Question

A team of researchers wanted to assess the effectiveness of the 4 components of a weight loss program by comparing the distribution of weight status across 4 different treatment groups. The data is summarized in the table below:

	Weight Status
Intervention Component	Normal Weight	Overweight	Obese
Control Group (n=100)	50	35	15
Calorie Counting (n=100)	70	20	10
Physical Activity (n=100)	73	15	12
Combination (n=100)	80	15	5

Is there a difference in weight status by intervention component? (Hint: Are intervention component and weight status independent?) Run the appropriate test at a 5% level of significance.

1. Compute the test statistic and indicate the outcome: (7.5pts)

1. 24.974
2. 0.428
3. 5.227
4. 11.73
5. Some other answer; indicate:

2. True or false: We DO NOT have statistically significant evidence at α=0.05 to show that intervention component and weight status are not independent. (1pt)

1. True
2. False

3. Indicate the most accurate p-value associated with the above-mentioned conclusion: (1pt)

1. p = not significant
2. p = 0.01
3. p = 0.005
4. p < 0.005
5. Some other answer; indicate:

Answer 1

Intervention Component	Normal Weight	Overweight	Obese	Total
Control Group (n=100)	50	35	15	100
Calorie Counting (n=100)	70	20	10	100
Physical Activity (n=100)	73	15	12	100
Combination (n=100)	80	15	5	100
Total	273	85	42	400

Null hypothesis : H_o : intervention component and weight status are independent

Alternate Hypothesis : H_a : intervention component and weight status are not independent

Test Statistic :

O : Observed Frequency

E: Expected Frequency

E :

Intervention Component	Normal Weight	Overweight	Obese	Total
Control Group (n=100)	(100*273)/400	(100*85)/400	(100*42)/400	100
Calorie Counting (n=100)	(100*273)/400	(100*85)/400	(100*42)/400	100
Physical Activity (n=100)	(100*273)/400	(100*85)/400	(100*42)/400	100
Combination (n=100)	(100*273)/400	(100*85)/400	(100*42)/400	100
Total	273	85	42	400

Intervention Component	Normal Weight	Overweight	Obese	Total
Control Group (n=100)	68.25	21.25	10.5	100
Calorie Counting (n=100)	68.25	21.25	10.5	100
Physical Activity (n=100)	68.25	21.25	10.5	100
Combination (n=100)	68.25	21.25	10.5	100
Total	273	85	42	400

O	E	(O-E)	(O-E)2	(O-E)2/E
50	68.25	-18.25	333.063	4.88
70	68.25	1.75	3.0625	0.0449
73	68.25	4.75	22.5625	0.3306
80	68.25	11.75	138.063	2.0229
35	21.25	13.75	189.063	8.8971
20	21.25	-1.25	1.5625	0.0735
15	21.25	-6.25	39.0625	1.8382
15	21.25	-6.25	39.0625	1.8382
15	10.5	4.5	20.25	1.9286
10	10.5	-0.5	0.25	0.0238
12	10.5	1.5	2.25	0.2143
5	10.5	-5.5	30.25	2.881
			Total	24.974

Compute the test statistic and indicate the outcome:

Ans : a. 24.974

Degrees of freedom =(Number of rows -1)(Number of columns -1) = (4-1)(3-1) = 3*2=6

p-value:

: 0.05

As p-value : 0.0000 less than the level of significance (0.05) Reject null hypothesis.;

Statistically significant evidence that intervention component and weight status are not independent.

We DO NOT have statistically significant evidence at α=0.05 to show that intervention component and weight status are not independent.

Ans:

a. True

p-value = 0.000

Most accurate p-value :

d. p < 0.005

even a p = not significant is also correct