Two Variances and F Distribution: Videos & Practice Problems

Two-sample hypothesis tests for variance use the F-distribution, an asymmetric distribution with two degrees of freedom, calculated as the ratio of sample variances $s{2}^{}/s{2}^{}$. The numerator's degrees of freedom are $n-1$ and the denominator's are $n-1$. Hypotheses test if population variances differ, using p-values from the F-distribution's right tail, found via graphing calculators. Assumptions include independent, random samples from normally distributed populations, essential for valid inference in variance comparison tests.

The F-Distribution

The F distribution is essential for conducting two-sample hypothesis tests for variances, extending the concepts learned from the chi-squared distribution used in one-sample variance tests. Like the chi-squared distribution, the F distribution is right-skewed and asymmetric, but it uniquely incorporates two degrees of freedom because it compares two independent samples. These degrees of freedom correspond to the sample sizes of each group minus one, denoted as \(d_1 = n_1 - 1\) for the numerator and \(d_2 = n_2 - 1\) for the denominator.

The core of the F test lies in the F statistic, which is the ratio of the two sample variances:

\[F = \frac{s_1^2}{s_2^2}\]

When performing hypothesis tests, it is standard practice to assign \(s_1^2\) as the larger sample variance to ensure the F statistic is greater than or equal to 1, simplifying interpretation and comparison against critical values.

Understanding how to find p-values for the F distribution is crucial. Unlike chi-squared tables, which can be used to find critical values, p-values for the F distribution are most accurately obtained using graphing calculators or statistical software. The process involves calculating the right-tailed probability starting from the observed F statistic to infinity, reflecting the distribution’s positive skewness.

To compute the p-value on a graphing calculator, use the cumulative distribution function for the F distribution, often labeled as Fcdf. This function requires four inputs: the lower bound (the observed F statistic), the upper bound (a very large number approximating infinity, such as \$1 \times 10^{99}\(), and the two degrees of freedom for the numerator and denominator. For example, if the sample sizes are \)n_1 = 11\( and \)n_2 = 12\(, the degrees of freedom are \)10\( and \)11$, respectively.

By inputting these values, the calculator returns the p-value, representing the probability of observing an F statistic as extreme or more extreme than the one calculated, assuming the null hypothesis is true. This p-value guides decision-making in hypothesis testing for equality of variances.

In summary, the F distribution serves as a powerful tool for comparing variances between two samples, relying on the ratio of sample variances and two degrees of freedom. Mastery of calculating p-values using graphing calculators enhances the ability to perform accurate two-sample variance hypothesis tests, building on foundational knowledge of the chi-squared distribution and variance analysis.

Which statement about the $F$-distribution is always true?

• A. It is symmetric like the normal distribution.
• B. It is centered at 0.
• C. It is always right-skewed and takes only positive values.
• D. It becomes negative when the denominator variance is very small.

Answer: (C)

Two machines produce metal rods. You take independent random samples of their lengths (shown below) as part of a hypothesis test. Calculate the F statistic for this test.
Sample A: $s^2=2.1,n=10$
Sample B: $s^2=4.2,n=12$

• A. 4
• B. 2
• C. 2.4
• D. 4.8

Answer: (B)

The F-Distribution Example 1

When comparing the variances of two independent samples, a two-sample hypothesis test can be used to determine if there is a significant difference between the variances. In this context, the test statistic follows an F-distribution, which requires calculating the F statistic as the ratio of the larger sample variance to the smaller sample variance. Given two sample variances, \(s_1^2\) and \(s_2^2\), where \(s_1^2\) is the larger variance, the F statistic is calculated as:

\[F = \frac{s_1^2}{s_2^2}\]

For example, if the variances are 16 and 9, the F statistic would be:

\[F = \frac{16}{9} \approx 1.78\]

The degrees of freedom for the numerator and denominator correspond to the sample sizes minus one, so if the sample sizes are \(n_1 = 12\) and \(n_2 = 8\), then the degrees of freedom are:

\[df_1 = n_1 - 1 = 11\]

\[df_2 = n_2 - 1 = 7\]

To find the p-value associated with this F statistic, technology such as a graphing calculator or statistical software is typically used because manual lookup in F-distribution tables can be tedious. The p-value represents the probability of observing an F statistic as extreme as, or more extreme than, the calculated value under the null hypothesis that the variances are equal.

When using a calculator, the cumulative distribution function for the F-distribution, often denoted as fcdf, requires four inputs: the lower bound, upper bound, numerator degrees of freedom, and denominator degrees of freedom. For a right-tailed test, the lower bound is the calculated F statistic, and the upper bound is set to a very large number (e.g., \$1 \times 10^{99}$) to capture the tail area.

