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Chapter 20. Inference about a Population Proportion
STAT 145
Tests and confidence intervals for a population proportion p when the data are an SRS of size n are based on the
sample proportion ^p :
or
We are interested in the unknown proportion
the outcome we are looking for a “success”.
p of a population that has some outcome. For convenience, call
Thus, let ^p be the sample proportion of successes.
^p is p .
•
The mean of the sampling distribution of
•
The standard deviation of the sampling distribution of
•
√
p(1−p)
.
n
As the sample size increases, the sampling distribution of ^p becomes approximately Normal.
That is, for large n, ^p has approximately the N ( p , √ p (1− p)/n) distribution.
^p is
Large-sample confidence interval for a population proportion
Draw an SRS of size n from a large population that contains an unknown proportion
approximate level C confidence interval for p is
✴
√
^p±z ⋅
✴
p of successes. An
^p (1− p^ )
n
where z is the critical value for the standard Normal density curve with area C between
(see Table C).
−z ✴ and
z✴
Use this interval only when the numbers of successes and failures in the sample are both 15 or greater.
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Chapter 20. Inference about a Population Proportion
STAT 145
Sample Size for desired margin of error
The level C confidence interval for a population proportion
to a specified value m when the sample size is
n=(
p will have margin of error approximately equal
z✴ 2 ✴
) ⋅p ⋅(1−p ✴ )
m
✴
where p is a guessed value for the sample proportion. The margin of error will always be less than or equal
✴
to m if you take the guess p to be 0.5.
✴
You can use the conservative guess p =0.5 if you expect the true ^p to be roughly between 0.3 and 0.7. If
✴
the true ^p is close to 0 or 1, using p =0.5 as your guess will give a sample much larger than you need.
Try to use a better guess from a pilot study when you suspect that ^p will be less than 0.3 or greater than 0.7.
Significance tests for a proportion
Draw an SRS of size n from a large population that contains an unknown proportion p of successes. To test
the hypothesis H 0 : p= p0 (a given value we are testing), compute the z statistic
z=
^p− p 0
√
p0 (1− p 0)
n
Find P-values from the standard Normal distribution (Table A).
Use this test when the sample size n is so large that both np0 and n(1−p 0) are 10 or more.
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Chapter 20. Inference about a Population Proportion
STAT 145
Problem 1.
A survey of 611 office workers investigated telephone answering practices, including how often each office
worker was able to answer incoming telephone calls and how often incoming telephone calls went directly to
voice mail. A total of 281 office workers indicated that they never need voice mail and are able to take every
telephone call.
a. What is the point estimate of the proportion of the population of office workers who are able to take every
telephone call?
b. At 90% confidence, what is the margin of error?
c. What is the 90% confidence interval for the proportion of the population of office workers who are able to
take every telephone call?
Problem 2.
The percentage of people not covered by health care insurance in 2003 was 15.6%. A congressional committee
has been charged with conducting a sample survey to obtain more current information.
a. What sample size would you recommend if the committee's goal is to estimate the current proportion of
individuals without health care insurance with a margin of error of 0.03? Use a 95% confidence level.
b. Repeat part a) assuming that you have no prior information (i.e.
p✴ =0.5 ).
c. Repeat part a) using a 99% confidence level.
d. Repeat part c) assuming that you have no prior information (i.e.
p✴ =0.5 ).
Problem 3.
The proportion of college students with student loans is thought to be between 0.3 and 0.7. How large a sample
is necessary to estimate the true proportion of college students with student loans within 3 percentage point (i.e.
±0.3 ) with 95% confidence?
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Chapter 20. Inference about a Population Proportion
STAT 145
Problem 4.
A study found that, in 2005, 12.5% of U.S. workers belonged to unions. Suppose a sample of 400 U.S. workers
is collected in 2006 to determine whether union efforts to organize have increased union membership.
a. Formulate the hypotheses that can be used to determine whether union membership increased in 2006.
b. If the sample results show that 52 of the workers belonged to unions, what is the p-value for your hypothesis
test?
c. At α=0.05 , what is your conclusion?
Problem 5.
In the 2010 US midterm elections 41% of registered voters actually voted. In the recent 2014 midterm election,
an SRS of 2500 US registered voters found that 915 registered voters actually voted. At the 1% level of
significance, is this evidence the proportion of registered voters who voted in the 2014 midterm election was
less than the proportion who voted in the 2010 midterm election?
a) What is the alternative hypothesis for the appropriate test of significance?
b) Calculate the test statistic.
c) Calculate the p-value.
d) What is the conclusion?
e) To check the conditions for inference for this test of significance, we check for an SRS from the population as
well as … (continue sentence).
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