An experiment to compare the spreading rates of five different brands of yellow interior latex paint available in a particular area used 4 gallons [tex](J=4)[/tex] of each paint. The sample average spreading rates [tex](ft^2/gal)[/tex] for the five brands were [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], and [tex]\bar{x}_5=532.1[/tex]. The computed value of [tex]F[/tex] was found to be significant at level [tex]\alpha=0.05[/tex]. With [tex]MSE=280.8[/tex], use Tukey's procedure to investigate significant differences between brands. (Round your answer to two decimal places.)

[tex]w=36.09[/tex]

Which means differ significantly from one another? (Select all that apply.)

- [tex]\square[/tex] [tex]\bar{x}_1[/tex] and [tex]\bar{x}_2[/tex]
- [tex]\square[/tex] [tex]\bar{x}_1[/tex] and [tex]\bar{x}_3[/tex]
- [tex]\square[/tex] [tex]\bar{x}_1[/tex] and [tex]\bar{x}_4[/tex]
- [tex]\square[/tex] [tex]\bar{x}_1[/tex] and [tex]\bar{x}_5[/tex]
- [tex]\square[/tex] [tex]\bar{x}_2[/tex] and [tex]\bar{x}_3[/tex]
- [tex]\square[/tex] [tex]\bar{x}_2[/tex] and [tex]\bar{x}_4[/tex]
- [tex]\square[/tex] [tex]\bar{x}_2[/tex] and [tex]\bar{x}_5[/tex]
- [tex]\square[/tex] [tex]\bar{x}_3[/tex] and [tex]\bar{x}_4[/tex]
- [tex]\square[/tex] [tex]\bar{x}_3[/tex] and [tex]\bar{x}_5[/tex]
- [tex]\square[/tex] [tex]\bar{x}_4[/tex] and [tex]\bar{x}_5[/tex]
- [tex]\square[/tex] There are no significant differences.



Answer :

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]