Thus, the p-value is computed as:

\[p = P(F \geq 1.78) = \text{fcdf}(1.78, 1 \times 10^{99}, 11, 7) \approx 0.228\]

This p-value, which lies between 0 and 1, indicates the probability of obtaining such a ratio of variances if the null hypothesis is true. A higher p-value suggests insufficient evidence to reject the null hypothesis, meaning the variances may not be significantly different.

Performing Hypothesis Tests for Two Variances

When conducting a two-sample hypothesis test for variances, the goal is to determine if there is a significant difference between the variances of two independent populations. This test is similar in structure to other hypothesis tests but uses the F-distribution instead of the chi-squared distribution, which is used for one-sample variance tests. The test statistic is calculated as the ratio of the larger sample variance to the smaller sample variance, ensuring the value is always greater than or equal to one. This ratio is expressed as \(F = \frac{s_1^2}{s_2^2}\), where \(s_1^2\) is the larger sample variance and \(s_2^2\) is the smaller one.

To set up the hypotheses, the null hypothesis (\(H_0\)) assumes that the population variances are equal, i.e., \(\sigma_a^2 = \sigma_b^2\). The alternative hypothesis (\(H_a\)) reflects the claim being tested, often that one variance is greater than the other, such as \(\sigma_a^2 > \sigma_b^2\). This directional alternative hypothesis determines whether the test is one-tailed or two-tailed.

After calculating the test statistic, the degrees of freedom for each sample are determined by subtracting one from the respective sample sizes: \(df_1 = n_1 - 1\) and \(df_2 = n_2 - 1\). These degrees of freedom are essential for referencing the F-distribution to find the p-value, which represents the probability of observing a test statistic as extreme as, or more extreme than, the calculated value under the null hypothesis.

The p-value is found by calculating the area to the right of the test statistic on the F-distribution curve with the corresponding degrees of freedom. If the p-value is less than the chosen significance level \(\alpha\) (commonly 0.05), the null hypothesis is rejected, indicating sufficient evidence that the variances differ as specified by the alternative hypothesis. If the p-value is greater, the null hypothesis is not rejected, meaning there is insufficient evidence to support the claim of unequal variances.

It is important to verify that the samples are independent and randomly selected, and that the populations from which the samples are drawn are approximately normally distributed. These assumptions ensure the validity of the F-test for comparing variances.

In summary, the two-sample variance test involves identifying the larger sample variance, calculating the F-statistic as the ratio of variances, determining degrees of freedom, and using the F-distribution to assess the p-value. This process helps determine whether there is statistically significant evidence to conclude that the variances of two populations differ.

In a two-sample Hypothesis Test for Variance, what does an $F$-statistic near 1 indicate?

• A. Both populations have zero variance.
• B. We reject the null hypothesis $H_0$.
• C. The numerator variance is smaller than the denominator variance.
• D. The two sample variances are approximately equal.

Answer: (D)

Performing Hypothesis Tests for Two Variances Example 2

When conducting a hypothesis test to compare the variation between two populations, such as cookie weights from two different ovens, the focus is on testing the equality of their standard deviations or variances. In this scenario, Oven A produces 16 cookies with a standard deviation of 2.45 grams, while Oven B produces 11 cookies with a standard deviation of 1.26 grams. The goal is to determine if the variation in cookie weights differs between the two ovens at a significance level of α = 0.05.

To begin, identify the larger standard deviation as s₁ and its corresponding sample size as n₁. Here, s₁ = 2.45 with n₁ = 16, and s₂ = 1.26 with n₂ = 11. The null hypothesis (H₀) states that the population standard deviations are equal:
\(H_0: \sigma_1 = \sigma_2\)

The alternative hypothesis (H₁) reflects the claim that the variations differ, so it is two-tailed:
\(H_1: \sigma_1 \neq \sigma_2\)

Since the test concerns variances, the test statistic is based on the ratio of sample variances, not standard deviations. The F-statistic is calculated as:
\[F = \frac{s_1^2}{s_2^2}\]

Squaring the standard deviations, we get:
\[F = \frac{(2.45)^2}{(1.26)^2} = \frac{6.0025}{1.5876} \approx 3.78\]

The degrees of freedom for the numerator and denominator are df₁ = n₁ - 1 = 15 and df₂ = n₂ - 1 = 10, respectively. Using these values, the p-value is found through the F-distribution. A p-value of approximately 0.0196 indicates the probability of observing such an extreme ratio of variances under the null hypothesis.

