Two Variances and F Distribution: Videos & Practice Problems

The F distribution, used for two sample hypothesis tests for variance, compares variances via the ratio $\frac{s^2}{s^2}$ with degrees of freedom $n-1$ for numerator and denominator. Hypotheses test if variances differ, using p-values from the right-tailed F distribution. Samples must be independent, random, and populations normally distributed. This method extends chi-square tests for one variance to compare two variances effectively in inferential statistics.

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 involves two degrees of freedom because it compares two independent sample variances.

The F statistic is calculated as the ratio of two sample variances, expressed as \(F = \frac{s_1^2}{s_2^2}\), where \(s_1^2\) and \(s_2^2\) are the variances of the first and second samples, respectively. To maintain consistency in hypothesis testing, the numerator variance (\(s_1^2\)) is typically the larger variance to ensure the F statistic is greater than or equal to 1. The degrees of freedom for the numerator and denominator correspond to the sample sizes minus one: \(df_1 = n_1 - 1\) and \(df_2 = n_2 - 1\).

Understanding how to find p-values for the F distribution is crucial. While F distribution tables exist, they are less commonly used for p-value calculations compared to chi-squared tables. Instead, graphing calculators or statistical software are preferred tools. Using a graphing calculator, the Fcdf function computes the cumulative distribution function for the F distribution, which helps determine the right-tailed p-value.

To calculate the p-value, input four parameters into the calculator: the lower bound (the observed F statistic), the upper bound (a very large number to approximate infinity, such as \$1 \times 10^{99}$), and the two degrees of freedom for the numerator and denominator. For example, with an F statistic of 1.5, sample sizes of 11 and 12, the degrees of freedom are 10 and 11, respectively. The calculator then returns the p-value, representing the probability of observing an F statistic as extreme or more extreme than the one calculated.

This p-value corresponds to the area under the F distribution curve to the right of the observed F statistic, reflecting the likelihood of the null hypothesis being true. Mastery of the F distribution and its application in hypothesis testing for two variances enables deeper analysis of variability between two populations, a fundamental skill in inferential statistics.

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.

Using the cumulative distribution function for the F-distribution, the p-value is calculated as the area to the right of the F statistic, which is a right-tailed test. The lower bound for the cumulative distribution function is the F statistic itself, and the upper bound is set to a very large number (e.g., \$1 \times 10^{99}$) to capture the entire tail.

In this example, plugging in the values:

\[\text{lower} = 1.78, \quad \text{upper} = 1 \times 10^{99}, \quad df_1 = 11, \quad df_2 = 7\]

yields a p-value of approximately 0.228. This p-value indicates the probability of obtaining an F statistic at least as large as 1.78 if the null hypothesis of equal variances is true.

Understanding how to calculate and interpret the F statistic and its corresponding p-value is essential for conducting hypothesis tests comparing variances. This process involves identifying the larger variance, computing the ratio, determining degrees of freedom, and using technology to find the p-value, which ultimately informs the decision to reject or fail to reject the null hypothesis.

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 particularly useful when comparing variability, such as assessing whether one supplier’s product measurements vary more than another’s. Unlike one-sample variance tests that use the chi-squared distribution, the two-sample variance test employs the F-distribution, which is based on the ratio of two sample variances.

To perform this test, the first step is to identify which sample variance is larger, as the test statistic is calculated as the ratio of the larger sample variance to the smaller one, expressed as \(F = \frac{s_1^2}{s_2^2}\). This ensures the test statistic is always greater than or equal to one, aligning with the properties of the F-distribution. The hypotheses are then formulated with the null hypothesis assuming equal population variances, \(\sigma_a^2 = \sigma_b^2\), and the alternative hypothesis reflecting the research question, often testing if one variance is greater than the other, such as \(\sigma_a^2 > \sigma_b^2\).

After calculating the test statistic, the degrees of freedom for each sample are determined by subtracting one from each sample size, \(df_1 = n_1 - 1\) and \(df_2 = n_2 - 1\). These degrees of freedom are essential for referencing the F-distribution to find the p-value, which indicates the probability of observing the test statistic 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, representing the likelihood of obtaining such a ratio if the variances were truly equal.

For example, if the calculated \(F\) statistic is 1.89 with degrees of freedom 21 and 19, and the p-value is 0.084, this p-value is compared to the significance level \(\alpha = 0.05\). Since 0.084 is greater than 0.05, there is insufficient evidence to reject the null hypothesis, meaning the data do not support the claim that one variance is greater than the other.

