You are part of a research team investigating the effects of aerobic cardiovascular and moderate weight resistance training on the BMI (Body Mass Index) of moderately overweight but not obese teens (BMI range: 25.0 - 29.9). Your team has carefully recruited four random samples of about fifty overweight adolescents each. All subjects were given access to a local gym, had an initial fitness session in which they were introduced to the facilities and their use, and passed a medical exam to ensure they were healthy enough for an exercise program. They were randomly assigned to one of four groups:

• Group A: Not assigned to any specific exercise program ('control' or 'no treatment' group)
• Group B: Asked to follow a cardiovascular exercise program 3 times per week
• Group C: Asked to follow a moderate weight resistance training program 3 times per week
• Group D: Asked to follow an exercise program that incorporated both cardiovascular activity and resistance training 3 times per week

The adolescents' BMI's were measured by their general practitioner or another medical professional before they began the study and again two months later. You have been asked to provide the top-line summary of preliminary results for your team to review based on the data file you have received, using a confidence level of 95%.

The data file for this project includes the following variables:

Id: The identification number given to each participant for confidentiality

Group: The treatment group the individual is assigned to (A, B, C or D)

BMI_1: BMI measurement of participants before they began the study

BMI_2: BMI measurement of participants 2 months later

1. a. What was the average change in BMI for subjects in each of the four groups? (These averages should be expressed as confidence intervals. i.e. Provide a 95% confidence interval for the change in BMI for subjects in each group)

b. Perform a hypothesis test to determine which groups showed a significant reduction in BMI. to include p-values for each group for this specific test

c. Compare the confidence intervals (from part a) and the hypothesis test results (from part b) and make connections between the two.

1. Which exercise program was more effective in reducing BMI: the cardiovascular exercise or weight resistance training programs? Can I assume equal variances.
2. What can you conclude about the variances of the change in BMI for the cardio and weight training group compared to the group that was not asked to follow a specific exercise program?
3. a. Verify or disprove that the average starting BMIs (BMI_1) of the four groups were the same.

Describe the test being used and verify its assumptions.

b. What impact does this have on the reliability of your results above?

