Question

1. A clinical trial is run to compare weight loss programs and participants are randomly assigned to one of the comparison programs and are counseled on the details of the assigned program. Participants follow the assigned program for 8 weeks. The outcome of interest is weight loss, defined as the difference in weight measured at the start of the study (baseline) and weight measured at the end of the study (8 weeks), measured in pounds. Three popular weight loss programs are considered. The first is a low calorie diet. The second is a low fat diet and the third is a low carbohydrate diet. For comparison purposes, a fourth group is considered as a control group. Participants in the fourth group are told that they are participating in a study of healthy behaviors with weight loss only one component of interest. The control group is included here to assess the placebo effect (i.e., weight loss due to simply participating in the study). A total of twenty patients agree to participate in the study and are randomly assigned to one of the four diet groups. Weights are measured at baseline and patients are counseled on the proper implementation of the assigned diet (with the exception of the control group). After 8 weeks, each patient's weight is again measured and the difference in weights is computed by subtracting the 8 week weight from the baseline weight. Positive differences indicate weight losses and negative differences indicate weight gains. For interpretation purposes, we refer to the differences in weights as weight losses and the observed weight losses are shown below.

LOW CALORIE	LOW FAT	LOW CARBOHYDRATE	CONTROL
8	2	3	2
9	4	5	2
6	3	4	31
7	5	2	0
3	1	3	3

Is there a statistically significant difference in the mean weight loss among the four diets?

Answer 1

the necessary calculation table be:-

LOW CALORIE ()	LOW FAT ()	LOW CARBOHYDRATE ()	CONTROL ()
8	2	3	2	1.96	1	0.16	31.36
9	4	5	2	5.76	1	2.56	31.36
6	3	4	31	0.36	0	0.36	547.56
7	5	2	0	0.16	4	1.96	57.76
3	1	3	3	12.96	4	0.16	21.16
sum=33	sum=15	sum=17	sum=38	sum=21.20	sum=10	sum=5.20	sum=689

[ i have considered the low calorie group as group 1, low fat group as group 2 , low carbohydrate group as group 3 and control group as group 4 ]

for low calorie group:-

for low fat group:-

for low carbohydrate group:-

for control group:-

hypothesis:-

mean group 1 = mean group 2 = mean group 3 = mean of group 4

mean group 1 ≠ mean group 2 ≠ mean group 3 ≠ mean group 4

necessary calculation:-

grand mean :-

sum of squares between the group be:-

sum of squares for error be:-

725.5992

number of groups (k) = 4

total number of subjects (N) = 5*4 =20

the ANOVA table be :-

source	sum of squares (SS)	df	mean of squares (MS)	F statistic	p value
treatments	78.95	(k-1) = (4-1) = 3	78.95 / 3 =26.3167	= 0.5803	= 0.6364 [ for alpha = 0.05, df = (3,16), F statistic = 0.5803,using calculator ]
error	725.5992	(N-k) =(20-4) = 16	725.5992 / 16 =45.35
total	804.5492	19

p value =0.6364 > 0.05 (alpha)

so, we fail to reject the null hypothesis.

[as nothing is mentioned , i am using level of significance = 0.05.

if you use 0.01 or 0.10 level of significance, then also you will get the same decision.]

conclusion:-

we conclude that,

there is no statistically significant difference in the mean weight loss among the four diets at 0.05 level of significance.

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