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"Four out of five music lovers agree, Brother Lucy rocks."
K. Murphy, Statistics Teacher & Fan
12.1 INFERENCE FOR A POPULATION PROPORTION (Pages 684 - 701)
OVERVIEW: In the past two sections we have been concerned with making
inferences about the population mean. In this chapter, we will be answering
questions about the proportion of some outcome in the population. Our focus
will be on ‘Yes or No’ data, do they have the characteristic of interest or not?
Remember in the previous two chapters we focused on the parameter,  ( population mean) and the
statistic, x (sample mean); inference about the mean.
Now we focus on the population proportion and the sample proportion:
p=
p̂ ( point estimate) =
=
p̂ =
The sample proportion, p̂ is known as the unbiased estimator of the population proportion, p.
The nice thing about Inference of Proportions is that it works just like the z test (with a few changes).
We will go back to the z test statistic and the normal distribution (which we should be quite good with
by now). The main changes are:
1. instead of  ( population mean) we will focus on
2. instead of x (sample mean) we will focus on
3. some of the assumptions have changed.
…otherwise the process is the same!
Confidence Interval for 1 Sample Proportions:
where
is the estimate and
Z test Statistic for 1 Sample Proportions:
is the standard error of p̂
z=
ASSUMPTIONS for Inference about a Proportion:
1)
2)
3)
Possible Null and Alternative Hypotheses:
1. H0:
versus Ha:
2. H0:
versus Ha:
3. H0:
versus Ha:
Example 1:
The National AIDS Behavioral Surveys used a random dialing of telephone numbers to contact a
sample of 2673 people. They are interested in the proportion of adult heterosexuals with multiple
sexual partners. Of those contacted 170 said they had multiple partners.
(a) Calculate a 99% confidence interval of the proportion of all adult heterosexuals with
multiple partners.
(b) Interpret this interval in words.
Response to (a):
p=
p̂ =
Assumptions:
1. data comes from an SRS from a population of interest
2. population at least 10n
3. n p̂  10 and n(1- p̂ )  10
-
The desired 99% confidence interval is
p̂  z *
p̂(1  p̂)
=
n
Response to (b):
Note: This survey was designed to study the AIDS threat. We may want to consider that some
groups at high risk for AIDS, such as IV drug users, often don’t live in settle households and are
underrepresented in this survey. Also some people may refuse to answer the questions b/c of the
topic or they may lie. A third problem with telephone surveys is nonresponse bias occurs. These
may want to be discussed along with results.
Example 2
Suppose that a pharmaceutical company wants to claim that certain drug side effects will
be experienced by fewer than 20% of the patients who use the medication. In a clinical
trial with n = 400 patients chosen randomly, they find that 68 patients experience the side
effects. Is this enough evidence to support their claim at the   .05 significance level?
KEY NUMBERS:
Assumptions:
p̂ =
p0 =
1. data comes from an SRS from a population of interest
2. population at least 10n
3. np0 and n(1- p0)  10
-
We will use a One-sample proportion test
p=
H0 :
H0 :
Ha :
Ha :
Test Statistic:
z=
P-value:
Since the P-value (
) > than   .05 we
H0 and conclude there
enough evidence to support the pharmaceutical companies claim that proportion of
patients experiencing side effects is less than 20%.
Choosing the Sample Size
To determine the sample size n that will yield a confidence interval for a population proportion with a
specific margin of error, m, set the expression for the margin of error to be less than or equal to m and
solve for n:
where p* is a guessed value for the sample proportion. If a p* is not given to you in the problem:
 use p* = 0.5
 if p̂ is closer to 0 or 1,
Gloria Chavez and Ronald Flynn are the candidates for mayor in a large city. You are planning a
sample survey to determine what percent of the voters plan to vote for Chavez. This is a
population proportion p. You will contact an SRS of registered voters in the city. You want to
estimate p with a 95% confidence and a margin of error no greater than 3% or 0.03. How large a
sample do you need?
z*
p * (1  p*)
m
n
You need a sample of
people to have a margin of error of 3%.
Helpful Hint: rewrite the key numbers – it will help with deciding on what test to run!
12.1 Exercises
(1) Consider a method currently used by doctors to screen women for possible breast cancer. This
method fails to detect cancer in 20% of the women who actually have the disease. Suppose a new
method has been developed that researchers hope will detect cancer more accurately. This new method
was used to screen a random sample of 140 women known to have breast cancer. Of these, the new
method failed to detect cancer in 12 of the women. Does this sample provide evidence that the failure
rate of the new method differs form the one currently used at the  = .05 significance level?
KEY NUMBERS: p̂ =
p0 =
Assumptions: 1.
2.
3.
We will use a One-sample proportion test
p=
H0 :
Ha :
Test Statistic:
H0 :
Ha :
z=
P-value:
(2) Pond’s Age-Defying Complex, a cream with alpha-hydroxy acid, advertises that it can reduce
wrinkles and improve the skin. In a study published in Archives of Dermatology (June 1996), 33
women over age 40 used a cream with alpha-hydroxy acid for twenty-two weeks. At the end of the
study period, 23 of the women exhibited skin improvement (as judged by a dermatologist). Is this
evidence that the cream will improve the skin of more than 60% of woman over 40?
KEY NUMBERS: p̂ =
p0 =
Assumptions: 1.
2.
3.
We will use a One-sample proportion test
p=
H0 :
Ha :
Test Statistic:
H0 :
Ha :
z=
P-value:
(3) A sporting foods manufacturer who produces both white and yellow golf balls claims that more
than 75% of all golf balls sold are white. A marketing study of the purchases of white and yellow
golf at a number of stores showed that of the 470 balls sold, 410 were white and 60 were yellow.
a) Is there sufficient evidence to support the manufacturers claim? Test using  = .01.
b) Calculate the probability, , of a type II error if in fact 80% of the golf balls sold are white.
Rule for accepting H0 in terms of p̂
Find the probability of accepting H0 assuming the alternative is true.
Prob(Type II Error) =
(4) The Gallup Poll asked a sample of 1785 adults, “Did you, yourself, happen to attend church or
synagogue in the last 7 days?” Of the respondents, 750 said “Yes.” Suppose that the Gallup sample
was an SRS of all American adults.
a) Give a 95% confidence interval for the proportion of all adults who attended church or synagogue
during the week preceding the poll.
KEY NUMBERS:
p̂ =
Assumptions: 1.
2.
3.
p̂  z *
p̂(1  p̂)
=
n
b) How large of a sample would be required to obtain a margin of error of 0.01 in a 99%
confidence interval for the proportion who attend church or synagogue? (Use p* = 0.5)
z*
p * (1  p*)
m
n
CALCULATOR DIRECTIONS FOR CONFIDENCE INTERVALS…
Example 4a)
On the TI-83+, STAT/TESTS/A:1-PropZInt
where x is the number in the sample that has the characteristic
of interest
When you do a 1-PropZInt on the calculator, you not only get
the interval but you can also double check your p̂ .
CALCULATOR DIRECTIONS FOR A 1 PROPORTION Z HYPOTHESIS TEST…
Example 3)
On the TI-83+, STAT/TESTS/5:1-PropZTest
From this screen you have 2 choices: you can Calculate or Draw. The
second calculator snapshot is what you get when you choose Calculate.