Performing a Two-Sample F-Test In Exercises 19–26, (a) identify the claim and state H0 and Ha, (b) find the critical value and identify the rejection region, (c) find the test statistic F, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.


U.S. History Assessment Tests A state school administrator claims that the standard deviations of U.S. history assessment test scores for eighth-grade students are the same in Districts 1 and 2. A sample of 10 test scores from District 1 has a standard deviation of 30.9 points, and a sample of 13 test scores from District 2 has a standard deviation of 27.2 points. At α=0.01, can you reject the administrator’s claim? (Adapted from National Center for Education Statistics)

Verified step by step guidance

1. Identify the claim and state the null and alternative hypotheses: The claim is that the standard deviations of U.S. history assessment test scores for eighth-grade students are the same in Districts 1 and 2. The null hypothesis (H₀) is that the variances are equal: σ₁² = σ₂². The alternative hypothesis (Hₐ) is that the variances are not equal: σ₁² ≠ σ₂².
2. Determine the critical value and rejection region: Since this is a two-tailed F-test, use the F-distribution table with degrees of freedom df₁ = n₁ - 1 = 10 - 1 = 9 and df₂ = n₂ - 1 = 13 - 1 = 12. The significance level is α = 0.01. Find the critical values for the upper and lower tails, and define the rejection region as F < F_lower or F > F_upper.
3. Calculate the test statistic F: Use the formula F = (s₁² / s₂²), where s₁ = 30.9 and s₂ = 27.2. First, square the standard deviations to get the variances, then divide the larger variance by the smaller variance to compute F.
4. Decide whether to reject or fail to reject the null hypothesis: Compare the calculated F value to the critical values. If F falls within the rejection region, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
5. Interpret the decision in the context of the original claim: Based on the decision in step 4, explain whether there is sufficient evidence to reject the administrator's claim that the standard deviations of test scores are the same in Districts 1 and 2 at the 0.01 significance level.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents no effect or no difference, and the alternative hypothesis (Ha), which indicates the presence of an effect or difference. In this context, the claim is that the standard deviations of test scores in two districts are equal, leading to specific hypotheses that need to be tested.

F-Test

The F-test is a statistical test used to compare the variances of two populations to determine if they are significantly different. It calculates the F-statistic, which is the ratio of the variances of the two samples. In this scenario, the F-test will help assess whether the standard deviations of U.S. history assessment test scores in the two districts are equal, as claimed by the administrator.

Critical Value and Rejection Region

The critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the significance level (α), which indicates the probability of making a Type I error. The rejection region is the range of values for the test statistic that leads to rejecting H0. In this case, with α=0.01, the critical value will help decide whether the calculated F-statistic falls within the rejection region, thus influencing the conclusion about the administrator's claim.