3. When to use the test:
• When the two samples are independent
• The sample sizes are small i.e. less than 30
and population standard deviation is not given
• We are assuming unequal variance of the two
populations
5. • There were two types of drugs (1 and 2) that were tried on some
patients for reducing weight. There were 8 adults who were subject
to drug 1 and seven adults who were administered drug 2. The
decrease in weight (in pounds) is given below:
• Do the drugs differ significantly in their effect on deceasing weight?
You may use 5 percent level of significance. Assume that the
variances of two populations are not same.
Drug 1 Drug 2
10 12
8 10
12 7
14 6
7 12
15 11
13 12
11
6. Solution:
X1 X2
10 12
8 10
12 7
14 6
7 12
15 11
13 12
11
Sum 90 70
Mean 11.25 10
1. Calculate mean of the two samples:
2
1
:
o
H
2
1
:
a
H
2
1
2
1
Drug
Drug
Let
7. • 2. Calculate standard deviation or variance
X1 X2 X1-11.25 X2-10 (X1-11.25)2 (X2-10)2
10 12 -1.25 2 1.5625 4
8 10 -3.25 0 10.5625 0
12 7 0.75 -3 0.5625 9
14 6 2.75 -4 7.5625 16
7 12 -4.25 2 18.0625 4
15 11 3.75 1 14.0625 1
13 12 1.75 2 3.0625 4
11 -0.25 0.0625
Sum 90 70 55.5 38
Mean 11.25 10
1
)
( 2
n
x
x
s
517
.
2
816
.
2
2
1
s
s
8.
9.
10.
11. Additional problems for practice:
• A marketing research firm tests the effectiveness of a new flavoring for a leading beverage using a
sample of 20 people, half of whom taste the beverage with the old flavoring and the other half who
taste the beverage with the new favoring. The people in the study are then given a questionnaire
which evaluates how enjoyable the beverage was. The scores are as in Table 1. Determine whether
there is a significant difference between the perception of the two flavorings at 5 per level of
significance. Assume that the variances of two populations are not same.
New Old
20 12
32 8
2 6
25 16
5 12
18 14
21 10
7 18
28 4
40 11
12. Answer:
Variable 1 Variable 2
Mean 19.8 11.1
Variance 150.6222 18.76667
Observations 10 10
Hypothesized Mean Difference 0
df 11
t Stat 2.113863
t Critical two-tail 2.200985
13. • The table below shows the observed pollution
indexes of air samples in two areas of a city. Test
the hypothesis that the mean pollution indexes
are the same for the two areas. Assume that the
variances of two populations are not same. (Use
α = 0.05)
Area A Area B
2.92 1.84
1.88 0.95
5.35 4.26
3.81 3.18
4.69 3.44
4.86 3.69
5.81 4.95
5.55 4.47
14. Answer:
Variable 1 Variable 2
Mean 4.35875 3.3475
Variance 1.912013 1.833593
Observations 8 8
Hypothesized Mean Difference 0
df 14
t Stat 1.477891
t Critical two-tail 2.144787
