Question

A researcher studied the flexibility of 11 women in an aerobic exercise class, 8 women in a modern dance class, and a control group of 10 women. One measurement she made was on the degree of spinal extension. Measurements were made before and after a 16 week training program and the difference between the two was calculated. Summary statistics for the data are given in Table 1 below. Let the significance level α = 0.01.

Table 1: Descriptive Statistics accompanying the flexibility study

Group	n	Mean	StDev
Aerobics	11	-0.05	0.79
Modern Dance	8	0.92	0.96
Control	10	0.11	0.5

A) Complete the ANOVA table by filling in the missing cells. Use 2-decimal precision where appropriate. MS_group has been reported for you.

Table 2: ANOVA results for the flexibility study

Source of Variation	Sum of Squares	df	Mean Square	F
Group			2.38
Error				Not meaningful
Total			Not meaningful	Not meaningful

B) The null/alternative pair appropriate for this ANOVA setting is:
    Note: You only get one submission for this problem.

Ho: At least one pair of σ_i differ
Ha: σ₁ = σ₂ = σ₃Ho: At least one pair of μ_i differ
Ha: μ₁ = μ₂ = μ₃     Ho: μ₁ = μ₂ = μ₃
Ha: μ₁ ≠ μ₂ ≠ μ₃Ho: σ₁ = σ₂ = σ₃
Ha: σ₁ ≠ σ₂ ≠ σ₃Ho: μ₁ = μ₂ = μ₃
Ha: At least one pair of μ_i differHo: σ₁ = σ₂ = σ₃
Ha: At least one pair of σ_i differHo: σ₁ ≠ σ₂ ≠ σ₃
Ha: σ₁ = σ₂ = σ₃Ho: μ₁ ≠ μ₂ ≠ μ₃
Ha: μ₁ = μ₂ = μ₃

C) The critical value for this hypothesis test is:  

D) The statistical decision and corresponding interpretation for this study is:
    Note: You only get one submission for this problem.

FTR Ho and do not conclude that there are differences in the mean difference in degree of spinal extension between the three groups.

Reject Ho and conclude that there are differences in the mean difference in degree of spinal extension between the three groups.    

Reject Ho and conclude that there are no differences in the mean difference in degree of spinal extension between the three groups.

FTR Ho and do not conclude that there are no differences in the mean difference in degree of spinal extension between the three groups.

E) Would you need to perform multiple comparisons?
    Note: You only get one submission for this problem.

Yes No   

Answer 1

A)

We are provided with the following information:

• n₁ = 11     x̅₁ = -0.05 s₁=0.79
• n₂ = 8     x̅₂ = 0.92 s₂=0.96
• n₃ = 10     x̅₃ = 0.11 s₃=0.5

The Total Mean is given by:

= 0.272758

So therefore the

=  

= 4.76218

&

= 14.9422

Therefore

=4.76218 + 14.9422

= 19.70438

There are three different groups (k=3) so degree of freedom is given by:

• df_Total = 29
• df_between = k - 1 = 2
• df_within = N - k = 29 - 3 = 26

Now

= 4.76218 / 2 = 2.38

= 14.9422/26

= 0.5747

F test statistics is

= 2.38 / 0.5747

= 4.12129

B)

Ha: At least one pair of μi differ Ho: μ1 = μ2 = μ3

C)

The critical value of F is F(2,26) = 5.5263

D)

Since the value of th test statistic(4.12129) is less than the critical value of F(5.5263) therefore we accept the null hypothesis.

a) FTR Ho and do not conclude that there are differences in the mean difference in degree of spinal extension between the three groups.

E)

Yes

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