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To determine which means differ significantly from one another using Tukey's procedure, we start by calculating the pairwise differences between the sample average spreading rates ([tex]\(\bar{x}_i\)[/tex]) of the five brands of paints. Given the threshold [tex]\(w = 36.09\)[/tex], we will assess if each pairwise difference is greater than [tex]\(w\)[/tex], indicating a significant difference.
Here are the given sample means:
- [tex]\(\bar{x}_1 = 462.0\)[/tex]
- [tex]\(\bar{x}_2 = 502.8\)[/tex]
- [tex]\(\bar{x}_3 = 427.5\)[/tex]
- [tex]\(\bar{x}_4 = 469.3\)[/tex]
- [tex]\(\bar{x}_5 = 532.1\)[/tex]
Let's calculate the absolute differences and compare them with [tex]\(w\)[/tex]:
1. [tex]\(\bar{x}_1\)[/tex] and [tex]\(\bar{x}_2\)[/tex]:
[tex]\[ |\bar{x}_1 - \bar{x}_2| = |462.0 - 502.8| = 40.8 \][/tex]
Since [tex]\(40.8 > 36.09\)[/tex], [tex]\(\bar{x}_1\)[/tex] and [tex]\(\bar{x}_2\)[/tex] are significantly different.
2. [tex]\(\bar{x}_1\)[/tex] and [tex]\(\bar{x}_3\)[/tex]:
[tex]\[ |\bar{x}_1 - \bar{x}_3| = |462.0 - 427.5| = 34.5 \][/tex]
Since [tex]\(34.5 < 36.09\)[/tex], [tex]\(\bar{x}_1\)[/tex] and [tex]\(\bar{x}_3\)[/tex] are not significantly different.
3. [tex]\(\bar{x}_1\)[/tex] and [tex]\(\bar{x}_4\)[/tex]:
[tex]\[ |\bar{x}_1 - \bar{x}_4| = |462.0 - 469.3| = 7.3 \][/tex]
Since [tex]\(7.3 < 36.09\)[/tex], [tex]\(\bar{x}_1\)[/tex] and [tex]\(\bar{x}_4\)[/tex] are not significantly different.
4. [tex]\(\bar{x}_1\)[/tex] and [tex]\(\bar{x}_5\)[/tex]:
[tex]\[ |\bar{x}_1 - \bar{x}_5| = |462.0 - 532.1| = 70.1 \][/tex]
Since [tex]\(70.1 > 36.09\)[/tex], [tex]\(\bar{x}_1\)[/tex] and [tex]\(\bar{x}_5\)[/tex] are significantly different.
5. [tex]\(\bar{x}_2\)[/tex] and [tex]\(\bar{x}_3\)[/tex]:
[tex]\[ |\bar{x}_2 - \bar{x}_3| = |502.8 - 427.5| = 75.3 \][/tex]
Since [tex]\(75.3 > 36.09\)[/tex], [tex]\(\bar{x}_2\)[/tex] and [tex]\(\bar{x}_3\)[/tex] are significantly different.
6. [tex]\(\bar{x}_2\)[/tex] and [tex]\(\bar{x}_4\)[/tex]:
[tex]\[ |\bar{x}_2 - \bar{x}_4| = |502.8 - 469.3| = 33.5 \][/tex]
Since [tex]\(33.5 < 36.09\)[/tex], [tex]\(\bar{x}_2\)[/tex] and [tex]\(\bar{x}_4\)[/tex] are not significantly different.
7. [tex]\(\bar{x}_2\)[/tex] and [tex]\(\bar{x}_5\)[/tex]:
[tex]\[ |\bar{x}_2 - \bar{x}_5| = |502.8 - 532.1| = 29.3 \][/tex]
Since [tex]\(29.3 < 36.09\)[/tex], [tex]\(\bar{x}_2\)[/tex] and [tex]\(\bar{x}_5\)[/tex] are not significantly different.
