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 3 exercise regimens for eight months: aerobic training, resistance training, 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 cm2) 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	37	25.6	31
Resistance training	41	8.6	51
Aerobic plus resistance training	31	28.7	35

Calculate the overall mean response x, the mean square for groups MSG, and the mean square for error MSE. (Round your answers to two decimal places.)

x= MSG= MSE=

Find the ANOVA F statistic and approximate P-value. (Round your answer to two decimal places)
F =

P-value:

Answer 1

Overall mean is:

\(\bar{x}=\frac{\sum_{i=1}^{k} n_{i} \bar{x}_{i}}{\sum_{i=1}^{k} n_{i}}=\frac{37 \times 25.6+41 \times 8.6+31 \times 28.7}{37+41+31}=19.93\)

\(\mathrm{We}\) are given \(n=37+41+31=109\) and \(k=3 .\) So, \(d f_{1}=k-1=3-1=2\) and \(d f_{2}=n-k=109-3=106 .\)

\(\begin{aligned} S S G &=\sum_{i=1}^{k} n_{i}\left(\bar{x}_{i}-\bar{x}\right)^{2}=37(25.6-19.93)^{2}+41(8.6-19.93)^{2}+31(28.7-19.93)^{2}=8834.24 \\ S S E &=\sum_{i=1}^{k}\left(n_{i}-1\right) s_{i}^{2}=(37-1) 31^{2}+(41-1) 51^{2}+(31-1) 35^{2}=175386.00 \\ M S G &=\frac{S S G}{d f_{1}}=\frac{8834.24}{2}=4417.12 \\ M S E &=\frac{S S E}{d f_{2}}=\frac{175386.00}{106}=1654.59 \\ F &=\frac{M S G}{M S E}=\frac{4417.12}{1654.59}=2.67 \end{aligned}\)

So, by using the MSEXCEL FDIST \((2.67,2,106)\), we get \(P\) -value \(=0.07\). (Answers are in red)