Performing a two-sample hypothesis test for variance or standard deviation can be streamlined using the two-sample F test function available on graphing calculators like the TI-84. This test evaluates whether there is significant evidence to support a claim about the relationship between two population standard deviations, typically denoted as σ₁ and σ₂. The test involves comparing sample standard deviations and sample sizes to determine if one population variance is greater than the other.
When conducting this test, the null hypothesis (H₀) generally states that the two population standard deviations are equal, i.e., σ₁ = σ₂, while the alternative hypothesis (H₁) reflects the claim being tested, such as σ₁ > σ₂ for a right-tailed test. It is important to note that although the test concerns variances, hypotheses can be expressed in terms of standard deviations without squaring, depending on the context.
To perform the test using sample statistics, input the sample standard deviations (s₁ and s₂) and their corresponding sample sizes (n₁ and n₂) into the calculator’s two-sample F test function under the statistics tab. Ensure that s₁ corresponds to the larger sample standard deviation, as the F statistic is calculated as the ratio of the larger variance to the smaller variance. The F statistic is computed as:
\[F = \frac{s_1^2}{s_2^2}\]The calculator then provides the F statistic and the p-value, which is used to make a decision. If the p-value is less than the significance level α (commonly 0.05), the null hypothesis is rejected, indicating sufficient evidence to support the claim that σ₁ is greater than σ₂. Conversely, if the p-value exceeds α, there is insufficient evidence to reject the null hypothesis.
When raw data is available instead of summary statistics, enter the data sets into separate lists (e.g., L1 and L2) on the calculator. Then, select the two-sample F test under the data tab, specify the lists, and choose the appropriate alternative hypothesis symbol. The calculator will compute the F statistic and p-value based on the data, allowing for the same decision-making process.
For example, if the calculated F statistic is 1.36 with a p-value of 0.026 and α = 0.05, the p-value being less than α leads to rejecting the null hypothesis, supporting the claim that σ₁ > σ₂. However, if the F statistic is 1.7 with a p-value of 0.286, the p-value exceeds α, so the null hypothesis is not rejected, indicating insufficient evidence to conclude that σ₁ is greater than σ₂.
Using the two-sample F test function simplifies the hypothesis testing process for variances by automating calculations and providing clear outputs for the F statistic and p-value. This approach enhances efficiency and accuracy when testing claims about population variances or standard deviations, whether working with summary statistics or raw data.