8. [tex]\(\bar{x}_3\)[/tex] and [tex]\(\bar{x}_4\)[/tex]:
[tex]\[ |\bar{x}_3 - \bar{x}_4| = |427.5 - 469.3| = 41.8 \][/tex]
Since [tex]\(41.8 > 36.09\)[/tex], [tex]\(\bar{x}_3\)[/tex] and [tex]\(\bar{x}_4\)[/tex] are significantly different.
9. [tex]\(\bar{x}_3\)[/tex] and [tex]\(\bar{x}_5\)[/tex]:
[tex]\[ |\bar{x}_3 - \bar{x}_5| = |427.5 - 532.1| = 104.6 \][/tex]
Since [tex]\(104.6 > 36.09\)[/tex], [tex]\(\bar{x}_3\)[/tex] and [tex]\(\bar{x}_5\)[/tex] are significantly different.
10. [tex]\(\bar{x}_4\)[/tex] and [tex]\(\bar{x}_5\)[/tex]:
[tex]\[ |\bar{x}_4 - \bar{x}_5| = |469.3 - 532.1| = 62.8 \][/tex]
Since [tex]\(62.8 > 36.09\)[/tex], [tex]\(\bar{x}_4\)[/tex] and [tex]\(\bar{x}_5\)[/tex] are significantly different.
So, the pairs that exhibit a significant difference are:
- [tex]\( \bar{x}_1 \)[/tex] and [tex]\( \bar{x}_2 \)[/tex]
- [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]\( \bar{x}_4 \)[/tex]
- [tex]\( \bar{x}_3 \)[/tex] and [tex]\( \bar{x}_5 \)[/tex]
- [tex]\( \bar{x}_4 \)[/tex] and [tex]\( \bar{x}_5 \)[/tex]
Hence, the selected pairs are:
- [tex]\( \bar{x}_1 \)[/tex] and [tex]\( \bar{x}_2 \)[/tex]
- [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]\( \bar{x}_4 \)[/tex]
- [tex]\( \bar{x}_3 \)[/tex] and [tex]\( \bar{x}_5 \)[/tex]
- [tex]\( \bar{x}_4 \)[/tex] and [tex]\( \bar{x}_5 \)[/tex]
Here are the given sample means:
- [tex]\(\bar{x}_1 = 462.0\)[/tex]
- [tex]\(\bar{x}_2 = 502.8\)[/tex]
- [tex]\(\bar{x}_3 = 427.5\)[/tex]
- [tex]\(\bar{x}_4 = 469.3\)[/tex]
- [tex]\(\bar{x}_5 = 532.1\)[/tex]
Let's calculate the absolute differences and compare them with [tex]\(w\)[/tex]:
1. [tex]\(\bar{x}_1\)[/tex] and [tex]\(\bar{x}_2\)[/tex]:
[tex]\[ |\bar{x}_1 - \bar{x}_2| = |462.0 - 502.8| = 40.8 \][/tex]
Since [tex]\(40.8 > 36.09\)[/tex], [tex]\(\bar{x}_1\)[/tex] and [tex]\(\bar{x}_2\)[/tex] are significantly different.
2. [tex]\(\bar{x}_1\)[/tex] and [tex]\(\bar{x}_3\)[/tex]:
[tex]\[ |\bar{x}_1 - \bar{x}_3| = |462.0 - 427.5| = 34.5 \][/tex]
Since [tex]\(34.5 < 36.09\)[/tex], [tex]\(\bar{x}_1\)[/tex] and [tex]\(\bar{x}_3\)[/tex] are not significantly different.
3. [tex]\(\bar{x}_1\)[/tex] and [tex]\(\bar{x}_4\)[/tex]:
[tex]\[ |\bar{x}_1 - \bar{x}_4| = |462.0 - 469.3| = 7.3 \][/tex]
Since [tex]\(7.3 < 36.09\)[/tex], [tex]\(\bar{x}_1\)[/tex] and [tex]\(\bar{x}_4\)[/tex] are not significantly different.
