In a completely randomized experimental design, three brands...

In a completely randomized experimental design, three brands of paper towels were tested for their ability to absorb water. Equal-size towels were used, with four sections of towels tested per brand. The absorbency rating data follow. At a .05 level of significance, does there appear to be a difference in the ability of the brands to absorb water? $$ \begin{array}{rcc} & \text { Brand } & \\ \mathbf{x} & \mathbf{y} & \mathbf{z} \\ \mathbf{9 1} & 99 & 83 \\ 100 & 96 & 88 \\ 88 & 94 & 89 \\ 89 & 99 & 76 \end{array} $$

In a completely randomized experimental design, three brands of paper towels were tested for their ability to absorb water. Equal-size towels were used, with four sections of towels tested per brand. The absorbency rating data follow. At a .05 level of significance, does there appear to be a difference in the ability of the brands to absorb water?
$$
\begin{array}{rcc}
& \text { Brand } & \\
\mathbf{x} & \mathbf{y} & \mathbf{z} \\
\mathbf{9 1} & 99 & 83 \\
100 & 96 & 88 \\
88 & 94 & 89 \\
89 & 99 & 76
\end{array}
$$

Key Concepts

Completely Randomized Experimental Design

This design involves randomly assigning experimental units to different treatments to ensure that each treatment is applied under similar conditions. It minimizes the effects of confounding variables by balancing them across treatment groups, making it easier to attribute any observed differences to the treatments themselves.

Analysis of Variance (ANOVA)

ANOVA is a statistical method used to compare the means of three or more groups to determine if at least one group mean is significantly different from the others. It partitions the total variability into variability between groups and within groups, and then uses the F-test to assess whether the group differences are greater than would be expected by random chance.

Hypothesis Testing

This is the process of making statistical decisions using experimental data. It involves setting up a null hypothesis, which typically posits no effect or no difference between groups, and an alternative hypothesis that indicates that at least one group differs. Statistical tests such as the F-test are then applied to determine whether there is enough evidence to reject the null hypothesis.

Significance Level

The significance level, commonly denoted as alpha (?), is the threshold for determining whether a statistical result is sufficiently unlikely under the null hypothesis to warrant rejection. A commonly used value is 0.05, implying a 5% risk of concluding that there is an effect when there is none.

F-test

The F-test is a statistical test used in the context of ANOVA to compare the variances between group means and the variances within the groups. It calculates an F-statistic which, when compared to a critical value from the F-distribution, helps determine whether observed differences among group means are statistically significant.

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