Question

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 jobjectivity and consistency of the judges. Assume scores are normally distrbuted.

	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 thier scoring?
2. At the 1% significance, can you conclude that average scores differ by gymnast?
3. If average scores differ by gymnast, use Tukey's HSD method at the 1% significance level to determine which gymnasts' perfomrances differ.

Answer 1

Using Minitab, (Stat -> ANOVA -> General Linear Model -> Fit General Linear Model), we get the following output -

To test whether average scores differ by judge,

The value of test statistic F = 2.55

and P-value = 0.139

Since P-value > 0.01, so we fail to reject H₀ at 1% level of significance and we can conclude that average scores do not significantly differ by judge.

To test whether average scores differ by gymnast,

The value of test statistic F = 44.75

and P-value = 0

Since P-value < 0.01, so we reject H₀ at 1% level of significance and we can conclude that average scores significantly differ by gymnast.

By Tukey's test, we can conclude that gymnast 2 and gymnast 3 are significantly different than others.