id	Group	BMI_1	BMI_2
1	A	27.5	27.9
2	A	29.2	29.4
3	A	28.7	29.6
4	A	26.3	27.5
5	A	28.2	29
6	A	25	24
7	A	27	26.1
8	A	28.1	26.6
9	A	27.8	27
10	A	27.5	27.5
11	A	27.8	28.3
12	A	28.7	30.1
13	A	27.7	27.5
14	A	28.1	27.7
15	A	27.3	26.3
16	A	27	27.2
17	A	26	25.3
18	A	26.5	26.6
19	A	28.5	28.3
20	A	28.3	28.1
21	A	26.1	27
22	A	27.1	25.9
23	A	27.5	27.7
24	A	27.2	26.6
25	A	25.8	27.4
26	A	26.1	26.5
27	A	27.2	26.5
28	A	28	28.1
29	A	27.9	26.6
30	A	27.3	26.9
31	A	28.5	27.3
32	A	28.1	27.1
33	A	29.3	28.1
34	A	27.9	27.3
35	A	25.9	25.1
36	A	26.8	27.2
37	A	26.6	25.6
38	A	27.1	28.2
39	A	27.6	27
40	A	27.8	28
41	A	28.2	27.6
42	A	26.6	26.2
43	A	29	29.6
44	A	26.6	28.1
45	A	28.4	28.6
46	A	27.7	28.5
47	A	28.8	27.1
48	A	28.2	26.7
49	A	27.6	27.9
50	A	27	26.2
51	A	26.1	25.9
52	A	26.2	26.9
53	A	29	28.5
54	A	28	28.1
55	B	29.3	28.6
56	B	26.7	26.3
57	B	27.4	26.6
58	B	28	27.7
59	B	26.2	25.1
60	B	27.8	27.1
61	B	29.6	29
62	B	28.4	27.5
63	B	26.4	25.6
64	B	27.2	26.5
65	B	27.8	27.7
66	B	26.7	26.3
67	B	27.4	26.7
68	B	28.6	28.3
69	B	25.6	25.2
70	B	28	27.4
71	B	27.2	26.6
72	B	26.4	25.4
73	B	25.3	24.2
74	B	26.8	26.1
75	B	28	26.9
76	B	28.6	28.3
77	B	27.8	27.1
78	B	26.9	26
79	B	26.7	26.3
80	B	28.8	28.8
81	B	27.2	26.6
82	B	27.1	26.4
83	B	27.4	26.6
84	B	26	25.4
85	B	26.9	26.3
86	B	28.1	28.2
87	B	25.7	25.4
88	B	26.5	26.4
89	B	26.3	25.5
90	B	29	29
91	B	28.7	27.9
92	B	27.9	26.8
93	B	29.5	29.1
94	B	27.4	26.5
95	B	25.7	24.6
96	B	26.6	26
97	B	26.6	26.2
98	B	27.7	26.2
99	B	27.8	26.8
100	B	27.5	26.9
101	B	28.3	27.9
102	B	28.5	28
103	B	27.5	26.7
104	B	28.9	28.4
105	B	28.7	28.5
106	B	27.2	26.2
107	B	27.9	27.1
108	B	27.8	26.7
109	B	26.8	25.8
110	B	27.6	27.3
111	C	27.7	27.6
112	C	27.3	26.7
113	C	27.1	26.6
114	C	27.9	27.2
115	C	26.9	27.3
116	C	29.2	29.6
117	C	27.5	26.6
118	C	27.6	27.1
119	C	26.9	26.2
120	C	27.4	27.6
121	C	27.8	27.8
122	C	28.6	28.8
123	C	27.6	26.5
124	C	26.9	26.9
125	C	29.6	28.9
126	C	26.8	26.5
127	C	27	27.1
128	C	27.1	26.1
129	C	27.4	27
130	C	27.3	27
131	C	27.7	27.5
132	C	29	28.4
133	C	26.5	25.8
134	C	28.7	28.5
135	C	29.3	28.5
136	C	27.5	27.1
137	C	28	27.4
138	C	26	26
139	C	26.9	27.5
140	C	27.6	27.6
141	C	27.4	27.4
142	C	28.3	27.6
143	C	27	26.9
144	C	28.9	28.7
145	C	26.4	26.2
146	C	28.1	27.5
147	C	28.5	28
148	C	26.3	25.3
149	C	27.5	27.1
150	C	27.8	26.6
151	C	28.3	27.6
152	C	26.8	26.6
153	C	29	29.1
154	C	26.3	25.6
155	C	28.5	28
156	C	27.2	27.5
157	C	28.1	28.2
158	C	27.7	27.1
159	C	26.2	25.9
160	C	27.3	27.6
161	C	27.5	27.2
162	D	28.7	27.8
163	D	27.8	26.7
164	D	27.9	27.6
165	D	28.7	27.8
166	D	26.1	25.9
167	D	27.5	27.1
168	D	26.6	25.3
169	D	27.7	28.1
170	D	28.3	27
171	D	27.3	26.7
172	D	28.4	27.7
173	D	26.7	26.6
174	D	27.7	26.8
175	D	26.7	26.3
176	D	27.9	26.6
177	D	29.7	29.8
178	D	27.6	26.4
179	D	25.8	24.8
180	D	26.6	25.4
181	D	28.2	27.6
182	D	29.2	28.2
183	D	28.3	27.4
184	D	27.9	27.7
185	D	27.2	26.5
186	D	25	24.6
187	D	27.5	26.7
188	D	26.7	25.8
189	D	27.1	26.6
190	D	28	27.7
191	D	27	26.7
192	D	27	26.2
193	D	28.6	28.5
194	D	27.8	27.4
195	D	26.1	25.3
196	D	28.2	27.5
197	D	25.7	25.8
198	D	26.3	25.3
199	D	29.3	28.4
200	D	27.6	26.9
201	D	27.5	27.6
202	D	28.6	28.1
203	D	26.3	26
204	D	27.9	28.1
205	D	25.5	25.5
206	D	28.5	28.2
207	D	28	27.5
208	D	26	25.8
209	D	27.9	27.3
210	D	27.1	25.9
211	D	26.5	25.8
212	D	26.8	25.9
213	D	27.3	27.2
214	D	25.6	25.1
215	D	25.5	24.6
216	D	28.5	28.2