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 objectivity and consistency of the judges. Assume scores are normally distributed. (You may find it useful to reference the q table.)

                                    Judge 1                        Judge 2                        Judge 3

Gymnast 1                   9.5                               8.0                               7.8

Gymnast 2                   9.5                               9.5                               9.2

Gymnast 3                   9.1                               8.4                               7.4

Gymnast 4                   7.9                               8.2                               8.6

Gymnast 5                   9.3                               7.9                               9.1

a-1. Construct an ANOVA table. (Round intermediate calculations to at least 4 decimal places. Round "SS", "MS", "p-value" to 4 decimal places and "F" to 3 decimal places.)

ANOVA

Source of Variation                 SS                    df                    MS                  F                      p-value

Rows                                   _______          ______             ______           ______             _______

Columns                              _______          ______             ______           ______             _______

Error                                    _______          ______             ______           ______             _______

Total                                    _______           ______

a-2. At the 5% significance level, can you conclude that average scores differ by judge?

• Yes, since the p-value for judge is less than significance level.
• Yes, since the p-value for judge is greater than significance level.
• No, since the p-value for judge is less than significance level.
• No, since the p-value for judge is less than significance level.

a-3. Can you conclude that the judges seem inconsistent with their scoring?

• No
• Yes

b. At the 5% significance level, can you conclude that average scores differ by gymnast?

• No, since the p-value for gymnast is greater than significance level.
• No, since the p-value for gymnast is less than significance level.
• Yes, since the p-value for gymnast is greater than significance level.
• Yes, since the p-value for gymnast is less than significance level.

c. If average scores differ by gymnast, use Tukey’s HSD method at the 5% significance level to determine which gymnasts’ performances differ. (Negative values should be indicated by a minus sign. Round intermediate calculations to at least 4 decimal places. Round your answers to 2 decimal places.)

Population Mean Difference              Confidence Interval    Does the mean score differ at the 5%

                                                                                                            Significance level?

µ1-µ2                                                  [            ,           ]

µ1-µ3                                                  [            ,           ]

µ1-µ4                                                  [            ,           ]

µ1-µ5                                                  [            ,           ]

µ2-µ3                                                  [            ,           ]

µ2-µ4                                                  [            ,           ]

µ2-µ5                                                  [            ,           ]

µ3-µ4                                                  [            ,           ]

µ3-µ5                                                  [            ,           ]

µ4-µ5                                                  [            ,           ]

Answer 1

MINTAB used

a-1. Construct an ANOVA table. (Round intermediate calculations to at least 4 decimal places. Round "SS", "MS", "p-value" to 4 decimal places and "F" to 3 decimal places.)

Analysis of Variance

Source	DF	SS	MS	F-Value	P-Value
Gymnast	4	2.7493	0.6873	1.702	0.2418
Judge	2	1.4093	0.7047	1.745	0.2350
Error	8	3.2307	0.4038
Total	14	7.3893

a-2. At the 5% significance level, can you conclude that average scores differ by judge?

• No, since the p-value for judge is greater than significance level.

a-3. Can you conclude that the judges seem inconsistent with their scoring?

• No

b. At the 5% significance level, can you conclude that average scores differ by gymnast?

• No, since the p-value for gymnast is greater than significance level.

c. If average scores differ by gymnast, use Tukey’s HSD method at the 5% significance level to determine which gymnasts’ performances differ. (Negative values should be indicated by a minus sign. Round intermediate calculations to at least 4 decimal places. Round your answers to 2 decimal places.)

Tukey Simultaneous Tests for Differences of Means

Difference of Gymnast Levels	Difference of Means	SE of Difference	Simultaneous 95% CI	T-Value	Adjusted P-Value
2 - 1	0.967	0.519	(-0.83, 2.76)	1.86	0.405
3 - 1	-0.133	0.519	(-1.93, 1.66)	-0.26	0.999
4 - 1	-0.200	0.519	(-1.99, 1.59)	-0.39	0.994
5 - 1	0.333	0.519	(-1.46, 2.13)	0.64	0.963
3 - 2	-1.100	0.519	(-2.89, 0.69)	-2.12	0.298
4 - 2	-1.167	0.519	(-2.96, 0.63)	-2.25	0.254
5 - 2	-0.633	0.519	(-2.43, 1.16)	-1.22	0.741
4 - 3	-0.067	0.519	(-1.86, 1.73)	-0.13	1.000
5 - 3	0.467	0.519	(-1.33, 2.26)	0.90	0.889
5 - 4	0.533	0.519	(-1.26, 2.33)	1.03	0.836

Individual confidence level = 99.14%

Tukey Simultaneous Tests for Differences of Means

	Difference of Means	Difference of Means	Simultaneous 95% CI	Significance
	µ1-µ2	-0.967	(-2.76, 0.83)	No
	µ1-µ3	0.133	(-1.66, 1.93)	No
	µ1-µ4	0.200	(-1.59, 1.99)	No
	µ1-µ5	-0.333	(-2.13, 1.46)	No
	µ2-µ3	1.100	(-0.69, 2.89)	No
	µ2-µ4	1.167	(-0.63, 2.96)	No
	µ2-µ5	0.633	(-1.16, 2.43)	No
	µ3-µ4	0.067	(-1.73, 1.86)	No
	µ3-µ5	-0.467	(-2.26, 1.33)	No
	µ4-µ5	-0.533	(-2.33, 1.26)	No