AP Statistics Section 9.2Sample Proportions

The objective of some statistical applications is

to reach a conclusion about a population proportion, p, by using the sample proportion, .

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

p̂

However, as we have seen, the value of will vary from sample to sample. The amount of variability

will depend upon the ________________

p̂

size sample

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.

count of successes in sample

ˆsize of sample

p n

x

Since values of X and will vary in repeated samples, both X and are

random variables.

Provided the population is at least 10 times the sample size, the count X will

follow a binomial distribution.

So, ____ and __________.

p̂p̂

x xnp )1( pnp

Now , , so use the

transformation rules:

If Y = a + bX, then

xnn

xp

1ˆ

xxy bba y and

Rule of Thumb 1This formula for the standard deviation of can only be used when the population is at least 10 times as large as the sample.p̂

________________________________

and _______________

p̂

ˆ

p

npn

10

p

)1(1

pnpn

2

)1(

n

pnp n

pp )1(

We saw with our simulations in Section 9.1, that our sampling

distribution of gets closer and closer to a Normal distribution

when the sample size, n, is large.

p̂

Rule of Thumb 2: Use the Normal approximation to the sampling distribution

of for values of n and p that satisfy ________ and ______________.

Note that these are the same conditions necessary to use a Normal distribution to

approximate a Binomial distribution.

p̂10np 101 pn

Summarizing the Sampling Distribution for Proportions

If we take repeated random samples of size n from a population, the sample proportion , will

have the following distribution and properties.p̂

p

n

pp 1

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?

109751500(.65) and 105251500(.35) because Normal approx. is p̂ of Dist.

15000or (10)(1500)students college all of pop. because 0123.1500

)65)(.35(.

and 35.

p̂

ˆ

p

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)?

1013351500(.89) and 101651500(.11) because Normal approx. is p̂ of Dist.

15000or (10)(1500)adults USall of pop. because 0081.1500

)89)(.11(.

11.

p̂

ˆ

p

sample. in the ageundercover

suspect reason to have weunlikely, so

is thisSince adults.black few so have

wouldsamplessuch all of 1.3%Only

0131.