4. [tex]\(\bar{x}_1\)[/tex] and [tex]\(\bar{x}_5\)[/tex]:
[tex]\[ |\bar{x}_1 - \bar{x}_5| = |462.0 - 532.1| = 70.1 \][/tex]
Since [tex]\(70.1 > 36.09\)[/tex], [tex]\(\bar{x}_1\)[/tex] and [tex]\(\bar{x}_5\)[/tex] are significantly different.
5. [tex]\(\bar{x}_2\)[/tex] and [tex]\(\bar{x}_3\)[/tex]:
[tex]\[ |\bar{x}_2 - \bar{x}_3| = |502.8 - 427.5| = 75.3 \][/tex]
Since [tex]\(75.3 > 36.09\)[/tex], [tex]\(\bar{x}_2\)[/tex] and [tex]\(\bar{x}_3\)[/tex] are significantly different.
6. [tex]\(\bar{x}_2\)[/tex] and [tex]\(\bar{x}_4\)[/tex]:
[tex]\[ |\bar{x}_2 - \bar{x}_4| = |502.8 - 469.3| = 33.5 \][/tex]
Since [tex]\(33.5 < 36.09\)[/tex], [tex]\(\bar{x}_2\)[/tex] and [tex]\(\bar{x}_4\)[/tex] are not significantly different.
7. [tex]\(\bar{x}_2\)[/tex] and [tex]\(\bar{x}_5\)[/tex]:
[tex]\[ |\bar{x}_2 - \bar{x}_5| = |502.8 - 532.1| = 29.3 \][/tex]
Since [tex]\(29.3 < 36.09\)[/tex], [tex]\(\bar{x}_2\)[/tex] and [tex]\(\bar{x}_5\)[/tex] are not significantly different.
8. [tex]\(\bar{x}_3\)[/tex] and [tex]\(\bar{x}_4\)[/tex]:
[tex]\[ |\bar{x}_3 - \bar{x}_4| = |427.5 - 469.3| = 41.8 \][/tex]
Since [tex]\(41.8 > 36.09\)[/tex], [tex]\(\bar{x}_3\)[/tex] and [tex]\(\bar{x}_4\)[/tex] are significantly different.
9. [tex]\(\bar{x}_3\)[/tex] and [tex]\(\bar{x}_5\)[/tex]:
[tex]\[ |\bar{x}_3 - \bar{x}_5| = |427.5 - 532.1| = 104.6 \][/tex]
Since [tex]\(104.6 > 36.09\)[/tex], [tex]\(\bar{x}_3\)[/tex] and [tex]\(\bar{x}_5\)[/tex] are significantly different.
10. [tex]\(\bar{x}_4\)[/tex] and [tex]\(\bar{x}_5\)[/tex]:
[tex]\[ |\bar{x}_4 - \bar{x}_5| = |469.3 - 532.1| = 62.8 \][/tex]
Since [tex]\(62.8 > 36.09\)[/tex], [tex]\(\bar{x}_4\)[/tex] and [tex]\(\bar{x}_5\)[/tex] are significantly different.
So, the pairs that exhibit a significant difference are:
- [tex]\( \bar{x}_1 \)[/tex] and [tex]\( \bar{x}_2 \)[/tex]
- [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]\( \bar{x}_4 \)[/tex]
- [tex]\( \bar{x}_3 \)[/tex] and [tex]\( \bar{x}_5 \)[/tex]
- [tex]\( \bar{x}_4 \)[/tex] and [tex]\( \bar{x}_5 \)[/tex]
Hence, the selected pairs are:
- [tex]\( \bar{x}_1 \)[/tex] and [tex]\( \bar{x}_2 \)[/tex]
- [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]\( \bar{x}_4 \)[/tex]
- [tex]\( \bar{x}_3 \)[/tex] and [tex]\( \bar{x}_5 \)[/tex]
- [tex]\( \bar{x}_4 \)[/tex] and [tex]\( \bar{x}_5 \)[/tex]
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