Do a complete one-way ANOVA. If the null hypothesis is...

Do a complete one-way ANOVA. If the null hypothesis is rejected, use either the Scheffé or Tukey test to see if there is a significant difference in the pairs of means. Assume all assumptions are met. The average weekly unemployment benefit for the entire United States is $\$ 314.74 .$ Three states are randomly selected, and a sample of weekly unemployment benefits is recorded for each. At $\alpha=0.05,$ is there sufficient evidence to conclude a difference in means? If so, perform the appropriate test to find out where the difference exists. $$\begin{array}{ccc}\text { Florida } & \text { Pennsylvania } & \text { Maine } \\\hline 200 & 300 & 250 \\187 & 350 & 195 \\192 & 295 & 275 \\235 & 362 & 260 \\260 & 280 & 220 \\175 & 340 & 290\end{array}$$

Do a complete one-way ANOVA. If the null hypothesis is rejected, use either the Scheffé or Tukey test to see if there is a significant difference in the pairs of means. Assume all assumptions are met.
The average weekly unemployment benefit for the entire United States is $\$ 314.74 .$ Three states are randomly selected, and a sample of weekly unemployment benefits is recorded for each. At $\alpha=0.05,$ is there sufficient evidence to conclude a difference in means? If so, perform the appropriate test to find out where the difference exists.
$$\begin{array}{ccc}\text { Florida } & \text { Pennsylvania } & \text { Maine } \\\hline 200 & 300 & 250 \\187 & 350 & 195 \\192 & 295 & 275 \\235 & 362 & 260 \\260 & 280 & 220 \\175 & 340 & 290\end{array}$$

Key Concepts

One-Way ANOVA

One-way ANOVA (Analysis of Variance) is a statistical method used to compare the means of three or more independent groups. It assesses whether any statistically significant differences exist among the group means by partitioning the overall variability into variability between groups and variability within groups, and then comparing these variabilities using the F-test.

Null and Alternative Hypotheses

In the context of one-way ANOVA, the null hypothesis states that all group means are equal, implying that any observed differences are due to random sampling variability. The alternative hypothesis argues that at least one group mean is different from the others, indicating that a real effect or difference exists among the groups.

Assumptions of ANOVA

ANOVA requires several assumptions to be met for valid results. These include independence of observations (each group's observations are independent of the others), normality of the distribution of residuals within each group, and homogeneity of variances across groups (equal variances). Meeting these assumptions ensures that the F-test used in ANOVA is reliable.

F-Test Statistic

The F-test statistic is calculated in one-way ANOVA by taking the ratio of the mean square between groups (variability due to the interaction between the different groups) to the mean square within groups (variability within the groups due to random error). A significantly high F-value suggests that the variability between the group means is greater than what would be expected by chance, leading to a rejection of the null hypothesis.

Post-Hoc Tests

When the overall one-way ANOVA test indicates a statistically significant difference among group means, post-hoc tests are used to identify which specific pairs of means differ. Common post-hoc tests such as the Tukey or Scheffé procedures control the overall Type I error rate when making multiple comparisons between groups. These tests provide confidence intervals and adjusted p-values to determine where the significant differences lie.

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