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AP Statistics Section 9.2
Sample Proportions
The objective of some statistical applications is
to reach a conclusion about a population
proportion, p, by using the sample proportion, p̂ .
For example, we may try to estimate an
approval rating through a survey or test a claim
about the proportion of defective light bulbs in a
shipment based on a random sample. Since p is
unknown to us, we must base our conclusion on
a sample proportion, p̂.
However, as we have seen, the
value of p̂ will vary from sample to
sample. The amount of variability
will depend upon the
________________
sample size
For example: A polling organization asks an SRS
of 1500 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 1500
students will give a result within 2 percentage
points of this true value?
Before we can answer this question,
we need to take a closer look at the
center, shape and spread of the
sampling distribution for p̂.
Take a SRS from the population of
interest.
x
count of successes in sample
pˆ 

size of sample
n
Since values of X and p̂ will vary in
repeated samples, both X and p̂ are
random variables.
Provided the population is at least 10
times the sample size, the count X
will follow a binomial distribution.
So,  x  ____
np (1  p )
np and  x  __________.
x 1
Now , pˆ   x , so use the
n n
transformation rules:
If Y = a + bX, then
 y  a  b x and  y  b x
1
0  np
p and
 pˆ  __________
 _____
n
p (1  p )
1
np (1  p )
np (1  p )
 p̂  __________
__  __________
n
n
n 2 __  ________
Rule of Thumb 1
This formula for the standard deviation of
p̂ can only be used when the population
is at least 10 times as large as the sample.
We saw with our simulations in
Section 9.1, that our sampling
distribution of p̂ gets closer and
closer to a Normal distribution
when the sample size, n, is large.
Rule of Thumb 2: Use the Normal
approximation to the sampling distribution
of p̂ for values of n and p that satisfy
________
np  10 and ______________.
n1  p   10
Note that these are the same conditions
necessary to use a Normal distribution to
approximate a Binomial distribution.
Summarizing the Sampling
Distribution for Proportions
If we take repeated random samples of size n from a
population, the sample proportion p̂ , will have the
following distribution and properties.
p1  p 
n
p
A polling organization asks an SRS of 1500 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 1500 students will give a result within
2 percentage points of this true value?
Dist. of p̂ is approx. Normal because 1500(.35)  525  10 and 1500(.65)  975  10
 pˆ  .35 and
 p̂ 
(.35)(.65)
 .0123 because pop. of all college students  (10)(1500) or 15000
1500
.896
Example: Based on Census data, we know 11% of US adults are black.
Therefore p = 0.11. We would expect an SRS to have roughly an 11% black
representation. Suppose a sample of 1500 adults contains 138 black
individuals. We would not expect to be exactly 0.11 because of sampling
variability, but, is this number lower than what would be expected by chance
(i.e. should we suspect “undercoverage” in the sample method)?
Dist. of p̂ is approx. Normal because 1500(.11)  165  10 and 1500(.89)  1335  10
 pˆ  .11
(.11)(.89)
 p̂ 
 .0081 because pop. of all US adults  (10)(1500) or 15000
1500
.0131
Only 1.3% of all such samples would
have so few black adults. Since this is
so unlikely, we have reason to suspect
undercoverage in the sample.