73

Aside:
Can you Visualize It?
Consider taking your one random sample of size
n
and computing your one
^
p
value.
(As in
the previous example, our one
^
p
= 460/1000 = 0.46.)
Suppose we did take another random
sample of the same size, we would get another value of
^
p
, say ___________.
Now repeat
that process over and over; taking one random sample after another; resulting in one
^
p
value after another.
Example picture showing the possible values when
n
= _______
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
^
p
values
Observations:
How would things change if the sample size
n
were even larger, say
n
= _____?
Suppose our
first sample proportion turned out to be
^
p
= _________ . Now imagine again repeating that
process over and over; taking one random sample after another; resulting in many
^
p
possible values.
Example picture showing the possible values when
n
= _______
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
^
p
values
Observations:
74

75

Let’s take a closer look at the sample proportion
^
p
.
The sample proportion is found by
taking the number of “successes” in the sample and dividing by the sample size.
So the count
variable
X
of the number of successes is directly related to the proportion of successes as
^
p
=
X
n
.
Earlier we studied the distribution of our first statistic, the count statistic
X
(the number of
successes in
n
independent trials when the probability of a success was
p
). We learned about its
exact distribution called the
Binomial Distribution
.
We also learned when the sample size
n
was large, the distribution of
X
could be approximated with a normal distribution.
Normal Approximation to the Binomial Distribution
If
X
is a binomial
random variable based on
n
trials with success probability
p
, and
n
is large
,
then the random variable
X
is also approximately
N
(
np,
√
np
(
1
−
p
)
)
.
Conditions
: The approximation works well when both
np
and
n
(1 –
p
) are at least 10.
So any probability question about a sample proportion could be converted to a probability
question about a sample count, and vice-versa.
If
n is small
, we would need to convert the question to a count and use the binomial
distribution to work it out.
If
n is large
, we could convert the question to a count and use the normal approximation for
a count, OR use a related normal approximation for a sample proportion (for large
n
).
The Stat 250 formula card summarizes this related normal approximation as follows:
Let’s put this result to work in our next Try It! Problem.
76

Try It!
Do Americans really vote when they say they do?
To answer this question, a telephone poll was taken two days an election. From the 800 adults
polled, 56% reported that they had voted. However, it was later reported in the press that, in
fact, only 39% of American adults had voted.
Suppose the 39% rate reported by the press is
the correct population proportion.
Also assume the responses of the 800 adults polled can be
viewed as a random sample.