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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](\left.ft^2 / gal \right)[/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 MSE [tex]= 400.8[/tex], use Tukey's procedure to investigate significant differences between brands. (Round your answer to two decimal places.)

[tex]
w = 46.39
[/tex]

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

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



Answer :

GinnyAnswer
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]

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