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Chapter 19: Confidence Intervals for
Proportions
Key Vocabulary:
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standard error
confidence level
confidence interval
Calculator Skills:
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margin of error
critical value
one-proportion z-interval
1-PropZInt
Suppose I want to know what proportion of teenagers typically goes to the movies on a Friday
night.
Suppose I take an SRS of 25 teenagers and calculate the sample proportion to be pˆ  0.40 .
The sample proportion _____ is an unbiased estimator of the unknown population proportion
_____, so I would estimate the population proportion to be approximately _____. However,
using a different sample would have given a different sample proportion, so I must consider the
amount of variation in the sampling model for _____.
Based on one sample, it would _____ be correct to conclude that 40% of all teenagers typically
go to the movies on a Friday night.
But don’t despair!… based on my one sample, I can come up with an _______________ that
**may** contain the true proportion of teenagers who typically go to the movies on a Friday
night.
Not only will I tell you what that interval is, but I will also tell you how _______________ I am
that the true proportion falls somewhere in that interval.
Remember…
▪
The sampling model for p̂ is ___________________________________ assuming
_______________ and ______________.
Chapter 19: Confidence Intervals for Proportions
▪
The mean of the sampling model is _____.
▪
The standard deviation of the sampling model is __________ assuming the population
size is at least _____ times larger than the sample size  N  10n  .
Draw the sampling model:
Since we don’t know p , we cannot calculate the standard deviation of the sampling model.
We can, however, use _____ to estimate the value of p and calculate the standard error
ˆˆ
pq
n
instead.
So the standard error for the sampling model for the proportion of teenagers who typically go
to the movies on a Friday night is:
_______________________________
According to the 68-95-99.7 Rule, _____ of all possible samples of size 25 will produce a statistic
p̂ that is within _____ standard errors of the mean of our sampling model.
This means that, in our example, 95% of the p̂ ’s will be between ___________ and
___________.
Chapter 19: Confidence Intervals for Proportions
So the distance between the actual _____ value and the statistic _____ will usually (95% of the
time) be less than or equal to ________.
Therefore, in 95% of our samples, the interval between pˆ  0.196 and pˆ  0.196 will contain
the parameter p .
We say that the _________________________ is 0.196
For our sample of 25 teenagers, pˆ  0.40 . Because the margin of error is 0.196, then we are
95% confident that the true population proportion lies somewhere in the interval
_______________, or ____________________.
The interval [0.21, 0.59] is called a _____ _______________ _______________ because we are
95% _______________ that the true proportion of teenagers who typically go to the movies on
a Friday night is between about _____ and _____.
CAUTION!!
This does NOT mean the probability that p is between 0.21 and 0.59 is 95%.
_______________ does not mean the same thing as _______________.
We _______________ calculate the probability that p is within a given interval without using a
Normal model, and we _____________ draw a Normal model because we don’t know the
center, p .
If you assume that p  pˆ , then P  x1  p  x2  is either ________ or ________.
Chapter 19: Confidence Intervals for Proportions
Okay… maybe you’re not satisfied with the interval we constructed. Is it too wide? Would you
prefer a more precise conclusion? One way of changing the length of the interval is to change
the _______________ __________.
So how do we construct 90% confidence intervals? 99% confidence intervals? C% confidence
intervals?
Since the sampling model of the sample proportion p̂ is _______________ _______________,
we can use normal calculations to construct confidence intervals.

For a 95% confidence interval, we want the interval corresponding to the
_______________ 95% of the normal curve.

For a 90% confidence interval, we want the interval corresponding to the
_______________ 90% of the normal curve.

