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To determine which means differ significantly from one another, we use Tukey's procedure, which involves comparing the absolute differences between each pair of sample means to a critical value [tex]\(w = 46.39\)[/tex]. Let's look at the pairwise comparisons between the sample means:
1. [tex]\(\bar{x}_1 = 462.0\)[/tex] vs. [tex]\(\bar{x}_2 = 502.8\)[/tex]
[tex]\[ |\bar{x}_1 - \bar{x}_2| = |462.0 - 502.8| = 40.8 \quad \text{(not significant)} \][/tex]
2. [tex]\(\bar{x}_1 = 462.0\)[/tex] vs. [tex]\(\bar{x}_3 = 427.5\)[/tex]
[tex]\[ |\bar{x}_1 - \bar{x}_3| = |462.0 - 427.5| = 34.5 \quad \text{(not significant)} \][/tex]
3. [tex]\(\bar{x}_1 = 462.0\)[/tex] vs. [tex]\(\bar{x}_4 = 469.3\)[/tex]
[tex]\[ |\bar{x}_1 - \bar{x}_4| = |462.0 - 469.3| = 7.3 \quad \text{(not significant)} \][/tex]
4. [tex]\(\bar{x}_1 = 462.0\)[/tex] vs. [tex]\(\bar{x}_5 = 532.1\)[/tex]
[tex]\[ |\bar{x}_1 - \bar{x}_5| = |462.0 - 532.1| = 70.1 \quad \text{(significant)} \][/tex]
5. [tex]\(\bar{x}_2 = 502.8\)[/tex] vs. [tex]\(\bar{x}_3 = 427.5\)[/tex]
[tex]\[ |\bar{x}_2 - \bar{x}_3| = |502.8 - 427.5| = 75.3 \quad \text{(significant)} \][/tex]
6. [tex]\(\bar{x}_2 = 502.8\)[/tex] vs. [tex]\(\bar{x}_4 = 469.3\)[/tex]
[tex]\[ |\bar{x}_2 - \bar{x}_4| = |502.8 - 469.3| = 33.5 \quad \text{(not significant)} \][/tex]
7. [tex]\(\bar{x}_2 = 502.8\)[/tex] vs. [tex]\(\bar{x}_5 = 532.1\)[/tex]
[tex]\[ |\bar{x}_2 - \bar{x}_5| = |502.8 - 532.1| = 29.3 \quad \text{(not significant)} \][/tex]
8. [tex]\(\bar{x}_3 = 427.5\)[/tex] vs. [tex]\(\bar{x}_4 = 469.3\)[/tex]
[tex]\[ |\bar{x}_3 - \bar{x}_4| = |427.5 - 469.3| = 41.8 \quad \text{(not significant)} \][/tex]
9. [tex]\(\bar{x}_3 = 427.5\)[/tex] vs. [tex]\(\bar{x}_5 = 532.1\)[/tex]
[tex]\[ |\bar{x}_3 - \bar{x}_5| = |427.5 - 532.1| = 104.6 \quad \text{(significant)} \][/tex]
10. [tex]\(\bar{x}_4 = 469.3\)[/tex] vs. [tex]\(\bar{x}_5 = 532.1\)[/tex]
[tex]\[ |\bar{x}_4 - \bar{x}_5| = |469.3 - 532.1| = 62.8 \quad \text{(significant)} \][/tex]
Comparing all of the absolute differences to [tex]\(w = 46.39\)[/tex], we find the following pairs show significant differences:
- [tex]\(\bar{x}_1\)[/tex] and [tex]\(\bar{x}_5\)[/tex]
- [tex]\(\bar{x}_2\)[/tex] and [tex]\(\bar{x}_3\)[/tex]
- [tex]\(\bar{x}_3\)[/tex] and [tex]\(\bar{x}_5\)[/tex]
- [tex]\(\bar{x}_4\)[/tex] and [tex]\(\bar{x}_5\)[/tex]
Therefore, the means that differ significantly from one another are:
