Question

1. In a study of the effectiveness of certain exercises in weight reduction, a group of 16 persons engaged in these exercises for three months and showed the weights, in pounds, before exercises (X) and after exercises (Y):  

Individual	X	Y
1	211	198
2	180	173
3	171	172
4	214	209
5	182	179
6	194	192
7	160	161
8	182	182
9	172	166
10	155	154
11	185	181
12	167	164
13	203	201
14	181	175
15	245	233
16	146	142

1. Perform a pooled two-sample t-test at 0.05 level of significance to see if the data provide sufficient evidence to show a weight reduction. What assumptions are required for this test to be valid?
2. Perform a paired t-test at 0.05 level of significance to see if the data provide sufficient evidence to show a weight reduction. What assumptions are required for this test to be valid?
3. Are the conclusions from (a) and (b) consistent, and why or why not?

Answer 1

In order to solve this question I used R software.

R codes and output:

> d=read.table('data.csv',header=T,sep=',')
> head(d)
Individual X Y
1 1 211 198
2 2 180 173
3 3 171 172
4 4 214 209
5 5 182 179
6 6 194 192
> attach(d)
The following objects are masked from d (pos = 3):

Individual, X, Y

> var.test(X,Y)

F test to compare two variances

data: X and Y
F = 1.2189, num df = 15, denom df = 15, p-value = 0.7064
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
0.4258673 3.4885237
sample estimates:
ratio of variances
1.218872

> t.test(X,Y,var.equal=T,alternative='greater')

Two Sample t-test

data: X and Y
t = 0.49095, df = 30, p-value = 0.3135
alternative hypothesis: true difference in means is greater than 0
95 percent confidence interval:
-10.13551 Inf
sample estimates:
mean of x mean of y
184.250 180.125

> t.test(X,Y,paired=T,,alternative='greater')

Paired t-test

data: X and Y
t = 4.06, df = 15, p-value = 0.0005132
alternative hypothesis: true difference in means is greater than 0
95 percent confidence interval:
2.34387 Inf
sample estimates:
mean of the differences
4.125

Que.a

Pooled two sample t test:

t = 0.49095, df = 30, p-value = 0.3135

Since p-value is greater than 0.05, we accept null hypothesis that two means are equal and conclude that data does not provide sufficient evidence to show a weight reduction.

Que.b

Paired t test:

t = 4.06, df = 15, p-value = 0.0005132

Since p-value is less than 0.05, we reject null hypothesis that difference between two means equal to zero and conclude that data provide sufficient evidence to show a weight reduction.

The assumptions of the two-sample t-test are:

1. The data are continuous (not discrete).

2. The data follow the normal probability distribution.

3. The variances of the two populations are equal. (If not, the Welch Unequal-Variance test is used.)

4. The two samples are independent. There is no relationship between the individuals in one sample as compared to the other (as there is in the paired t-test).

5. Both samples are simple random samples from their respective populations

Que.c

Conclusion from a and b are not consistent. Because these two are different tests. One for independent sample and other is for dependent sample.