Chapters 19 - 21: Statistical Inference – Confidence Interval
and Hypothesis Testing for One Population Proportion
In these chapters, we will talk about:
Estimation of a population proportion,
Large-sample confidence interval for a population proportion,
How to choose the sample size for a study about a population proportion,
One-sample hypothesis test for a population proportion when sample size is large.
One population proportion, p
Two population proportions, p p
One population mean,
Two population means,
More than two population proportions
More than two population means Notations:
Suppose Z has a standard normal distribution. Then, for given
(0 1) , z(or for simplisity is defined as:
PZ() z Definition: a100(1)% confidence interval (C.I.) for a parameeis an
interval (L, U) such that
1 PL ( )
L and U are called lower and upper bound of the confidence interval,
respectively. The percentage 100(1)% is called confidence level.
The presence of bacteria in a urine sample (bacteriuria) is sometimes
associated with symptoms of kidney disease. It was found in a study 30 out
of 500 were positive for bacteriuria. Find a 95% confidence interval for the
proportion of people who are positive for bacteriuria. Confidence Interval for One Population Proportion:
A 100(1)%approximate confidence interval for populatiop, is givenn,
1 where , pz ˆ*p(1-ppz ˆ ˆ pqq p
ME z *
The quantity n is called the margin of error. Example 1: An analyst wishes to estimate the market share captured by
Brand X detergent - that is, the proportion of Brand X sales compared to the
total sales of all detergents. From data supplied by several stores, the analyst
finds that out of a total of 325 boxes of detergent sold, 120 were BranX.
(a) Estimate the market share captured by BrandX.
(b) Estimate the standard error of the above estimate.
(c) Find a 95% confidence interval for . p Example 2: A sample of 78 university students revealed that 49 carried their
books and notes in a backpack.
(a) Estimate the population proportion of students who carry their books
and notes in a backpack.
(b) Obtain the estimated standard error of the above estimate.
(a) Give an approximate 95.4% error margin.
(b) Find an approximate 95.4% confid ence interval for the population
proportion of students who carry their books and notes in a backpack. Top-Hat Question (Review 9-1):
Example 3: The table below summarizes the result of a survey of opinions
of both smokers and nonsmokers on an ordinance prohibiting smoking in
Smokers 16 20 111
Nonsmokers 90 349 114
Construct a 99% confidence interval for the proportion of smokers who are
against the ban in bars. What is upper limit of the confidence interval? Sample Size for a Study about a Population Proportion: To determine how
large a sample is needed for estimating alation proportion, we must specify:
ME = the desired margin of error
1 = the probability associated with the error margin.
Then, the formula for calculating the sample nize, , is given by
n pz (1 )
Note that: pis unknown. Hence
pn p ˆˆ (1 )
If the value of p is known to be roughly in the neighborhood of a
value p , thenncan be determined from
pp z * (1 )
We may take a small-scale preliminary sampling is to obtain an
estimate of p to be used in the formula to computen. Therefore,
pn p z ˆ(1 )
Without prior knowledge of p, the value of p(1 p) can be replaced by
its maximum possiblevalue 0.5(1-0.5) and can bn determined from
n z 0.5 (1 0.5)
E Example 4: A study is conducted to evaluate the proportion of infected
needles used by injection drug users.
(a) If there is no preliminary study is available, how large a random sample
is needed to form a 95 percent conf idence interval with an error bound
of 5 percent?
(b) A preliminary study of a random sample of 160 needles that were used
before a needle exchange program was established found 108 to be
HIV-positive. In the context of this study, how large a sample is needed
to form a 95 percent confidence interval on the population proportion
of pre-exchange-program needles that are HIV- positive, with an error
bound of 5 percent? Example 5: A political pollster wants to estimate the percentage of voters
who will vote for the Democratic candidate in a presidential campaign. The
pollster wishes to have 90% confidence that her prediction is correct to
within 4% of the population proportion.
(a) What sample size is needed?
Top-Hat Question (Review 9-2):
(b) If the pollster wants to have 95% confidence, what sample size is
(c) If she wants to have a 95% confid ence and a sampling error of 3%,
what sample size is needed? The presence of bacteria in a urin e sample (bacteriuria) is sometimes
associated with symptoms of kidney disease.
Assume that a determination of bacteriuria has been made over a large
population at one point in time and that 4% of the population are
positive for bacteriuria.
However, the general feeling is that the rate of bacteriuria has
To test this feeling, a new study by mail questionnaire was performed
and it was found that 30 out of 500 were positive for bacteriuria.
Can we conclude that the true rate of bacteriuria has increased?