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	7.8	7.8	8.4
Gymnast 2	7.4	8.2	7.9
Gymnast 3	9.3	9.8	8.9
Gymnast 4	8.4	9.3	9.3
Gymnast 5	9.4	8.3	9.5

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 7.8 Judge 2 Judge 3 Gymnast 1 Gymnast 2 Gymnast 3 Gymnast 4 Gymnast 5 8.9 9.4 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 "P' to 3 decimal places.) ANOVA Source of Variation Rows Columns Error Total df MS p-value 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 greater 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? b. At the 5% significance level, can you conclude that average scores differ by gymnast? Yes since the p-value for gymnast is less than significance level Yes since the p-value for gymnast is greater than significance level No since the p-value for gymnast is less than significance level 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.) Population Mean Difference Does the mean score differ at the 5% significance level? Confidence Interval H2 -H4 H3 - H5

Answer 1

Here, H₀ : There is no significance difference in the average scores by the judges.

H₁ : There is a significance difference in the average score by the judges.

a1) Step 1: Feed the data in excel worksheet.

Step 2: Click on data analysis in data tab.

Step 3: Select ANOVA-one factor from the dialog box.

Step 4: Select the values as the input the range.

Step 5: Now, since we have to answer two types of questions like hypothesis for gymnasts and hypothesis for judges, we will be finding anova table under the two categories that is grouped by columns and grouped by rows.

Select these two options one by one.

If grouped by rows i.e. grouped by gymnasts then

Anova: Single Factor
SUMMARY
Groups	Count	Sum	Average	Variance
Row 1	3	24	8	0.12
Row 2	3	23.5	7.833333	0.163333
Row 3	3	28	9.333333	0.203333
Row 4	3	27	9	0.27
Row 5	3	27.2	9.066667	0.443333
ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Between Groups	5.557333	4	1.389333	5.788889	0.011211	3.47805
Within Groups	2.4	10	0.24
Total	7.957333	14

If grouped by columns i.e. grouped by judges then

Anova: Single Factor
SUMMARY
Groups	Count	Sum	Average	Variance
Column 1	5	42.3	8.46	0.788
Column 2	5	43.4	8.68	0.697
Column 3	5	44	8.8	0.43
ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Between Groups	0.297333	2	0.148667	0.232898	0.795733	3.885294
Within Groups	7.66	12	0.638333
Total	7.957333	14

.a2) No, the average scores by the judges does not differ as the p-value = 0.795733 is much more than the significance level 0.05.
When the p-value is more than the significance level, we accept the null hypothesis

b) Yes, the average scores of the gymnasts differ as the p-value = 0.011 is much less than the significance level 0.05.

When the p-value is less than the significance level, we reject the null hypothesis.

c)

Gymnasts		Difference	n(group 1)	n(group 2)	SE	q
Gymnast 1	Gymnast 2	0.16666667	3	3	0.282843	0.589256
Gymnast 1	Gymnast 3	1.33333333	3	3	0.282843	4.714045
Gymnast 1	Gymnast 4	1	3	3	0.282843	3.535534
Gymnast 1	Gymnast 5	1.06666667	3	3	0.282843	3.771236
Gymnast 2	Gymnast 3	1.5	3	3	0.282843	5.303301
Gymnast 2	Gymnast 4	1.16666667	3	3	0.282843	4.12479
Gymnast 2	Gymnast 5	1.23333333	3	3	0.282843	4.360492
Gymnast 3	Gymnast 4	0.33333333	3	3	0.282843	1.178511
Gymnast 3	Gymnast 5	0.26666667	3	3	0.282843	0.942809
Gymnast 4	Gymnast 5	0.06666667	3	3	0.282843	0.235702