It is important to ensure that the samples are independent and randomly selected, and that the populations from which the samples are drawn are approximately normally distributed. These assumptions underpin the validity of the F-test for variances. When these conditions are met, the two-sample variance test provides a robust method for comparing variability between two groups, aiding in quality control, manufacturing processes, and other applications where understanding variance differences is critical.

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, two samples are taken: Oven A produces 16 cookies with a standard deviation of 2.45 grams, and 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 sample 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:
\[F = \frac{s_1^2}{s_2^2} = \frac{(2.45)^2}{(1.26)^2} = \frac{6.0025}{1.5876} \approx 3.78\]

The degrees of freedom for the numerator and denominator are df₁ = n₁ - 1 = 15 and df₂ = n₂ - 1 = 10, respectively. Using these degrees of freedom and the calculated F-statistic, the p-value can be found through F-distribution tables or technology. In this case, the p-value is approximately 0.0196.

Comparing the p-value to the significance level α = 0.05, since 0.0196 < 0.05, the null hypothesis is rejected. This indicates 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 distinguishing between standard deviation and variance in hypothesis testing. The F-test for equality of variances requires squaring the standard deviations to obtain variances before calculating the test statistic. Understanding this distinction ensures accurate hypothesis testing when comparing variability between two populations.

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, $s_1^2$ and $s_2^2$, where the numerator variance is always the larger one to ensure the F statistic is greater than or equal to 1. The distribution has two degrees of freedom: one for the numerator ($n\_1\; -\; 1$) and one for the denominator ($n\_2\; -\; 1$), corresponding to the sample sizes of the two groups. The F distribution helps determine if the variances of two populations are significantly different by calculating p-values or critical values, often using graphing calculators or F tables. This test assumes that the populations are normally distributed and samples are independent and random.

How do you calculate the F statistic for comparing two variances?

To calculate the F statistic for comparing two variances, you take the ratio of the larger sample variance to the smaller sample variance. Mathematically, this is expressed as $F=\frac{s_1}{^}\frac{s_2}{^}$, where $s_1^2$ is the larger sample variance and $s_2^2$ is the smaller one. This ensures the F statistic is always greater than or equal to 1. The degrees of freedom for the numerator is $n\_1\; -\; 1$ and for the denominator is $n\_2\; -\; 1$, where $n\_1$ and $n\_2$ are the sample sizes. This ratio is then used with the F distribution to test hypotheses about the equality of variances.

What are the steps to perform a two-sample hypothesis test for variances using the F distribution?

Performing a two-sample hypothesis test for variances using the F distribution involves several steps: First, identify which sample variance is larger and assign it as $s_1^2$ and the smaller as $s_2^2$. Second, state the null hypothesis $H\_0:\; \backslash ,\; \backslash \backslash sigma\_1^2\; =\; \backslash \backslash sigma\_2^2$ (variances are equal) and the alternative hypothesis depending on the claim (e.g., $H\_a:\; \backslash \backslash sigma\_1^2\; >\; \backslash \backslash sigma\_2^2$). Third, calculate the F statistic as the ratio of the larger to smaller sample variance. Fourth, determine the degrees of freedom for numerator ($n\_1\; -\; 1$) and denominator ($n\_2\; -\; 1$). Fifth, find the p-value using the F distribution, often with a graphing calculator or F tables. Finally, compare the p-value to the significance level $\backslash alpha$ to decide whether to reject or fail to reject the null hypothesis.

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, follow these steps: Access the distribution menu (usually by pressing the '2nd' button then 'VARS'). Select the function for the F cumulative distribution function (often labeled as 'Fcdf'). 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 degrees of freedom for the numerator and denominator ($n\_1\; -\; 1$ and $n\_2\; -\; 1$). The calculator will output the area to the right of the F statistic, which is the p-value for a right-tailed test. This p-value helps determine the significance of the test result.

What assumptions must be met to validly perform a two-sample F test for variances?

To validly perform a two-sample F test for variances, several assumptions must be met: First, each sample should be independently and randomly drawn from its population. 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 departures from normality. Third, the samples should be independent of each other. Meeting these assumptions ensures the validity of the test results and the accuracy of p-values derived from the F distribution.

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 that the test statistic is always greater than or equal to 1. This standardization simplifies interpretation and aligns with the right-skewed nature of the F distribution, which only takes positive values. By placing the larger variance on top, the F distribution tables and p-value calculations are consistent, and the test focuses on detecting whether one variance is significantly larger than the other. This convention also helps avoid confusion when determining critical values and p-values.