If average scores differ by the gymnast, use Tukey’s HSD method at the 1% significance level to determine which gymnasts’ performances differ.

Case:

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 job objectivity and consistency of the judges. Assume scores are normally distributed.

Judge 1 Judge 2 Judge 3
Gymnast 1 8.0 8.5 8.2
Gymnast 2 9.5 9.2 9.7
Gymnast 3 7.3 7.5 7.7
Gymnast 4 8.3 8.7 8.5
Gymnast 5 8.8 9.2 9.0
1.)At the 1% significance level, can you conclude that average scores differ by judge? Can you conclude that the judges seem inconsistent with their scoring?

2.)At the 1% significance, can you conclude that average scores differ by gymnast?

3.)If average scores differ by the gymnast, use Tukey’s HSD method at the 1% significance level to determine which gymnasts’ performances differ.

Please use your own words.