A t a g ym nastics m e e t, th re e ju d g e s e v a lu a te t h e balance b e a m p e rfo rm a n c e s o f five gym nasts. T h e ju d g e s use a scale o f 1 to 10, w h e re 10 is a pe rfec t score.

Gymnast Judge Means 1 2 3 1 8.0 8.5 8.2 2 9.5 9.2 9.7 3 7.3 7.5 7.7 4 8.3 8.7 8.5 A statistician w ants to exam in e th e objectivity and consistency o f th e ju d g e s. She p e rfo rm s a tw o -w a y ANO VA analysis for α = 0.01 and obtains th e follow ing results:
ANOVA Source of Variation SS df MS F p-value F crit Rows 6.742 4 1.686 44.75 1.62E-05 7.006 Columns 0.192 2 0.096 2.55 0.1392 8.649 Error 0.301 8 0.038 Total 7.235 14

a. A t th e 1% significance level, can you conclu de th a t average scores differ by judge?

b. At th e 1% significance level, can you conclude th at average scores differ by gym nast?

c. If a v e ra g e scores d iffer by g y m n a s t, use Tukey's HSD te s t a t th e 1 % significance level to d e te rm in e w h ic h g y m nasts' p e rfo rm an c e s differ.