Since the p-value (0.0196) is less than the significance level (0.05), the null hypothesis is rejected. This provides sufficient evidence to conclude that the variation in cookie weights between Oven A and Oven B is statistically different.

This process highlights the importance of correctly identifying and squaring standard deviations to obtain variances when performing hypothesis tests for variability. Understanding how to calculate the F-statistic and interpret p-values within the context of degrees of freedom is essential for comparing population variances effectively.

Here’s what students ask on this topic:

What is the F distribution and how is it used in two-sample hypothesis tests for variance?

The F distribution is a right-skewed, asymmetric probability distribution used primarily in two-sample hypothesis tests for variance. It compares the ratio of two sample variances, $s_1^2$ over $s_2^2$. The distribution has two degrees of freedom: one for the numerator ($n\_1\; -\; 1$) and one for the denominator ($n\_2\; -\; 1$), where $n\_1$ and $n\_2$ are the sample sizes. It helps test if two population variances are equal by analyzing the ratio of sample variances. The F statistic is calculated as $F\; =\; \backslash frac\{\backslash text\{squared\; variance\; of\; sample\; 1\}\}\{\backslash text\{squared\; variance\; of\; sample\; 2\}\}$, with the larger variance always placed in the numerator to ensure the F statistic is greater than or equal to 1. This distribution is essential for determining p-values and critical values in variance comparison tests.

How do you calculate the degrees of freedom for the F distribution in a two-sample variance test?

In a two-sample variance test using the F distribution, the degrees of freedom correspond to the sample sizes of each group minus one. Specifically, the numerator degrees of freedom is $d\_1\; =\; n\_1\; -\; 1$, where $n\_1$ is the sample size of the group with the larger sample variance (placed in the numerator). The denominator degrees of freedom is $d\_2\; =\; n\_2\; -\; 1$, where $n\_2$ is the sample size of the other group. These two degrees of freedom define the shape of the F distribution and are used to find critical values or p-values when performing hypothesis tests comparing two variances.

What are the assumptions required for conducting a two-sample hypothesis test for variances using the F distribution?

When conducting a two-sample hypothesis test for variances using the F distribution, several assumptions must be met for valid results. First, both samples should be independent and randomly selected from their respective populations. Second, the populations from which the samples are drawn must be normally distributed. This normality assumption is crucial because the F test is sensitive to deviations from normality. Lastly, the samples should be independent of each other. Meeting these assumptions ensures the F distribution accurately models the ratio of variances, allowing reliable hypothesis testing about population variances.

How do you find the p-value for an F statistic using a graphing calculator?

To find the p-value for an F statistic using a graphing calculator, you use the calculator's distribution functions. First, access the distribution menu (often by pressing the '2nd' button then 'VARS'). Select the function for the F cumulative distribution function (F CDF). You will input four values: the lower bound (the calculated F statistic), the upper bound (a very large number, e.g., $1\; \backslash times\; 10^\{99\}$), and the two degrees of freedom ($d\_1$ and $d\_2$). The calculator then computes the area to the right of the F statistic, which is the p-value for the right-tailed test. This p-value helps determine whether to reject the null hypothesis about equal variances.

How do you set up the null and alternative hypotheses for a two-sample test comparing variances?

In a two-sample test comparing variances, the null hypothesis ($H\_0$) typically states that the population variances are equal: $\backslash sigma\_1^2\; =\; \backslash sigma\_2^2$. The alternative hypothesis ($H\_a$) reflects the claim being tested and can take one of three forms depending on the research question: $\backslash sigma\_1^2\; >\; \backslash sigma\_2^2$ (right-tailed), (left-tailed), or $\backslash sigma\_1^2\; \backslash neq\; \backslash sigma\_2^2$ (two-tailed). Before setting hypotheses, ensure the sample variance with the larger value is assigned as $s\_1^2$ to maintain the F statistic greater than or equal to 1. This setup guides the direction of the test and interpretation of results.

Why must the larger sample variance be placed in the numerator when calculating the F statistic?

The larger sample variance must be placed in the numerator when calculating the F statistic to ensure the test statistic is always greater than or equal to 1. This convention simplifies the interpretation of the F distribution, which is right-skewed and defined only for positive values. By placing the larger variance on top, the F statistic avoids values less than 1, which would complicate the use of F tables and p-value calculations. This approach standardizes the test and aligns with the distribution's properties, making hypothesis testing for variance comparisons more straightforward.