Question

1. In a study of the effectiveness of physical exercise in weight reduction, a group of 16 people (randomly selected) engaged in a prescribed program of physical exercise for one month. Each subject’s weight (in pounds) is measured two times (before and after the physical exercise). The data can be found in the excel file Exercise with variables name Subject, Before, and After. At the 5% significance level, do we have sufficient evidence to conclude that the physical exercise program is effective? (21 points total)

Hint: The physical program will be considered effective, if before weight is greater than after weight. Difference = Before – After

1. Label the Parameters: (3 points)

2. State null and alternative hypotheses: (2 points)

H₀:

H_a:

3. Propose an appropriate hypothesis test. Explain! (4 points)
4. Verify the required conditions for the proposed hypothesis test: (4 points)

Randomization assumption:

Normality assumption: 

Note: Attach R codes and normal QQ plot in the following space:

5. Report a test statistic, df, and p-value. (3 points)

Test statistic =

df =

p-value =

Note: Attach R codes and outputs in the following space:

6. Make a decision of hypothesis test: (2 points)
7. Make a conclusion of hypothesis test in context: (3 points)

   Subject	Before	After
   1	201	196
   2	178	171
   3	169	170
   4	212	207
   5	180	177
   6	192	190
   7	158	159
   8	180	180
   9	170	164
   10	153	152
   11	183	179
   12	165	162
   13	201	199
   14	179	173
   15	236	231
   16	144	140

Answer 1

a) The average weight before physical exercise is mu1
The average weight after physical exercise is mu2

b) H0: The physical program will be considered not effective
H1: The physical program will be considered effective

c) Paired t - test is used since the samples are not independent

d) Normality assumption:

The distribution is follows normal distribution

e) From the given data

Test Statistic t = 5.0814

P-value = 0.000

df = 15

f) since p-value < alpha 0.05 so we reject H0

g) Thus we conclude that  the physical program will be considered effective