AP Statistics Chapter
19 Notes
“Confidence Intervals for Sample Proportions”
p vs. pˆ
◦
p
- the population’s proportion
◦
p̂
- the sample’s proportion
◦ In the last chapter, we knew the population
proportion
◦ In this chapter, we DO NOT know the
population proportion and are going to use the
sample proportion to ESTIMATE it
Standard Error (not standard
deviation this time)

Standard Error =

We call it a “standard error” here instead
of a “standard deviation” because we don’t
know the population proportion “p” and we
are trying to ESTIMATE it with a certain
amount of room for error.
pˆ qˆ
n
Example 1

Police set up an auto checkpoint at which drivers
are stopped and their cars inspected for safety
problems. They find that 14 of 134 cars stopped
have at least one safety violation. They want to
estimate the percentage of all cars that may be
unsafe.

What is the population?

What is the sample size?

What does p represent?

What does p̂ represent?
All cars in the US
The 134 cars at this checkpoint
The % of all cars in the US with at
least one safety violation
The 10.4% of cars with at least one
safety violation found in this sample
More about this example

Based on this sample, we want to
estimate the true percentage of cars that
are unsafe. Draw the normal model up to
three “standard errors” away from the
mean.
Mean = 10.4%
10.4%
7.8%
13%
5.2%
15.6%
2.6%
18.2%

ˆˆ
SE = pq  10.4%  89.6%  2.6%
n
134
Make a prediction about the true value of
p with 95% confidence.
Based on this sample, we are 95% confident that between
5.2% and 15.6% of all US cars have a safety violation.
Confidence Intervals
When the true proportion p is unknown,
we use p̂ to predict it to some level of
confidence. This is called a confidence
interval.
 A confidence interval is the interval of
values that you are C% confident
contains the true value of p.
( p̂ +/- margin of error)

Margin of Error:





The extent of the interval on either side of p̂ is called
margin of error
As the confidence level improves (e.g. 90% … 95% … 99%),
the margin of error widens.
As the margin of error tightens up, the confidence level will
decrease.
The balance between certainty and precision is somewhat
subjective, but a 90% or 95% confidence interval is usually
standard
Know that the larger the sample size, the smaller the
standard error and therefore the smaller the margin of error.
Example


A consumer group hoping to estimate the percentage of
college students who have cell phones surveyed 2883
students as they entered a football stadium. 2781
indicated they had a cell phone.
Based on this sample, find a 95% confidence interval for
the true proportion of all cell-phone-carrying college
students.
Mean = 96.5%
96.5%
96.16% 96.84%
95.82%
95.48%

ˆˆ
SE = pq  96.5%  3.5%  0.34%
97.18%
n
2883
97.52%
What is the margin of error for your confidence interval?
96.5% - 95.82% = 0.68 or 96.5% - 97.18% = -0.68 so the ME = +/- 0.68%
Confidence Intervals in the
Calculator



STAT
TESTS
A. 1 – PropZInt
◦ x = number of successes in your sample 2781
◦ n = the sample size 2883
◦ C – level = the confidence level you want .95

Press “Calculate”

For margin of error – subtract either end
of the confidence interval from p̂
What we can say and what we
cannot say
Say this:
◦ Based on this sample, we can be 95%
___% confident
that the true proportion of college students
97.18
carrying cell phones is between 95.82
___% and ____%.
Don’t say this:
Between 95.82
___% and 97.18
___% of all college students carry cell
phones. (We don’t have 100% certainty.)
Don’t Forget the Conditions
Independence – We assume the data values do
not affect each other
 Random sample – We assume the data are
sampled randomly and adequately represent the
population
 Population must be at least 10 times the size of
the sample
 10 Successes/10 Failures condition – np > 10
and nq > 10 to ensure that the sample size is big
enough.