And so on…
If we are using the standard normal curve, we want to find the interval using
_______________.
Suppose we want to find a 90% confidence interval for a standard normal curve. If the middle
90% lies within our interval, then the remaining _______ lies _______________ our interval.
Because the curve is symmetric, there is _______ below the interval and _______ above the
interval. Find the _______________ with area 5% below and 5% above.
These z-values are denoted __________. Because they come from the standard normal curve,
they are centered at mean _____.
Chapter 19: Confidence Intervals for Proportions
__________ is called the ______________________________, with probability p lying to its
__________ under the standard normal curve.
To find the upper critical p value, we find the complement of C and divide it in half, or find:
___________________________________
For a 95% confidence interval, we want the z-values with upper p critical value __________.
Found how: ________________________________
For a 99% confidence interval, we want the z-values with upper p critical value __________.
Found how: ________________________________
Your Turn:
Find the critical value z* for a confidence level of 73%?
____________________________________________________________________
Remember that z-values tell us how many _______________ _______________ we are above
or below the mean.
To construct a 95% confidence interval, we want to find the values __________ standard
deviations below the mean and 1.96 standard deviations above the mean, or:
_____________________________________
Using our sample data, this is pˆ  1.96
ˆˆ
pq
, assuming the population is at least _______ times
n
as large as the sample size, _______.
Chapter 19: Confidence Intervals for Proportions
In general, to construct a level C confidence interval using our sample data, we want to find:
_________________________________
Assumptions and Conditions:
________________________________: We first need to Think about whether the
Independence Assumption is plausible. It’s not one you can check by looking at the data.
Instead, we check two conditions to decide whether independence is reasonable.
–
__________________________________: Were the data sampled at random or
generated from a properly randomized experiment? Proper randomization can
help ensure independence.
–
______________________________: Is the sample size no more than 10% of the
population?
__________________________________: The sample needs to be large enough for us to be
able to use the CLT.
–
____________________________________: We must expect at least 10
“successes” and at least 10 “failures.”
Procedure: Confidence Interval for a Population Proportion (Don’t PANIC)
P ________________________: Identify the population of interest and the parameter you want
to draw conclusions about (population proportion p).
A ________________________: Verify the conditions for using the selected procedure.
N ___________________: One sample proportions z interval.
I ______________________: Calculate the confidence interval.
C ______________________________________________: Answer the question in the context
of the problem.
Chapter 19: Confidence Intervals for Proportions
Your Turn:
In a random sample of 50 Philadelphia families with children of preschool age, 35 had children
enrolled in preschool. Find a 95% confidence interval for the true proportion of Philadelphia
families with children enrolled in preschool.
P____________________________:
A______________________________:
N_____________________:
I__________________________:
C___________________________________________:
Chapter 19: Confidence Intervals for Proportions
Your Turn: Using TI -84.
In 2000, the GSS asked subjects if they would be willing to pay much higher prices in order to
protect the environment. Of n = 1154 respondents, 518 indicated a willingness to do so.
a) Find a 95% confidence interval for the population proportion of adult Americans
willing to do so at the time of that survey.
_________________________________
b) Interpret the results.
______________________________________________________________________________
______________________________________________________________________________
The margin of error is z *
ˆˆ
pq
. Note that the margin of error is a positive __________. It is
n
not an interval.
We would like __________ confidence and a __________ margin of error.
A higher confidence level means a higher percentage of all samples produce a statistic close to
the true value of the parameter. Therefore we want a __________ level of confidence.
A smaller margin of error allows us to get closer to the true value of the parameter (length of
the interval is small), so we want a __________ margin of error.
So how do we reduce the margin of error?
▪
__________ the confidence level (by decreasing the value of z*)
▪
__________ the standard deviation
Chapter 19: Confidence Intervals for Proportions
▪
__________ the sample size. To cut the margin of error in half, increase the sample size
by __________ times the previous size.
You can have __________ confidence and a __________ margin of error if you choose the right
sample size.
To determine the sample size n that will yield a confidence interval for a population proportion
with a specified margin of error m, set the expression for the margin of error to be equal to m
and solve for n. Always round n up to the next greatest integer.
______________________________________
Your Turn:
Suppose the U.S. President wants an estimate of the proportion of the population who support
his current policy toward revisions in the Social Security System. The president wants the
estimate to be within .04 of the true proportion. Assume a 90 percent level confidence. How
large a sample is required?
Sample Size:______________________________
CAUTION!!
These methods only apply to certain situations. In order to construct a level C confidence
interval using the formula pˆ  z*
ˆˆ
pq
, for example, the data must come from a
n
_______________ sample. Also, we want to eliminate (if possible) any _______________.
Chapter 19: Confidence Intervals for Proportions
The margin of error only covers random sampling errors. Things like ____________________,
____________________, and ____________________ can cause additional errors.
*Remember, if you are asked to construct a Confidence Interval, you must not PANIC!!
P:
A:
N:
I:
C:
______
Questions:
1. What is the standard error of p̂ ? When would you use standard error in place of standard
deviation?
2. What is meant by a confidence interval?
3. What is meant by a confidence level?
4. What is the general form of a confidence interval for a one-proportion z-interval?
Chapter 19: Confidence Intervals for Proportions
5. Explain how to calculate margin of error.
6. As the confidence level increases, what happens to the margin of error? What happens to the
confidence interval?
7. What is the critical value z* for a 90% confidence interval? Draw a sketch.
8. What is the critical value z* for a 95% confidence interval? Draw a sketch.
9. What is the critical value z* for a 99% confidence interval? Draw a sketch.
10. What assumptions and conditions must you consider before creating a
confidence interval for a proportion?
Chapter 19: Confidence Intervals for Proportions