(a)
To explain do the standard deviations satisfy the rule of thumb for safe use of ANOVA.
(a)
Answer to Problem 24.10AYK
Yes, the standard deviations satisfy the rule of thumb for safe use of ANOVA.
Explanation of Solution
In the question, it is given that a study examined the impact of exercise type on visceral and subcutaneous fat. Overweight but otherwise disease free adults were assigned to three exercise regiments for eight months. The study report contains the information given in the question about the visceral fat reduction achieved by the subjects in each group. Thus, the standard deviation rule of thumb is that the largest sample standard deviation is no more than twice as large as the smallest standard deviation. Thus, the standard deviation is already given in the question, so let us find the ratio as:
Thus, the standard deviations satisfy the rule of thumb for safe use of ANOVA.
(b)
To explain why ANOVA is nonetheless safe for these data.
(b)
Explanation of Solution
In the question, it is given that a study examined the impact of exercise type on visceral and subcutaneous fat. Overweight but otherwise disease free adults were assigned to three exercise regiments for eight months. The study report contains the information given in the question about the visceral fat reduction achieved by the subjects in each group. Thus, the standard deviation rule of thumb is that the largest sample standard deviation is no more than twice as large as the smallest standard deviation. And the standard deviations satisfy the rule of thumb for safe use of ANOVA. The report does not provide the distributions of visceral fat reduction. But ANOVA nonetheless safe for these data because as we look at the means and standard deviations given then we can assume that they are approximately
(c)
To calculate the overall
(c)
Answer to Problem 24.10AYK
The overall mean response
Explanation of Solution
In the question, it is given that a study examined the impact of exercise type on visceral and subcutaneous fat. Overweight but otherwise disease free adults were assigned to three exercise regiments for eight months. The study report contains the information given in the question about the visceral fat reduction achieved by the subjects in each group. Thus, the standard deviation rule of thumb is that the largest sample standard deviation is no more than twice as large as the smallest standard deviation. And the standard deviations satisfy the rule of thumb for safe use of ANOVA. Thus, the overall mean response
The calculations are as:
Treatment | n | x bar | n*x bar | n*(x-x total)^2 | s^2 | SS=(n-1)*s^2 |
1 | 36 | 15.9 | =BO50*BP50 | =BO50*(BP50-$BP$54)^2 | =34^2 | =(BO50-1)*BS50 |
2 | 39 | 0.8 | =BO51*BP51 | =BO51*(BP51-$BP$54)^2 | =19^2 | =(BO51-1)*BS51 |
3 | 35 | 10.9 | =BO52*BP52 | =BO52*(BP52-$BP$54)^2 | =33^2 | =(BO52-1)*BS52 |
Total | =SUM(BO50:BO52) | =SUM(BQ50:BQ52) | =SUM(BR50:BR52) | =SUM(BT50:BT52) | ||
Grand mean | =BQ53/BO53 | SStr | SSE |
Source of variation | df | SS | MS | F |
Groups | =3-1 | 4462.452 | =BP60/BO60 | =BQ60/BQ61 |
Error | =110-3 | 91204 | =BP61/BO61 | |
Total | =BO60+BO61 | =SUM(BP60:BP61) |
The result will be as:
Treatment | n | x bar | n*x bar | n*(x-x total)^2 | s^2 | SS=(n-1)*s^2 |
1 | 36 | 15.9 | 572.4 | 1736.162 | 1156 | 40460 |
2 | 39 | 0.8 | 31.2 | 2593.946 | 361 | 13718 |
3 | 35 | 10.9 | 381.5 | 132.344 | 1089 | 37026 |
Total | 110 | 985.1 | 4462.452 | 91204 | ||
Grand mean | 8.955455 | SStr | SSE |
Source of variation | df | SS | MS | F |
Groups | 2 | 4462.452 | 2231.226 | 2.617661 |
Error | 107 | 91204 | 852.3738 | |
Total | 109 | 95666.45 |
Thus, the overall mean response
(d)
To obtain the ANOVA F statistic and the test P-value and explain is there evidence that the mean visceral fat reduction in overweight adults depends on which three exercise programs they follow.
(d)
Answer to Problem 24.10AYK
The ANOVA F statistic is
Explanation of Solution
In the question, it is given that a study examined the impact of exercise type on visceral and subcutaneous fat. Overweight but otherwise disease free adults were assigned to three exercise regiments for eight months. The study report contains the information given in the question about the visceral fat reduction achieved by the subjects in each group. Thus, the standard deviation rule of thumb is that the largest sample standard deviation is no more than twice as large as the smallest standard deviation. And the standard deviations satisfy the rule of thumb for safe use of ANOVA. And from part (d) we have the ANOVA table as:
Source of variation | df | SS | MS | F |
Groups | 2 | 4462.452 | 2231.226 | 2.617661 |
Error | 107 | 91204 | 852.3738 | |
Total | 109 | 95666.45 |
Thus, the P-value is
Thus, we do not have sufficient evidence to conclude that the mean visceral fat reduction in overweight adults depends on which three exercise programs they follow.
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