F-distribution quiz: Test your knowledge on properties and critical values

📚 Quick Study Guide

• 📐 The F-distribution is a continuous probability distribution that arises frequently as the null distribution of a test statistic, most notably in the analysis of variance (ANOVA).
• 🔢 It is defined by two parameters: degrees of freedom for the numerator ($df_1$) and degrees of freedom for the denominator ($df_2$).
• 📊 The F-statistic is calculated as the ratio of two chi-square variables, each divided by its degrees of freedom: $F = \frac{\chi^2_1 / df_1}{\chi^2_2 / df_2}$.
• 📈 The F-distribution is always non-negative and is skewed to the right.
• 🔍 Critical values for the F-distribution are used in hypothesis testing to determine whether to reject the null hypothesis. These values depend on the chosen significance level ($\alpha$) and the degrees of freedom.
• 💡 To find the critical value, you typically use an F-table or statistical software, looking up the value corresponding to the desired $\alpha$, $df_1$, and $df_2$.

Practice Quiz

1. Which of the following is NOT a property of the F-distribution?

   1. It is continuous.
   2. It is symmetric.
   3. It is non-negative.
   4. It is defined by two degrees of freedom.

2. The F-distribution is primarily used in:

   1. Regression analysis.
   2. T-tests.
   3. Analysis of Variance (ANOVA).
   4. Chi-square tests.

3. What parameters define an F-distribution?

   1. Mean and standard deviation.
   2. Variance and skewness.
   3. Degrees of freedom for numerator and denominator.
   4. Sample size and population size.

4. If $df_1 = 5$ and $df_2 = 10$, which variable is in the numerator of the F-statistic?

   1. A chi-square variable with 5 degrees of freedom.
   2. A chi-square variable with 10 degrees of freedom.
   3. A t-statistic with 5 degrees of freedom.
   4. A t-statistic with 10 degrees of freedom.

5. What happens to the shape of the F-distribution as the degrees of freedom increase?

   1. It becomes more skewed.
   2. It becomes more symmetric.
   3. It remains the same.
   4. It becomes bimodal.

6. What does the critical value of an F-distribution represent?

   1. The mean of the distribution.
   2. The median of the distribution.
   3. The value beyond which we reject the null hypothesis.
   4. The standard deviation of the distribution.

7. In ANOVA, if the calculated F-statistic is greater than the critical F-value, what decision do you make?

   1. Fail to reject the null hypothesis.
   2. Reject the null hypothesis.
   3. Increase the sample size.
   4. Decrease the significance level.