[tex]\[ \begin{align*} \boxed{\bar{x}_1 \text{ and }\bar{x}_5} \\ \boxed{\bar{x}_2 \text{ and }\bar{x}_3} \\ \boxed{\bar{x}_3 \text{ and }\bar{x}_5} \\ \boxed{\bar{x}_4 \text{ and }\bar{x}_5} \end{align*} \][/tex]
1. [tex]\(\bar{x}_1 = 462.0\)[/tex] vs. [tex]\(\bar{x}_2 = 502.8\)[/tex]
[tex]\[ |\bar{x}_1 - \bar{x}_2| = |462.0 - 502.8| = 40.8 \quad \text{(not significant)} \][/tex]
2. [tex]\(\bar{x}_1 = 462.0\)[/tex] vs. [tex]\(\bar{x}_3 = 427.5\)[/tex]
[tex]\[ |\bar{x}_1 - \bar{x}_3| = |462.0 - 427.5| = 34.5 \quad \text{(not significant)} \][/tex]
3. [tex]\(\bar{x}_1 = 462.0\)[/tex] vs. [tex]\(\bar{x}_4 = 469.3\)[/tex]
[tex]\[ |\bar{x}_1 - \bar{x}_4| = |462.0 - 469.3| = 7.3 \quad \text{(not significant)} \][/tex]
4. [tex]\(\bar{x}_1 = 462.0\)[/tex] vs. [tex]\(\bar{x}_5 = 532.1\)[/tex]
[tex]\[ |\bar{x}_1 - \bar{x}_5| = |462.0 - 532.1| = 70.1 \quad \text{(significant)} \][/tex]
5. [tex]\(\bar{x}_2 = 502.8\)[/tex] vs. [tex]\(\bar{x}_3 = 427.5\)[/tex]
[tex]\[ |\bar{x}_2 - \bar{x}_3| = |502.8 - 427.5| = 75.3 \quad \text{(significant)} \][/tex]
6. [tex]\(\bar{x}_2 = 502.8\)[/tex] vs. [tex]\(\bar{x}_4 = 469.3\)[/tex]
[tex]\[ |\bar{x}_2 - \bar{x}_4| = |502.8 - 469.3| = 33.5 \quad \text{(not significant)} \][/tex]
7. [tex]\(\bar{x}_2 = 502.8\)[/tex] vs. [tex]\(\bar{x}_5 = 532.1\)[/tex]
[tex]\[ |\bar{x}_2 - \bar{x}_5| = |502.8 - 532.1| = 29.3 \quad \text{(not significant)} \][/tex]
8. [tex]\(\bar{x}_3 = 427.5\)[/tex] vs. [tex]\(\bar{x}_4 = 469.3\)[/tex]
[tex]\[ |\bar{x}_3 - \bar{x}_4| = |427.5 - 469.3| = 41.8 \quad \text{(not significant)} \][/tex]
9. [tex]\(\bar{x}_3 = 427.5\)[/tex] vs. [tex]\(\bar{x}_5 = 532.1\)[/tex]
[tex]\[ |\bar{x}_3 - \bar{x}_5| = |427.5 - 532.1| = 104.6 \quad \text{(significant)} \][/tex]
10. [tex]\(\bar{x}_4 = 469.3\)[/tex] vs. [tex]\(\bar{x}_5 = 532.1\)[/tex]
[tex]\[ |\bar{x}_4 - \bar{x}_5| = |469.3 - 532.1| = 62.8 \quad \text{(significant)} \][/tex]
Comparing all of the absolute differences to [tex]\(w = 46.39\)[/tex], we find the following pairs show significant differences:
- [tex]\(\bar{x}_1\)[/tex] and [tex]\(\bar{x}_5\)[/tex]
- [tex]\(\bar{x}_2\)[/tex] and [tex]\(\bar{x}_3\)[/tex]
- [tex]\(\bar{x}_3\)[/tex] and [tex]\(\bar{x}_5\)[/tex]
- [tex]\(\bar{x}_4\)[/tex] and [tex]\(\bar{x}_5\)[/tex]
Therefore, the means that differ significantly from one another are:
[tex]\[ \begin{align*} \boxed{\bar{x}_1 \text{ and }\bar{x}_5} \\ \boxed{\bar{x}_2 \text{ and }\bar{x}_3} \\ \boxed{\bar{x}_3 \text{ and }\bar{x}_5} \\ \boxed{\bar{x}_4 \text{ and }\bar{x}_5} \end{align*} \][/tex]
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