Comparing Two Population Means Using Independent Samples
Pooled-t test
Definitions Two samples are independent if the sample values selected from one population are not related to or
somehow paired or matched with the sample values selected from the other population. If there is some relationship so
that each value in one sample is paired with a corresponding value in the other sample, the samples are dependent.
Dependent samples are often referred to as matched pairs, or paired samples.
Independent Samples: One group of subjects is treated with the cholesterol reducing drug Lipitor, while a second and
separate group of subjects is given a placebo. These two groups are independent because the individuals in the treatment
group are in no way paired or matched with corresponding members in the placebo group.
Dependent Samples: The effectiveness of a diet is tested using weights of subjects measured before and after the diet
treatment. Each “before” value is matched with the “after” value because each before/after pair of measurements comes
from the same person.
Pooled-t test
Ho: 1 - 2 = 0
Ha:
1. 1 - 2 > 0
2. 1 - 2 < 0
3. 1 - 2  0
t
Test Statistic:
x1  x 2  0
sP
Where
sP
=
1
1

n1 n 2
(n1  1) s12  (n2  1) s 22
n1  n2  2
Rejection Region: For a specified , we can reject Ho if
1. t  t ,n1  n2 2
2. t  t ,n1  n2 2
3. t  t / 2,n1  n2 2 or t  t / 2,n1  n2 2
ASSUMPTIONS:
1. The two samples are independent.
2. The two samples are randomly selected from normally distributed populations.
3.
 12   22
Example British health officials have recently expressed concern about problems associated with vitamin D deficiency
among certain immigrants. Doctors have conjectured that such a deficiency could be related to the amount of fiber in a
persons diet. The chemical compound H-labeled 25-hydroxy-vitamin D3 (25(OH)D3) is the major circulating form of
vitamin D, and its plasma half-life is intimately related to the body's vitamin D level. An experiment to compare plasma
half-lives for two groups of health individuals, one placed on a normal diet and the other placed on a high-fiber diet,
resulted in the following data (from "Reduced Plasma Half-Lives of Radio-Labeled 25(OH)D3 in Subjects Receiving a
High-Fiber Diet" Brit. J. of Nutrit. (1983): 213-216).(8 pts.)
Normal Diet:
High-fiber
19.1
12.0
24.0
13.0
28.6
13.6
29.7
20.5
30.0
22.7
34.8
23.7
24.8
Does this data suggest that the average plasma half-life for persons placed on a normal diet exceeds the average half-life
for those on a high fiber diet? Test the appropriate hypothesis at =.05. Use output below.
Minitab Commands for pooled-t test: > stat> basic statistics > two-sample t > click assume equal variances box
********************Minitab Output********************
Two sample T for normal vs high
N Mean StDev SE Mean
normal 6
27.70 5.44
2.2
high
7
18.61 5.55
2.1
95% CI for mu normal - mu high: ( 2.3, 15.8)
T-Test mu normal = mu high (vs >): T= 2.97 P=0.0064 DF= 11
Both use Pooled StDev = 5.50
******************************************************
Comparing Two Population Means Using Dependent Samples
Paired-t Test
Ho: d = 1 - 2 = 0
1. d > 0
2. d < 0
3. d  0
Ha:
Test Statistic:
t
d
sd
n
Rejection Region: For a specified , we can reject Ho if
1. t  t , n 1
2. t  t ,n 1
3. t  t / 2,n 1 or t  t / 2,n 1
Assumption: Differences are randomly selected from a normal population of such differences.
Example: A teacher randomly selects ten students to participate in a week of training designed to improve their typing
speed. The teacher times their speed before and after the course, to see if the course is worth the time and expense. State
the appropriate null and alternative hypothesis and conduct test of hypothesis at =.05
After Course
Before Course
Difference
55
50
5
46
42
4
78
70
8
61
63
-2
52
58
-6
45
35
10
47
46
1
57
52
5
71
60
11
58
49
9
Minitab Commands for a paired-t test: > stat > basic statistics > 1-sample-t
Minitab Output:
T-Test of the Mean
Test of mu = 0.00 vs mu > 0.00
Variable
DIFF
N
Mean
StDev
SE Mean
T
P
10
4.50
5.48
1.73
2.60
0.014