A.P. STATISTICS
LESSON 9 - 2
SAMPLE PROPORTIONS
ESSENTIAL QUESTION:
What are the tests used in order to use
normal calculations for a sample?
Objectives:
 To define the rules of thumb so that
the normal curve may be used for
calculating proportions from a
sample.
 When a sample can use normal
calculations in the formula for the
p and σ.
The Sampling Distribution of p
The sample proportion p is the statistic that we
use to gain information about the unknown
population parameter p.
Since values of X and p will vary in repeated
samples, both X and p are random variables.
Provided that the population is much larger
than the sample (say at least 10 times ), the
count X will follow a binomial distribution.
Sampling Distribution of a
Sample Proportion
Choose an SRS of size n from a large
population proportion p having some
characteristic of interest.
Let p be the proportion of the sample having
that characteristic.
Then:
The mean of the sampling distribution is
exactly p.
Sampling Distribution of a
Sample Proportion
The standard deviation of the
sample distribution is:
√ p( 1 – p)/n
Rule of Thumb #1
Use the recipe for the standard
deviation of p only when the population
is at least 10 times as large as the
sample.
Rule of Thumb #2
We will use the normal approximation
to the sampling distribution of p for
values of n and p that satisfy np ≥ 10
and N( 1- p) ≥ 10
Example 9.7
page 507
Applying to College
A polling organization asks an SRS of 1,500
first-year college students whether they
applied for admission to any other college.
In fact, 35% of all first-year students applied
to colleges besides the one they are
attending.
What is the probability
that the random sample
of 1,500 students will give
a result within 2
percentage points of this
The normal approximation to the
true value?
sampling distribution of p.
Example 9.8
page 509
Survey Undercoverage?
One way of checking the effect of undercoverage,
nonresponse, and other sources of error in a sample
is to compare the sample with known facts about the
population.
About 11% of American adults are black. The
proportion p of blacks in a SRS of 1,500 adults should
therefore be close to .11.
It is unlikely to be exactly .11 because of sampling
variability.
If a national sample contains only 9.2 % blacks,
should we suspect that the sampling procedure is
somehow underrepresenting blacks?