Question

Obesity and being overweight is caused by excessive adipose tissues, or body fat. Visceral fat, surrounding internal organs, is clearly associated with heart disease and diabetes. Subcutaneous fat, found just below the skin, often in the buttocks and thighs, is not.

A study examined the impact of exercise type on visceral and subcutaneous fat. Overweight, sedentary, but otherwise disease‑free adults were randomly assigned to three exercise regimens for eight months, aerobic training, resistance training, and aerobic plus resistance training. All exercise sessions were supervised to ensure correct completion. The subjects’ body fat amount was assessed with computed tomography imaging, in squared centimeters, cm^{2 at the beginning and end of the experiment. The study report contains the following information about the visceral fat reduction in cm2 achieved by the subjects in each group. Note that the reduction is indicated as a positive value.}

Treatment	n	¯¯¯x	s
Aerobic training	36	15.9	34
Resistance training	39	0.8	19
Aerobic plus resistance training	35	10.9	33

(a) Do the standard deviations satisfy the rule of thumb for safe use of ANOVA?

- Yes

- No

(b) The report does not provide the distributions of visceral fat reduction. ANOVA is nonetheless safe for these data. Select the correct response.

- The sample sizes are large, and the F test is robust against non‑Normality with large samples.

- None of the options are correct.

- The sample standard deviations are very large, and the F test is robust against wildy varying standard deviations.

- The samples are all obviously Normally distributed, and the F test is robust against Normal distributions.

(c) Calculate the overall mean response ¯x, the sum of squares for groups (SSG), the mean square for groups (MSG), the sum of squares for error (SSE), and the mean square for error, MSE. (Enter all of your answers rounded to two decimal places.)

Overall mean =

SSG=

MSG=

SSE=

MSE=

(d) Obtain the ANOVA F statistic and the test P‑value. Is there evidence that the mean visceral fat reduction in overweight adults depends on which three exercise programs they follow? (Enter your answer rounded to two decimal places.)

F=

With degrees of freedom 2 and 107, or 100 in Table F, we find that Answer 1<P‑value<Answer 2 . (Enter your answers rounded to two decimal places.)

Answer 1:

Answer 2:

Answer 1

a)

Largest SD = 34 < 2* Smallest SD = 2*19 = 38

Yes, the standard deviations satisfy the rule of thumb for safe use of ANOVA

b)

The sample sizes are large

c)

Treatment	n	X bar	s		n*(X bar - Overall X bar)^2	(n-1)*S^2
Aerobic training	36	15.9	34		1736.138889	40460
Resistance training	39	0.8	19		2593.97503	13718
Aerobic plus resistance training	35	10.9	33		132.3378088	37026
Total	110			Total	4462.451728	91204
k	3
Overall Mean or Overall X bar	8.955454545
	SS	df	MS	F
Groups	4462.451728	2	2231.226	2.617661
Error	91204	107	852.3738
Total	95666.45173	109
	SS	df	MS	F
Groups	SUM(n*(X bar - Overall X bar)^2)	k-1	SSG/dfG	MSG/MSE
Error	SUM((n-1)*S^2)	n-k	SSE/dfE
Total	SSG+SSE	n-1

d)

F stat = 2.6177

P value = 0.0776, With degrees of freedom 2 and 107 (Use F table)

P value > 0.05

there is not enough evidence that the mean visceral fat reduction in overweight adults depends on which three exercise programs they follow at 5% significance level