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Chapter 8: Estimation
Section 5: Estimating 1  2 and p1  p2
►Two
samples are independent
if the data values obtained
from one are unrelated to the
values from the other.
►The samples are dependent if
each data value from one
sample is paired in a natural
way with a data value from the
other sample.
Examples
► In
a medical experiment, a sample of subjects is
randomly divided into two groups. One group is
given a specific treatment and the other group is
given a placebo. After a certain period of time,
both groups are measured for the same condition.
► A group of students in an English composition
course is given a pretest. After the course, the
same students are given a post-test.
Confidence Interval (large samples)
2
2
2
2
s1 s2
s1 s2
( x1  x2 )  zc
  1   2  ( x1  x2 )  zc

n1 n2
n1 n2
Example (large sample)
► Suppose a biologist is studying data from Yellowstone
streams. Results are summarized below:
YEAR
1983-1988
1990-1993
SAMPLE
SIZE
n1 = 167
n2 = 125
MEAN
x1 = 5.2
x2 = 6.8
-Are the samples dependent or independent?
-Compute a 95% confidence interval for 1  2
STANDARD
DEVIATION
s1 = 1.9
s2 = 2.3
Confidence Interval (small samples)
1 1
1 1
( x1  x2 )  tc  s
  1  2  ( x1  x2 )  tc  s

n1 n2
n1 n2
(n1  1) s1  (n2  1) s2
n1  n2  2
2
s
df
2
(pooled variance)
= n1  n2  2
Example (small sample)
► Suppose
brain waves are being analyzed for a sleep
study. Results are summarized below:
Group
SAMPLE
SIZE
MEAN
STANDARD
DEVIATION
alcohol
n1 = 15
x1 = 19.65
s1 = 1.86
no alcohol
n2 = 14
x2 = 6.59
s2 = 1.91
Are the samples dependent or independent?
Compute a 90% confidence interval for 1  2
Confidence Interval for p1 – p2
r1
pˆ 1 
n1
r2
pˆ 2 
n2
ˆ 
qˆ1  1  pˆ1
qˆ 2  1  pˆ 2
pˆ1qˆ1 pˆ 2 qˆ2

n1
n2
 pˆ1  pˆ 2   zc  ˆ  p1  p2   pˆ1  pˆ 2   zc  ˆ
Example
► Suppose
two groups are randomly chosen for a
sleep study. In group 1, the subjects watch a
movie before going to sleep. In this group, there
were a total of n1 = 175 dreams recorded, of
which r1 = 49 were dreams with feelings of
anxiety, fear, or aggression. In group 2, the
subjects did not watch a movie. In this group,
there were a total of n2 = 180 dreams recorded,
of which r = 632 were dreams with feelings of
anxiety, fear, or aggression. Compute a 95%
confidence interval for p1 – p2.