# 11.1: The F-Distribution and Testing Two Variances

**At Grade**Created by: CK-12

## Learning Objectives

- Understand the differences between the
F -distribution and Student’st -distribution. - Calculate a test statistic as a ratio of values derived from sample variances.
- Use random samples to test hypotheses about multiple independent population variances.
- Understand the limits of inferences derived from these methods.

## Introduction

In previous lessons, we learned how to conduct hypothesis tests that examined the relationship between two variables. Most of these tests simply evaluated the relationship of the means of two variables. However, sometimes we also want to test the variance, or the degree to which observations are spread out within a distribution. In the figure below, we see three samples with identical means (the samples in red, green, and blue) but with very difference variances:

So why would we want to conduct a hypothesis test on variance? Let’s consider an example. Suppose a teacher wants to examine the effectiveness of two reading programs. She randomly assigns her students into two groups, uses a different reading program with each group, and gives her students an achievement test. In deciding which reading program is more effective, it would be helpful to not only look at the mean scores of each of the groups, but also the “spreading out” of the achievement scores. To test hypotheses about variance, we use a statistical tool called the

In this lesson, we will examine the difference between the

###
The F -Distribution

The * F-distribution* is actually a family of distributions. The specific

###
F -Max Test: Calculating the Sample Test Statistic

We use the * F-ratio test statistic* when testing the hypothesis that there is no difference between population variances. When calculating this ratio, we really just need the variance from each of the samples. It is recommended that the larger sample variance be placed in the numerator of the

*Example:* Suppose a teacher administered two different reading programs to two groups of students and collected the following achievement score data:

What is the

###
F -Max Test: Testing Hypotheses about Multiple Independent Population Variances

When we test the hypothesis that two variances of populations from which random samples were selected are equal, * F-Max test*. Since we have a null hypothesis of

Establishing the critical values in an

*On the Web*

http://www.statsoft.com/textbook/sttable.html#f01

*Example:* Suppose we are trying to determine the critical values for the scenario in the preceding section, and we set the level of significance to 0.02. Because we have a two-tailed test, we assign 0.01 to the area to the right of the positive critical value. Using the

Once we find our critical values and calculate our test statistic, we perform the hypothesis test the same way we do with the hypothesis tests using the normal distribution and Student’s

*Example:* Using our example from the preceding section, suppose a teacher administered two different reading programs to two different groups of students and was interested if one program produced a greater variance in scores. Perform a hypothesis test to answer her question.

For the example, we calculated an

###
The Limits of Using the F -Distribution to Test Variance

The test of the null hypothesis,

## Lesson Summary

We use the

The

When testing the variances from independent samples, we calculate the \begin{align*}F\end{align*}-ratio test statistic, which is the ratio of the variances of the independent samples.

When we reject the null hypothesis, \begin{align*}H_0:\sigma^2_1=\sigma^2_2\end{align*}, we conclude that the variances of the two populations are not equal.

The test of the null hypothesis, \begin{align*}H_0: \sigma^2_1=\sigma^2_2\end{align*}, using the \begin{align*}F\end{align*}-distribution is only appropriate when it can be safely assumed that the population is normally distributed.

## Review Questions

- We use the \begin{align*}F\end{align*}-Max test to examine the differences in the ___ between two independent samples.
- List two differences between the \begin{align*}F\end{align*}-distribution and Student’s \begin{align*}t\end{align*}-distribution.
- When we test the differences between the variances of two independent samples, we calculate the ___.
- When calculating the \begin{align*}F\end{align*}-ratio, it is recommended that the sample with the ___ sample variance be placed in the numerator, and the sample with the ___ sample variance be placed in the denominator.
- Suppose a guidance counselor tested the mean of two student achievement samples from different SAT preparatory courses. She found that the two independent samples had similar means, but also wants to test the variance associated with the samples. She collected the following data:

\begin{align*}& \text{SAT Prep Course} \ \# 1 && \text{SAT Prep Course} \ \# 2\\ & n=31 && n=21\\ & s^2=42.30 && s^2=18.80\end{align*}

(a) What are the null and alternative hypotheses for this scenario?

(b) What is the critical value with \begin{align*}\alpha=0.10\end{align*}?

(c) Calculate the \begin{align*}F\end{align*}-ratio.

(d) Would you reject or fail to reject the null hypothesis? Explain your reasoning.

(e) Interpret the results and determine what the guidance counselor can conclude from this hypothesis test.

- True or False: The test of the null hypothesis, \begin{align*}H_0:\sigma^2_1=\sigma^2_2\end{align*}, using the \begin{align*}F\end{align*}-distribution is only appropriate when it can be safely assumed that the population is normally distributed.

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