away from the true proportion. How many senior citizens do we need for
our study?
Example:
Suppose we think that the 1953 proportion might be signifi
cantly different from the current proportion. What is the largest sample
size we’ll need to guarantee that the confidence interval will be no more
than
0
.
03
away from the true proportion?
21
Hypothesis tests on population proportion:
We use the following as
our test statistic.
z
=
ˆ
p

p
r
p
(1

p
)
n
Note:
Be careful!
In the confidence interval, we used the
estimated
standard error
p
ˆ
p
(1

ˆ
p
)
/n
since we did not have a value of
p
to use.
For hypothesis tests, since we
do
have a hypothesized value of
p
, we use
the
standard error
p
p
(1

p
)
/n
.
22
Example:
At a hospital, the administrators have decided that the ac
ceptable proportion of correctly interpreted scans must be
98%
or better.
In a random sample of
500
scans, it is found that
487
were interpreted
correctly.
Is there evidence that the proportion of correctly interpreted
scans is below the acceptable level?
Test your hypothesis at the level
α
= 0
.
15
.
23
Sets 24
Instead of drawing samples from one population, we may take random
samples from two populations so that we can carry out some sort of
comparison.
We may wish to compare
p
1
and
p
2
, the population proportions for pop
ulations
1
and
2
. We do so by examining the difference,
p
1

p
2
.
Example:
Suppose we wish to compare the
p
1
, the proportion of people
in BC with diabetes, with
p
2
, the proportion of people in Alberta with
diabetes.
•
If the proportions are equal, then
p
1

p
2
= 0
.
•
If the proportions are different, then
p
1

p
2
6
= 0
.
•
If the proportion is higher in BC, then
p
1
> p
2
, which means that
p
1

p
2
>
0
.
•
If the proportion is higher in Alberta, then
p
1
< p
2
, which means
that
p
1

p
2
<
0
.
•
If the proportion in BC is higher than in Alberta by at least
0
.
05
,
then
p
1
> p
2
+ 0
.
05
, which means that
p
1

p
2
>
0
.
05
•
If the proportion in Alberta is higher than in BC by at least
0
.
02
,
then
p
1
+ 0
.
02
< p
2
, which means that
p
1

p
2
<

0
.
02
24
We use
ˆ
p
1

ˆ
p
2
as the point estimate for
p
1

p
2
Distribution of
ˆ
p
1

ˆ
p
2
:
Suppose we have
n
1
samples from a first population, and
n
2
samples from
a second population.
1.
ˆ
p
1

ˆ
p
2
has a normal distribution.
2.
ˆ
p
1

ˆ
p
2
has mean
p
1

p
2
and has standard error:
s
p
1
(1

p
1
)
n
1
+
p
2
(1

p
2
)
n
2
25
Confidence interval for difference in population proportions:
( ˆ
p
1

ˆ
p
2
)
±
z
α/
2
·
s
ˆ
p
1
(1

ˆ
p
1
)
n
1
+
ˆ
p
2
(1

ˆ
p
2
)
n
2