results actually observed, assuming that the null hypothesis is correct.
• ⚖️ Decision Rule: If the p-value is less than or equal to the significance level (p ≤ α), reject the null hypothesis. If the p-value is greater than the significance level (p > α), fail to reject the null hypothesis.
• 🧪 F-Test: Used to compare variances of two or more populations. In a two-sample F-test, the null hypothesis is that the variances of the two populations are equal. The test 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: For a two-sample F-test, the degrees of freedom are $df_1 = n_1 - 1$ and $df_2 = n_2 - 1$, where $n_1$ and $n_2$ are the sample sizes.

Practice Quiz

1. What does the significance level (α) represent in hypothesis testing?
   1. The probability of accepting the null hypothesis when it is false.
   2. The probability of rejecting the null hypothesis when it is true.
   3. The probability of making a correct decision.
   4. The standard deviation of the sample.
2. Which of the following is the correct decision rule when comparing the p-value to the significance level (α)?
   1. Reject the null hypothesis if p > α.
   2. Fail to reject the null hypothesis if p ≤ α.
   3. Reject the null hypothesis if p ≤ α.
   4. Always reject the null hypothesis.
3. In a two-sample F-test, what is the null hypothesis?
   1. The means of the two populations are equal.
   2. The variances of the two populations are equal.
   3. The standard deviations of the two populations are different.
   4. There is no difference between the two populations.
4. What is the formula for the F-statistic in a two-sample F-test, where $s_1^2$ and $s_2^2$ are the sample variances?
   1. $F = s_1^2 - s_2^2$
   2. $F = \frac{s_1^2}{s_2^2}$
   3. $F = s_1^2 + s_2^2$
   4. $F = \sqrt{\frac{s_1^2}{s_2^2}}$
5. What are the degrees of freedom for a two-sample F-test with sample sizes $n_1 = 15$ and $n_2 = 20$?
   1. $df_1 = 15, df_2 = 20$
   2. $df_1 = 14, df_2 = 19$
   3. $df_1 = 16, df_2 = 21$
   4. $df_1 = 20, df_2 = 15$
6. If the p-value of an F-test is 0.02 and the significance level (α) is 0.05, what is the correct conclusion?
   1. Fail to reject the null hypothesis.
   2. Reject the null hypothesis.
   3. Accept the null hypothesis.
   4. The test is invalid.
7. If the calculated F-statistic is 3.5, and the critical F-value at α = 0.05 is 2.8, what should you conclude?
   1. Fail to reject the null hypothesis.
   2. Reject the null hypothesis.
   3. Accept the null hypothesis.
   4. The test is inconclusive.