IDNLearn.com: Your trusted platform for finding precise and reliable answers. Our experts provide accurate and detailed responses to help you navigate any topic or issue with confidence.
Sagot :
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]
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thanks for visiting IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more helpful information.
