At a gymnastics meet, three judges evaluate the balance beam performances of five gymnasts. The judges use a scale of 1 to 10, where 10 is a perfect score.

A statistician wants to examine the objectivity and consistency of the judges. She performs a two-way ANOVA analysis without interaction using α = 0.01 and obtains the following results:

a. At the 1% significance level, can you conclude that average scores differ by judge?
b. At the 1% significance level, can you conclude that average scores differ by gymnast?
c. If average scores differ by gymnast, use Tukey's HSD method at the 1% significance level to determine which gymnasts' performances differ.

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Gymnast Judge Means 8.0 9.5 7.3 8.3 8.8 8.5 9.2 7.5 8.7 9.2 8.2 9.7 7.7 8.5 9.0 Xoymnast8.2 Xaymnast 2 9.5 XGymnast 3 75 4. Xaymnast 59.0 Means Rjudge 1 8.4 Rudgs 2 8.6 Rudge 8.6 Source of Variation SS Rows Columns 0.1922 0.096 2.55 0.1392 8.649 Error p-value Fcrit 6.742 4 1.686 44.75 1.62E-05 7006 df MS 0.30180.038 Total 7.235 14