The following data represent a simple random sample of n=5...

The following data represent a simple random sample of $n=5$ from three populations that are known to be normally distributed. Verify that the $F$ -test statistic is 2.599 $$\begin{array}{ccc} \text { Sample 1 } & \text { Sample 2 } & \text { Sample 3 } \\ \hline 73 & 67 & 72 \\ \hline 82 & 77 & 80 \\ \hline 82 & 66 & 87 \\ \hline 81 & 67 & 77 \\ \hline 97 & 83 & 96 \end{array}$$

The following data represent a simple random sample of $n=5$ from three populations that are known to be normally distributed. Verify that the $F$ -test statistic is 2.599
$$\begin{array}{ccc}
\text { Sample 1 } & \text { Sample 2 } & \text { Sample 3 } \\
\hline 73 & 67 & 72 \\
\hline 82 & 77 & 80 \\
\hline 82 & 66 & 87 \\
\hline 81 & 67 & 77 \\
\hline 97 & 83 & 96
\end{array}$$

Key Concepts

Degrees of Freedom

Degrees of freedom are a fundamental concept in statistics that represent the number of independent values or quantities which can vary in the analysis. In the context of ANOVA, degrees of freedom are associated with both the numerator (number of groups minus one) and the denominator (total number of observations minus the number of groups) of the F-test statistic. Correct determination of degrees of freedom is essential for accurately computing the mean squares and the F statistic.

Between-Group and Within-Group Variance

In the ANOVA framework, the total variability is divided into between-group variance (which measures the variation due to the interaction between the groups) and within-group variance (which measures the variation within each group). The F-test statistic is obtained by taking the ratio of these variances after adjusting for their respective degrees of freedom. This ratio helps determine if the differences among the group means are statistically significant.

Normal Distribution Assumption

Many inferential statistical methods, including the F-test and ANOVA, rely on the assumption that the data are drawn from normally distributed populations. This assumption is crucial because the validity of the F-test statistic depends on the normality of the underlying distributions. When data are normally distributed, the test results are more robust and the statistical properties of the F distribution can be appropriately applied.

F-Test for Equality of Variances

The F-test is used to compare two variances by taking the ratio of the larger sample variance to the smaller one. In the context of verifying an F-test statistic, the calculation involves assessing whether the variability between groups is significantly different from the variability within groups. This is central to testing the hypothesis that the populations have equal variances, assuming normally distributed samples.

Analysis of Variance (ANOVA)

ANOVA is a statistical method used for comparing means across multiple groups by partitioning total variability into components between and within groups. Although commonly used to compare means, the underlying calculations involve comparisons of variances, where the F-test statistic is computed as the ratio of mean square values. This helps in determining if there is more variability between group means than would be expected by chance.

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