Question

The accompanying data resulted from a flammability study in which specimens of five different fabrics were tested to determine burn times. $$\begin{array}{lllllll} & \mathbf{1} & 17.8 & 16.2 & 15.9 & 15.5 & \\ & \mathbf{2} & 13.2 & 10.4 & 11.3 & & \\ \text { Fabric } & \mathbf{3} & 11.8 & 11.0 & 9.2 & 10.0 & \\ & \mathbf{4} & 16.5 & 15.3 & 14.1 & 15.0 & 13.9 \\ & \mathbf{5} & 13.9 & 10.8 & 12.8 & 11.7 & \end{array}$$ \(\begin{aligned} \mathrm{MSTr} &=23.67 \\ \mathrm{MSE} &=1.39 \\ F &=17.08 \\ P \text { -value } &=.000 \end{aligned}\) The accompanying output gives the T-K intervals as calculated by MINITAB. Identify significant differences and give the underscoring pattern. $$\begin{array}{lrrr} & 1 & 2 & 3 & 4 \\ 2 & 1.938 & & & \\ & 7.495 & & & \\ 3 & 3.278 & -1.645 & & \\ & 8.422 & 3.912 & & \\ 4 & 3.830 & -0.670 & -2.020 & \\ & 1.478 & -3.445 & -4.372 & 0.220 \\ 5 & 6.622 & 2.112 & 0.772 & 5.100 \end{array}$$

Short answer

There are significant differences among the five fabrics' burn times according to the ANOVA results. Fabrics 1 and 5 are significantly different from the others according to the Tukey-Kramer intervals. The underscoring pattern can be represented as: Fabric 1: __, Fabric 2: _, Fabric 3: _, Fabric 4: __, Fabric 5: ____.

Step-by-step solution

Understand Provided ANOVA Results

The first step is to understand the provided ANOVA results. The mean sum of squares due to treatment (\(MSTr\)) is 23.67, while the mean sum of squares due to error (\(MSE\)) is 1.39. The F statistic is 17.08 and the P value is 0. This indicates that there are significant differences among the group means because the P value is less than 0.05.

Interpretation of T-K Intervals

The second step is to interpret the provided Tukey-Kramer (T-K) intervals. A fabric is significantly different from another if their interval does not contain zero. For example, the interval for fabric 1 and 2 (1.938 to 7.495) does not contain zero, which means fabric 1 and 2 are significantly different. But the interval for fabric 2 and 4 (-0.670 to -2.020) does contain zero, so there's no significant difference between fabric 2 and 4.

Identify UnderScoring Pattern

Finally, the underscoring pattern represents the significant differences among the groups. Fabrics that have significant differences receive different underscores. Based on the T-K intervals interpreted in the previous step, the underscoring pattern would be: Fabric 1: __, Fabric 2: _, Fabric 3: _, Fabric 4: __, Fabric 5: ____.

Key concepts

Mean Sum of Squares

The 'mean sum of squares’ (often abbreviated as MSS) is a critical component in the ANOVA flammability study that helps us understand the variability within a set of data points. To envision this, imagine how different fabrics may resist flames differently, so when subjected to burning, each type will take various times to burn. The MSS can be calculated for both the treatments (MSTr) and the error (MSE).

Specifically, the MSTr reflects the variance between the group means (comparing the burn times across different fabrics), while the MSE reflects the variance within the groups (the burn times within a single fabric type). A large MSTr relative to MSE, as we see in the study with an F statistic of 17.08, suggests that not all fabrics behave the same when lit – some are more flammable than others. Therefore, the MSS is a foundational marker in discerning significant differences between fabric types in this study.

Tukey-Kramer Intervals

One method of comparing means from multiple treatments is through the use of 'Tukey-Kramer (T-K) intervals'. These intervals help us to determine if the differences observed in mean burn times between any two fabrics are statistically significant or just due to random chance.

In simple terms, a T-K interval that does not cross zero indicates that the two fabrics have significantly different mean burn times; these are fabrics we would not group together when considering flammability. On the other hand, intervals spanning zero show no significant difference, indicating similar flammability characteristics. Through careful inspection of these intervals, we can separate fabrics into categories based on their similarity in burn time and evaluate which fabrics might be more suitable for certain applications where fire resistance is a concern.

F Statistic

The 'F statistic' is a number obtained during an ANOVA test and is used to compare variances. It's calculated by dividing MSTr by MSE, and it tells us whether the treatment factor (in this case, different fabric types) has a significant effect. A high F statistic, as obtained in this study (F=17.08), is a beacon that alerts us – the chance that all fabric types have identical average burn times is so low that we should reject this assumption.

Essentially, the F statistic serves as a way to judge whether observed differences in an experiment are due to the treatment effects or merely by chance. Its value acts as a guide, steering the analysis towards investigating specific differences between group means with confidence.

P-value Analysis

Lastly, the 'P-value' offers a threshold to decide if our findings from the experiment can be deemed significant. Traditionally, a P-value of less than 0.05 is considered the tipping point. In our flammability study, the P-value is .000, which falls far below the conventional line. This whispers to us a message loud and clear – coincidence is not at play here; rather, the fabric types do, in fact, have significantly different burn times.

The P-value, in essence, tells us the probability of observing our results (or something more extreme) if, in the realm of theoretical mathematics, the fabrics would have identical resistance to fire. The closer this value is to zero, the stronger the evidence against this world of sameness, prompting us to confidently assert the reality of variation among fabric types.